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Dmitry Faddeev Dmitry Konstantinovich Faddeev (Russian: Дми́трий Константи́нович Фадде́ев, IPA: [ˈdmʲitrʲɪj kənstɐnʲˈtʲinəvʲɪtɕ fɐˈdʲe(j)ɪf]; 30 June 1907 – 20 October 1989) was a Soviet mathematician. Biography Dmitry was born June 30, 1907, about 200 kilometers southwest of Moscow on his father's estate. His father Konstantin Tikhonovich Faddeev was an engineer while his mother was a doctor and appreciator of music who instilled the love for music in Dmitry. Friends found his piano playing entertaining. In 1928 he graduated from Petrograd State University, as it was then called. His teachers included Ivan Matveyevich Vinogradov and Boris Nicolaevich Delone. In 1930 he married Vera Nicolaevna Zamyatina and in 1934 she gave birth to Lyudvig Dmitrievich Faddeev who grew up to be a physicist. Contributions Dmitry and his wife co-authored Numerical Methods in Linear Algebra in 1960, followed by an enlarged edition in 1963. For instance, they developed an idea of Urbain Leverrier to produce an algorithm to find the resolvent matrix $(A-sI)^{-1}$ of a given matrix A. By iteration, the method computed the adjugate matrix and characteristic polynomial for A.[1] Dmitry was committed to mathematics education and aware of the need for graded sets of mathematical exercises. With Iliya Samuilovich Sominskii he wrote Problems in Higher Algebra. He was one of the founders of the Russian Mathematical Olympiads. He was one of the founders of the a Physics-Mathematics secondary school later named after him.[2] See also • Faddeev–LeVerrier algorithm References 1. Hou, Shui-Hung (January 1998). "Classroom Note:A Simple Proof of the Leverrier--Faddeev Characteristic Polynomial Algorithm". SIAM Review. Society for Industrial and Applied Mathematics. 40 (3): 706–709. Bibcode:1998SIAMR..40..706H. doi:10.1137/S003614459732076X. ISSN 1095-7200. 2. Sokolova, N. N. (16 October 2003). "К 40-летию физико-математической и химико-биологической школы-интерната №45 при ЛГУ". Saint Petersburg University. • Aleksandrov, A. D.; Bashmakov, M. I.; Borevich, Z. I.; Kublanovskaya, V. N.; Nikulin, M. S.; Skopin, A. I.; Yakovlev, A. V. (1989). Translated by Lofthouse, A. "Dmitrii Konstantinovich Faddeev (on his eightieth birthday)". Russian Mathematical Surveys. Russian Academy of Sciences. 44 (3): 223–231. doi:10.1070/RM1989v044n03ABEH002126. ISSN 0042-1316. S2CID 250913337 – via Saint Petersburg Mathematical Society. • Borevich, Z. I.; Linnik, Yu. V.; Skopin, A. I. (1968). "Дмитрий Константинович Фаддеев (к шестидесятилетию со дня рождения)" [Dmitrii Konstantinovich Faddeev (on his sixtieth birthday)]. Russian Mathematical Surveys (in Russian). Russian Academy of Sciences. 23 (3): 169–175. doi:10.1070/RM1968v023n03ABEH003777. ISSN 0042-1316. S2CID 250895230. External links • O'Connor, John J.; Robertson, Edmund F., "Dmitry Faddeev", MacTutor History of Mathematics Archive, University of St Andrews Authority control International • ISNI • VIAF National • Germany • Israel • United States • Czech Republic • Netherlands • Poland Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
Isaak Yaglom Isaak Moiseevich Yaglom[1] (Russian: Исаа́к Моисе́евич Ягло́м; 6 March 1921 – 17 April 1988)[2][3] was a Soviet mathematician and author of popular mathematics books, some with his twin Akiva Yaglom. Isaak Yaglom Born(1921-03-06)6 March 1921 Kharkov Died17 April 1988(1988-04-17) (aged 67) Moscow, Soviet Union NationalitySoviet Alma materMoscow State University Scientific career FieldsMathematics InstitutionsYaroslavl State University Doctoral advisorBoris Delaunay Veniamin Kagan Yaglom received a Ph.D. from Moscow State University in 1945 as student of Veniamin Kagan.[4] As the author of several books, translated into English, that have become academic standards of reference, he has an international stature. His attention to the necessities of learning (pedagogy) make his books pleasing experiences for students. The seven authors of his Russian obituary recount "…the breadth of his interests was truly extraordinary: he was seriously interested in history and philosophy, passionately loved and had a good knowledge of literature and art, often came forward with reports and lectures on the most diverse topics (for example, on Alexander Blok, Anna Akhmatova, and the Dutch painter M. C. Escher), actively took part in the work of the cinema club in Yaroslavl and the music club at the House of Composers in Moscow, and was a continual participant of conferences on mathematical linguistics and on semiotics."[5] University life Yaglom started his higher education at Moscow State University in 1938. During World War II he volunteered, but due to myopia he was deferred from military service. In the evacuation of Moscow he went with his family to Sverdlovsk in the Ural Mountains. He studied at Sverdlovsk State University, graduated in 1942, and when the usual Moscow faculty assembled in Sverdlovsk during the war, he took up graduate study. Under the geometer Veniamin Kagan he developed his Ph.D. thesis which he defended in Moscow in 1945. It is reported that this thesis "was devoted to projective metrics on a plane and their connections with different types of complex numbers $a+jb$ (where $jj=-1$, or $jj=+1$, or else $jj=0$)."[5] Institutes and titles During his career, Yaglom was affiliated with these institutions:[5] • Moscow Energy Institute (1946) – lecturer in mathematics • Moscow State University (1946 – 49) – lecturer, department of analysis and differential geometry • Orekhovo-Zuevo Pedagogical Institute (1949–56) – lecturer in mathematics • Lenin State Pedagogical Institute (Moscow) (1956–68) – obtained D.Sc. 1965 • Moscow Evening Metallurgical Institute (1968–74) – professor of mathematics • Yaroslavl State University (1974–83) – professor of mathematics • Academy of Pedagogical Sciences (1984–88) – technical consultant Affine geometry In 1962 Yaglom and Vladimir G. Ashkinuse published Ideas and Methods of Affine and Projective Geometry, in Russian. The text is limited to affine geometry since projective geometry was put off to a second volume that did not appear. The concept of hyperbolic angle is developed through area of hyperbolic sectors. A treatment of Routh's theorem is given at page 193. This textbook, published by the Ministry of Education, includes 234 exercises with hints and solutions in an appendix. English translations Isaac Yaglom wrote over 40 books and many articles. Several were translated, and appeared in the year given: Complex numbers in geometry (1968) Translated by Eric J. F. Primrose, published by Academic Press (N.Y.). The trinity of complex number planes is laid out and exploited. Topics include line coordinates in the Euclidean and Lobachevski planes, and inversive geometry. Geometric Transformations (1962, 1968, 1973, 2009) The first three books were originally published in English by Random House as part of the series New Mathematical Library (Volumes 8, 21, and 24). They were keenly appreciated by proponents of the New Math in the U.S.A., but represented only a part of Yaglom’s two-volume original published in Russian in 1955 and 56. More recently the final portion of Yaglom's work was translated into English and published by the Mathematical Association of America. All four volumes are now available from the MAA in the series Anneli Lax New Mathematical Library (Volumes 8, 21, 24, and 44). A simple non-euclidean geometry and its physical basis (1979) Subtitle: An elementary account of Galilean geometry and the Galilean principle of relativity. Translated by Abe Shenitzer, published by Springer-Verlag. In his prefix, the translator says the book is "a fascinating story which flows from one geometry to another, from geometry to algebra, and from geometry to kinematics, and in so doing crosses artificial boundaries separating one area of mathematics from another and mathematics from physics." The author’s own prefix speaks of "the important connection between Klein’s Erlanger Program and the principles of relativity." The approach taken is elementary; simple manipulations by shear mapping lead on page 68 to the conclusion that "the difference between the Galilean geometry of points and the Galilean geometry of lines is just a matter of terminology". The concepts of the dual number and its "imaginary" ε, ε2 = 0, do not appear in the development of Galilean geometry. However, Yaglom shows that the common slope concept in analytic geometry corresponds to the Galilean angle. Yaglom extensively develops his non-Euclidean geometry including the theory of cycles (pp. 77–79), duality, and the circumcycle and incycle of a triangle (p. 104). Yaglom continues with his Galilean study to the inversive Galilean plane by including a special line at infinity and showing the topology with a stereographic projection. The Conclusion of the book delves into the Minkowskian geometry of hyperbolas in the plane, including the nine-point hyperbola. Yaglom also covers the inversive Minkowski plane. Probability and information (1983) Co-author: A. M. Yaglom. Russian editions in 1956, 59 and 72. Translated by V. K. Jain, published by D. Reidel and the Hindustan Publishing Corporation, India. The channel capacity work of Claude Shannon is developed from first principles in four chapters: probability, entropy and information, information calculation to solve logical problems, and applications to information transmission. The final chapter is well-developed including code efficiency, Huffman codes, natural language and biological information channels, influence of noise, and error detection and correction. Challenging Mathematical Problems With Elementary Solutions (1987) Co-author: A. M. Yaglom. Two volumes. Russian edition in 1954. First English edition 1964-1967 Felix Klein and Sophus Lie (1988) Subtitle: The evolution of the idea of symmetry in the 19th century. In his chapter on "Felix Klein and his Erlangen Program", Yaglom says that "finding a general description of all geometric systems [was] considered by mathematicians the central question of the day."[6] The subtitle more accurately describes the book than the main title, since a great number of mathematicians are credited in this account of the modern tools and methods of symmetry. In 2009 the book was republished by Ishi Press as Geometry, Groups and Algebra in the Nineteenth Century. The new edition, designed by Sam Sloan, has a foreword by Richard Bozulich. See also • Anti-cosmopolitan campaign[3] References 1. His last name is sometimes transliterated as "Jaglom", "Iaglom", "IAglom", or "I-Aglom". The double capitalization in the latter cases indicates that IA transliterates a single capital letter Я (Ya). 2. Russian Jewish Encyclopedia 3. Rosenfeld, Boris Abramowitsch [in Russian] (2003). "Ob Isaake Moiseyeviche Yaglome" Об Исааке Моисеевиче Ягломе [About Isaak Moiseevich Yaglom]. Мат. просвещение (Mat. enlightenment) (in Russian): 25–2. Retrieved 2022-01-12 – via math.ru. […] во время антисемитской кампании, известной как «борьба с космополитизмом», был уволен вместе с И.М. Гельфандом и И. С. Градштейном […] [during the antisemitic campaign known as the "fight against cosmopolitanism", he was fired along with I. M. Gelfand and I. S. Gradstein.] 4. Isaak Yaglom at the Mathematics Genealogy Project 5. Boltyansky , et al. 6. Chapter 7, pp. 111–24. Further reading • V. G. Boltyansky, L. I. Golovina, O. A. Ladyzhenskaya, Yu. I. Manin, S. P. Novikov, B. A. Rozenfel'd, A. M. Yaglom (1989). "Isaak Moiseevich Yaglom (obituary)". Russian Mathematical Surveys. 44 (1): 225–227. Bibcode:1989RuMaS..44..225B. doi:10.1070/RM1989v044n01ABEH002018. S2CID 250907488.{{cite journal}}: CS1 maint: uses authors parameter (link) • Yaglom, Isaak M. (c. 1979). A simple non-Euclidean geometry and its physical basis : an elementary account of Galilean geometry and the Galilean principle of relativity. Abe Shenitzer (translator). New York: Springer-Verlag. ISBN 0-387-90332-1. via Internet Archive • А. Д. Мышкис, "Исаак Моисеевич Яглом — выдающийся просветитель" (transl.: "Isaak Moiseevich Yaglom, prominent educator"), Матем. просв., сер. 3, 7, МЦНМО, М., 2003, pp. 29–34. (in Russian) • В. М. Тихомиров, "Вспоминая братьев Ягломов" (transl.: "Remembering the Yaglom brothers"), Матем. просв., сер. 3, 16, Изд-во МЦНМО, М., 2012, pp. 5–13. (in Russian) External links • About Isaak Moiseevich Yaglom by B. A. Rozenfel'd (in Russian) • Isaak Yaglom at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Spain • France • BnF data • Germany • Israel • United States • Sweden • Japan • Czech Republic • Netherlands • Poland • Vatican Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
Mikhail Lavrentyev Mikhail Alekseyevich Lavrentyev (or Lavrentiev, Russian: Михаи́л Алексе́евич Лавре́нтьев) (November 19, 1900 – October 15, 1980) was a Soviet mathematician and hydrodynamicist. Mikhail Lavrentyev Born Mikhail Alekseyevich Lavrentyev (1900-11-19)November 19, 1900 Kazan, Russian Empire DiedOctober 15, 1980(1980-10-15) (aged 79) Moscow, Soviet Union NationalitySoviet Alma materMoscow State University AwardsLomonosov Gold Medal (1977) Scientific career FieldsMathematics InstitutionsMoscow State University Steklov Institute of Mathematics Doctoral advisorNikolai Luzin Doctoral studentsMstislav Keldysh Aleksei Markushevich Early years Lavrentyev was born in Kazan, where his father was an instructor at a college (he later became a professor at Kazan University, then Moscow University). He entered Kazan University, and, when his family moved to Moscow in 1921, he transferred to the Department of Physics and Mathematics of Moscow University. He graduated in 1922. He continued his studies in the university in 1923-26 as a graduate student of Nikolai Luzin. Although Luzin was alleged to plagiarize in science and indulge in anti-Sovietism by some of his students in 1936, Lavrentyev did not participate in the notorious political persecution of his teacher which is known as the Luzin case or Luzin affair. In fact Luzin was a friend of his father.[1]: 5 Mid career In 1927, Lavrentyev spent half a year in France, collaborating with French mathematicians, and upon returned took up a position with Moscow University. Later he became a member of the staff of the Steklov Institute. His main contributions relate to conformal mappings and partial differential equations. Mstislav Keldysh was one of his students. In 1939, Oleksandr Bogomolets, the president of the Ukrainian Academy of Sciences, asked Lavrentyev to become director of the Institute of Mathematics at Kyiv. One of Lavrentyev's scientific interests was the physics of explosive processes, in which he had become involved when doing defense work during World War II. A better understanding of the physics of explosions made it possible to use controlled explosions in construction, the best-known example being the construction of the Medeu Mudflow Control Dam outside of Almaty in Kazakhstan. In Siberia Mikhail Lavrentyev was one of the main organizers and the first Chairman of the Siberian Division of the Russian Academy of Sciences (in his time the Academy of Sciences of the USSR) from its founding in 1957 to 1975. The foundation of the Siberia's "Academic Town" Akademgorodok (now a district of Novosibirsk) remains his most widely known achievement. Six months after the decision to found the Siberian Division of the USSR Academy of Sciences Novosibirsk State University was established. The Decree of the Council of Ministers of the USSR was signed January 9, 1958.[2] From 1959 to 1966 he was a professor at Novosibirsk State University.[3] Lavrentyev was also a founder of the Institute of Hydrodynamics of the Siberian Division of the Russian Academy of Sciences which since 1980 has been named after Lavrentyev.[4] Lavrentyev was awarded the honorary title of Hero of Socialist Labour, a Lenin and 2 Stalin Prizes, and a Lomonosov Gold Medal. He was elected a member of several world-renowned academies, and an honorable citizen of Novosibirsk. Mikhail A. Lavrentyev's son, also named Mikhail (Mikhail M. Lavrentyev, 1930-2010), also became a mathematician and was a member of the leadership of Akademgorodok.[5] Eponyms • Street in Kazan • Street in Dolgoprudny • Academician Lavrentyev Avenue in Novosibirsk • Lavrentyev Institute of Hydrodynamics in Novosibirsk • SESC affiliated with NSU • Novosibirsk Lavrentyev Lyceum 130 • RV Akademik Lavrentyev • Aiguilles in Altai and Pamir References 1. Josephson, Paul R. (1997). New Atlantis Revisited: Akademgorodok, the Siberian City of Science. Colby College: Faculty Books. Retrieved 20 August 2017. 2. NSU history Archived 2014-09-03 at the Wayback Machine 3. Mikhail Lavrentev's biography 4. Lavrentyev Institute of Hydrodynamics Archived 2008-05-03 at the Wayback Machine 5. Fortune April 2, 2007 p.36 External links • Mikhail Lavrentyev at the Mathematics Genealogy Project • O'Connor, John J.; Robertson, Edmund F., "Mikhail Lavrentyev", MacTutor History of Mathematics Archive, University of St Andrews • Mikhail Lavrentiev's biography, at the site of Lavrentiev Hydrodynamics Institute. (in Russian) Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Israel • Belgium • United States • Sweden • Czech Republic • Netherlands • Poland • Portugal Academics • CiNii • Leopoldina • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie • Trove Other • IdRef
Mikhail Menshikov Mikhail Vasilyevich Menshikov (Russian: Михаи́л Васи́льевич Ме́ньшиков; born January 17, 1948) is a Russian-British mathematician with publications in areas ranging from probability to combinatorics. He currently holds the post of Professor in the University of Durham. Mikhail Menshikov Born (1948-01-17) January 17, 1948 Moscow, Russian SFSR, Soviet Union NationalityRussian and British Alma materMoscow State University Scientific career FieldsMathematician InstitutionsUniversity of Durham Thesis (1976) Doctoral advisorVadim Malyshev Menshikov has made a substantial contribution to percolation theory and the theory of random walks. Menshikov was born in Moscow and went to school in Kharkov, Ukrainian SSR, Soviet Union. He studied at Moscow State University earning all his degrees up to Candidate of Sciences (1976) and Doctor of Sciences (1988). After briefly working in Zhukovsky, Menshikov worked in Moscow State University for many years. His career then took him to the University of Sao Paulo before becoming a professor at the University of Durham, where he currently lives. External links • Personal webpage • Mikhail Menshikov at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Israel • United States • Netherlands • Poland Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
Sergei Sobolev Prof Sergei Lvovich Sobolev (Russian: Серге́й Льво́вич Со́болев) FRSE (6 October 1908 – 3 January 1989) was a Soviet mathematician working in mathematical analysis and partial differential equations. Sergei Lvovich Sobolev Sobolev in Nice in 1970 Born(1908-10-06)6 October 1908 Saint Petersburg, Russian Empire Died3 January 1989(1989-01-03) (aged 80) Moscow, Soviet Union Alma materLeningrad State University, 1929 Known forGeneralized functions Riesz–Sobolev inequality Sobolev conjugate Sobolev embedding theorem Sobolev generalized derivative Sobolev inequality Sobolev space AwardsLomonosov Gold Medal (1988) USSR State Prize (1983) Hero of Socialist Labor (1951) Stalin Prize (1941, 1951, 1953) Scientific career FieldsMathematics InstitutionsSteklov Mathematical Institute, Lomonosov Moscow State University, Kurchatov Institute, Novosibirsk State University, Sobolev Institute Doctoral advisorNikolai Günther InfluencedFunctional analysis, partial differential equations Sobolev introduced notions that are now fundamental for several areas of mathematics. Sobolev spaces can be defined by some growth conditions on the Fourier transform. They and their embedding theorems are an important subject in functional analysis. Generalized functions (later known as distributions) were first introduced by Sobolev in 1935 for weak solutions, and further developed by Laurent Schwartz. Sobolev abstracted the classical notion of differentiation, so expanding the range of application of the technique of Newton and Leibniz. The theory of distributions is considered now as the calculus of the modern epoch.[1] Life He was born in St. Petersburg as the son of Lev Alexandrovich Sobolev, a lawyer, and his wife, Natalya Georgievna.[2] His city was renamed Petrograd in his youth and then Leningrad in 1924. Sobolev studied Mathematics at Leningrad University and graduated in 1929, having studied under Professor Nikolai Günther. After graduation, he worked with Vladimir Smirnov, whom he considered as his second teacher. He worked in Leningrad from 1932, and at the Steklov Mathematical Institute in Moscow from 1934. He headed the institute in evacuation to Kazan during World War II. He was a Moscow State University Professor of Mathematics from 1935 to 1957 and also a deputy director of the Institute for Atomic Energy from 1943 to 1957 where he participated in the A-bomb project of the USSR. In 1958, he led with Nikolay Brusentsov the development of the ternary computer Setun. In 1956, Sobolev joined a number of scientists in proposing a large-scale scientific and educational initiative for the Eastern parts of the Soviet Union, which resulted in the creation of the Siberian Division of the Academy of Sciences.[3] He was the founder and first director of the Institute of Mathematics at Akademgorodok near Novosibirsk, which was later to bear his name, and played an important role in the establishment and development of Novosibirsk State University. In 1962, he called for a reform of the Soviet education system.[4] He died in Moscow.[5] Family In 1930 he married Ariadna Dmitrievna.[2] Publications In 1955 he co-wrote The Main Features of Cybernetics with Alexey Lyapunov and Anatoly Kitov which was published in Voprosy filosofii. See also • Mollifier • Sobolev mapping Notes 1. e.g. Friedman, A. (1970). Foundations of modern analysis. Courier Corporation, p. iii 2. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X. 3. "The Siberian Branch, an overview Siberian Branch of the Russian Academy of Sciences (SB RAS)". sbras.ru. Retrieved 1 March 2018. 4. Berg A., (1964), 'Cybernetics and Education' in The Anglo-Soviet Journal, March 1964, pp. 13–20 5. O'Connor, J J. "Sergei Lvovich Sobolev". References • Sobolev, Sergei L. (1936), "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales", Matematicheskii Sbornik, 1(43) (1): 39–72, Zbl 0014.05902 (in French). In this paper Sergei Sobolev introduces generalized functions, applying them to the problem of solving linear hyperbolic partial differential equations. • Sobolev, Sergei L. (1938), "Sur un théorème d'analyse fonctionnelle", Recueil Mathématique de la Société Mathématique de Moscou, 4(46) (3): 471–497, Zbl 0022.14803 (in Russian, with French summary). In this paper Sergei Sobolev proved his embedding theorem, introducing and using integral operators very similar to mollifiers, without naming them. Bibliography • O'Connor, John J.; Robertson, Edmund F., "Sergei Sobolev", MacTutor History of Mathematics Archive, University of St Andrews • Sergei Lvovich Sobolev (1908-1989). Bio-Bibliography (S.S. Kutateladze, editor) Novosibirsk, Sobolev Institute (2008), ISBN 978-5-86134-196-7 • Sergei Lvovich Sobolev., in: Russian Mathematicians in the 20th Century (Yakov Sinai, editor), pp. 381–382. World Scientific Publishing, 2003. ISBN 978-981-238-385-3 • Jean Leray. La vie et l'œuvre de Serge Sobolev. [The life and works of Sergeĭ Sobolev]. Comptes Rendus de l'Académie des Sciences. Série Générale. La Vie des Sciences, vol. 7 (1990), no. 6, pp. 467–471. • G. V. Demidenko. A GREAT MATHEMATICIAN OF 20th CENTURY. On the occasion of the centenary from the birthdate of Sergei Lvovich Sobolev. Science in Siberia, no. 39 (2674), 2 October 2008 • M. M. Lavrent'ev, Yu. G. Reshetnyak, A. A. Borovkov, S. K. Godunov, T. I. Zelenyak and S. S. Kutateladze. Remembrances of Sergei L'vovich Sobolev. Siberian Mathematical Journal, vol. 30 (1989), no. 3, pp. 502–504 doi:10.1007/BF00971511 External links • Media related to Sergei Lvovich Sobolev (mathematician) at Wikimedia Commons • Sergei Sobolev at the Mathematics Genealogy Project • Sobolev Institute of Mathematics • Kutateladze S.S.,Sobolev and Schwartz: Two Fates and Two Fames • Kutateladze S.S.,Sobolev of the Euler school • ru:Соболев, Сергей Львович • Sobolev's Biography in Russian Authority control International • FAST • ISNI • VIAF National • France • BnF data • Catalonia • Germany • Israel • United States • Japan • Czech Republic • Netherlands • Poland Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
Kelvin transform The Kelvin transform is a device used in classical potential theory to extend the concept of a harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic and superharmonic functions. In order to define the Kelvin transform f* of a function f, it is necessary to first consider the concept of inversion in a sphere in Rn as follows. It is possible to use inversion in any sphere, but the ideas are clearest when considering a sphere with centre at the origin. Given a fixed sphere S(0,R) with centre 0 and radius R, the inversion of a point x in Rn is defined to be $x^{}={\frac {R^{2}}{\|x\|^{2}}}x.$ A useful effect of this inversion is that the origin 0 is the image of $\infty $, and $\infty $ is the image of 0. Under this inversion, spheres are transformed into spheres, and the exterior of a sphere is transformed to the interior, and vice versa. The Kelvin transform of a function is then defined by: If D is an open subset of Rn which does not contain 0, then for any function f defined on D, the Kelvin transform f of f with respect to the sphere S(0,R) is $f^{}(x^{})={\frac {\|x\|^{n-2}}{R^{2n-4}}}f(x)={\frac {1}{\|x^{}\|^{n-2}}}f(x)={\frac {1}{\|x^{}\|^{n-2}}}f\left({\frac {R^{2}}{\|x^{}\|^{2}}}x^{}\right).$ One of the important properties of the Kelvin transform, and the main reason behind its creation, is the following result: Let D be an open subset in Rn which does not contain the origin 0. Then a function u is harmonic, subharmonic or superharmonic in D if and only if the Kelvin transform u* with respect to the sphere S(0,R) is harmonic, subharmonic or superharmonic in D. This follows from the formula $\Delta u^{}(x^{})={\frac {R^{4}}{\|x^{}\|^{n+2}}}(\Delta u)\left({\frac {R^{2}}{\|x^{}\|^{2}}}x^{}\right).$ See also • William Thomson, 1st Baron Kelvin • Inversive geometry • Spherical wave transformation References • William Thomson, Lord Kelvin (1845) "Extrait d'une lettre de M. William Thomson à M. Liouville", Journal de Mathématiques Pures et Appliquées 10: 364–7 • William Thompson (1847) "Extraits deux lettres adressees à M. Liouville, par M. William Thomson", Journal de Mathématiques Pures et Appliquées 12: 556–64 • J. L. Doob (2001). Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag. p. 26. ISBN 3-540-41206-9. • L. L. Helms (1975). Introduction to potential theory. R. E. Krieger. ISBN 0-88275-224-3. • O. D. Kellogg (1953). Foundations of potential theory. Dover. ISBN 0-486-60144-7. • John Wermer (1981) Potential Theory 2nd edition, page 84, Lecture Notes in Mathematics #408 ISBN 3-540-10276-0
Elon Lindenstrauss Elon Lindenstrauss (Hebrew: אילון לינדנשטראוס, born August 1, 1970) is an Israeli mathematician, and a winner of the 2010 Fields Medal.[1][2] Elon Lindenstrauss Born (1970-08-01) August 1, 1970 Jerusalem, Israel NationalityIsraeli Alma materHebrew University of Jerusalem AwardsBlumenthal Award (2001) Salem Prize (2003) EMS Prize (2004) Fermat Prize (2009) Erdős Prize (2009) Fields Medal (2010) Scientific career FieldsMathematics InstitutionsHebrew University of Jerusalem Princeton University Doctoral advisorBenjamin Weiss Since 2004, he has been a professor at Princeton University. In 2009, he was appointed to Professor at the Mathematics Institute at the Hebrew University. Biography Lindenstrauss was born into an Israeli-Jewish family with German Jewish origins, the son of the mathematician Joram Lindenstrauss, the namesake of the Johnson–Lindenstrauss lemma, and computer scientist Naomi Lindenstrauss, both professors at the Hebrew University of Jerusalem. His sister Ayelet Lindenstrauss is also a mathematician. He attended the Hebrew University Secondary School. In 1988 he was awarded a bronze medal at the International Mathematical Olympiad. He enlisted to the IDF's Talpiot program, and studied at the Hebrew University of Jerusalem, where he earned his BSc in Mathematics and Physics in 1991 and his master's degree in mathematics in 1995. In 1999 he finished his Ph.D., his thesis being "Entropy properties of dynamical systems", under the guidance of Prof. Benjamin Weiss. He was a member at the Institute for Advanced Study in Princeton, New Jersey, then a Szego Assistant Prof. at Stanford University. From 2003 to 2005, he was a Long Term Prize Fellow at the Clay Mathematics Institute. Academic career In Fall 2014, he was Visiting Miller Professor at the University of California, Berkeley.[3] Lindenstrauss is an editor for Duke Mathematical Journal and Journal d'Analyse Mathématique.[4][5] Lindenstrauss works in the area of dynamics, particularly in the area of ergodic theory and its applications in number theory. With Anatole Katok and Manfred Einsiedler, he made progress on the Littlewood conjecture.[6] In a series of two papers (one co-authored with Jean Bourgain) he made major progress on Peter Sarnak's Arithmetic Quantum Unique Ergodicity conjecture. The proof of the conjecture was completed by Kannan Soundararajan. Recently with Manfred Einsiedler, Philippe Michel and Akshay Venkatesh, he studied distributions of torus periodic orbits in some arithmetic spaces, generalizing theorems by Hermann Minkowski and Yuri Linnik. Together with Benjamin Weiss he developed and studied systematically the invariant of mean dimension[7] introduced in 1999 by Mikhail Gromov.[8] In related work he introduced and studied the small boundary property and stated fundamental conjectures.[9] Among his co-authors are Jean Bourgain, Manfred Einsiedler, Philippe Michel, Shahar Mozes, Akshay Venkatesh and Barak Weiss. Awards and recognition • In 1988, Lindenstrauss represented Israel in the International Mathematical Olympiad and won a bronze medal. • During his service in the IDF, he was awarded the Israel Defense Prize. • In 2003, he was awarded the Salem Prize jointly with Kannan Soundararajan. • In 2004, he was awarded the European Mathematical Society Prize. • In 2008, he received the Michael Bruno Memorial Award. • In 2009, he was awarded the Erdős Prize. • In 2009, he received the Fermat Prize. • In 2010, he became the first Israeli to be awarded the Fields Medal, for his results on measure rigidity in ergodic theory, and their applications to number theory.[10] References 1. "Israeli wins world's most prestigious math prize". ynet. 19 August 2010. Retrieved 19 August 2010. 2. "Israeli Mathemetician Elon Lindenstrauss Wins Field Medal — Pictures". Zimbio. Retrieved 2010-08-21. 3. "Elon Lindenstrauss". Simons Institute for the Theory of Computing. 17 March 2014. Retrieved 24 May 2017. 4. "Editorial board". Duke Mathematical Journal. Retrieved 16 October 2022. 5. "Editorial board". Journal d'Analyse Mathématique. Retrieved 16 October 2022. 6. Einsiedler, Manfred; Katok, Anatole; Lindenstrauss, Elon (2006-09-01). "Invariant measures and the set of exceptions to Littlewood's conjecture". Annals of Mathematics. 164 (2): 513–560. arXiv:math.DS/0612721. Bibcode:2006math.....12721E. doi:10.4007/annals.2006.164.513. MR 2247967. S2CID 613883. Zbl 1109.22004. 7. Lindenstrauss, Elon; Weiss, Benjamin (2000-12-01). "Mean topological dimension". Israel Journal of Mathematics. 115 (1): 1–24. CiteSeerX 10.1.1.30.3552. doi:10.1007/BF02810577. ISSN 0021-2172. 8. Gromov, Misha (1999). "Topological invariants of dynamical systems and spaces of holomorphic maps: I". Mathematical Physics, Analysis and Geometry. 2 (4): 323–415. doi:10.1023/A:1009841100168. S2CID 117100302. 9. Lindenstrauss, Elon (1999-12-01). "Mean dimension, small entropy factors and an embedding theorem". Publications Mathématiques de l'Institut des Hautes Études Scientifiques. 89 (1): 227–262. doi:10.1007/BF02698858. ISSN 0073-8301. 10. "Fields Medal – Elon Lindenstrauss, ICM2010". External links • Homepage at Hebrew University • Elon Lindenstrauss at the Mathematics Genealogy Project • Elon Lindenstrauss's results at International Mathematical Olympiad Fields Medalists • 1936 Ahlfors • Douglas • 1950 Schwartz • Selberg • 1954 Kodaira • Serre • 1958 Roth • Thom • 1962 Hörmander • Milnor • 1966 Atiyah • Cohen • Grothendieck • Smale • 1970 Baker • Hironaka • Novikov • Thompson • 1974 Bombieri • Mumford • 1978 Deligne • Fefferman • Margulis • Quillen • 1982 Connes • Thurston • Yau • 1986 Donaldson • Faltings • Freedman • 1990 Drinfeld • Jones • Mori • Witten • 1994 Bourgain • Lions • Yoccoz • Zelmanov • 1998 Borcherds • Gowers • Kontsevich • McMullen • 2002 Lafforgue • Voevodsky • 2006 Okounkov • Perelman • Tao • Werner • 2010 Lindenstrauss • Ngô • Smirnov • Villani • 2014 Avila • Bhargava • Hairer • Mirzakhani • 2018 Birkar • Figalli • Scholze • Venkatesh • 2022 Duminil-Copin • Huh • Maynard • Viazovska • Category • Mathematics portal Authority control International • ISNI • VIAF National • Germany • Israel Academics • DBLP • MathSciNet • Mathematics Genealogy Project • ORCID • zbMATH
Oded Schramm Oded Schramm (Hebrew: עודד שרם; December 10, 1961 – September 1, 2008) was an Israeli-American mathematician known for the invention of the Schramm–Loewner evolution (SLE) and for working at the intersection of conformal field theory and probability theory.[1][2] Oded Schramm Schramm in 2008 Born(1961-12-10)December 10, 1961 Jerusalem, Israel DiedSeptember 1, 2008(2008-09-01) (aged 46) Washington, US CitizenshipIsraeli and US Alma mater • Hebrew University of Jerusalem • Princeton University Known for • Schramm–Loewner evolution • Circle packing Awards • Erdős Prize (1996) • Salem Prize (2001) • Clay Research Award (2002) • Loève Prize (2003) • Henri Poincaré Prize (2003) • George Pólya Prize (2006) • Ostrowski Prize (2007) Scientific career Institutions • University of California, San Diego • Weizmann Institute • Microsoft Research Doctoral advisorWilliam Thurston Biography Schramm was born in Jerusalem.[3] His father, Michael Schramm, was a biochemistry professor at the Hebrew University of Jerusalem. He attended Hebrew University, where he received his bachelor's degree in mathematics and computer science in 1986 and his master's degree in 1987, under the supervision of Gil Kalai. He then received his PhD from Princeton University in 1990 under the supervision of William Thurston. After receiving his doctorate, he worked for two years at the University of California, San Diego, and then had a permanent position at the Weizmann Institute from 1992 to 1999. In 1999 he moved to the Theory Group at Microsoft Research in Redmond, Washington, where he remained for the rest of his life. He and his wife had two children, Tselil and Pele.[3] Tselil is an assistant professor of statistics at Stanford University.[4] On September 1, 2008, Schramm fell to his death while scrambling Guye Peak, north of Snoqualmie Pass in Washington.[3][5][6] Research A constant theme in Schramm's research was the exploration of relations between discrete models and their continuous scaling limits, which for a number of models turn out to be conformally invariant. Schramm's most significant contribution was the invention of Schramm–Loewner evolution, a tool which has paved the way for mathematical proofs of conjectured scaling limit relations[7][8] on models from statistical mechanics such as self-avoiding random walk and percolation. This technique has had a profound impact on the field.[3][9] It has been recognized by many awards to Schramm and others, including a Fields Medal to Wendelin Werner, who was one of Schramm's principal collaborators, along with Gregory Lawler. The New York Times wrote in his obituary: If Dr. Schramm had been born three weeks and a day later, he would almost certainly have been one of the winners of the Fields Medal, perhaps the highest honor in mathematics, in 2002. Schramm's doctorate[10] was in complex analysis, but he made contributions in many other areas of pure mathematics, although self-taught in those areas. Frequently he would prove a result by himself before reading the literature to obtain an appropriate credit. Often his proof was original or more elegant than the original.[11] Besides conformally invariant planar processes and SLE, he made fundamental contributions to several topics:[9] • Circle packings and discrete conformal geometry. • Embeddings of Gromov hyperbolic spaces. • Percolation, uniform and minimal spanning trees and forests, harmonic functions on Cayley graphs of infinite finitely generated groups (especially non-amenable groups) and the hyperbolic plane. • Limits of sequences of finite graphs. • Noise sensitivity of Boolean functions, with applications to dynamical percolation. • Random turn games (e.g. random turn hex) and the infinity Laplacian equation. • Random permutations. Awards and honors • Erdős Prize (1996)[12] • Salem Prize (2001)[13] • Clay Research Award (2002),[14] for his work in combining analytic power with geometric insight in the field of random walks, percolation, and probability theory in general, especially for formulating stochastic Loewner evolution. His work opens new doors and reinvigorates research in these fields. [14] • Loève Prize (2003) • Henri Poincaré Prize (2003),[15] For his contributions to discrete conformal geometry, where he discovered new classes of circle patterns described by integrable systems and proved the ultimate results on convergence to the corresponding conformal mappings, and for the discovery of the Stochastic Loewner Process as a candidate for scaling limits in two dimensional statistical mechanics.[16] • SIAM George Pólya Prize (2006),[17] with Gregory Lawler and Wendelin Werner, for groundbreaking work on the development and application of stochastic Loewner evolution (SLE). Of particular note is the rigorous establishment of the existence and conformal invariance of critical scaling limits of a number of 2D lattice models arising in statistical physics. [18] • Ostrowski Prize (2007) • Elected in 2008 as a foreign member of the Royal Swedish Academy of Sciences.[19] Selected publications • Schramm, Oded (2000), "Scaling limits of loop-erased random walks and uniform spanning trees", Israel Journal of Mathematics, 118: 221–288, arXiv:math.PR/9904022, doi:10.1007/BF02803524, MR 1776084, S2CID 17164604. Schramm's paper introducing the Schramm–Loewner evolution. • Schramm, Oded (2007), "Conformally invariant scaling limits: an overview and a collection of problems", International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, pp. 513–543, arXiv:math/0602151, Bibcode:2006math......2151S, doi:10.4171/022-1/20, ISBN 978-3-03719-022-7, MR 2334202 • Schramm, Oded (2011), Benjamini, Itai; Häggström, Olle (eds.), Selected works of Oded Schramm, Selected Works in Probability and Statistics, New York: Springer, doi:10.1007/978-1-4419-9675-6, ISBN 978-1-4419-9674-9, MR 2885266 References 1. "Clay Mathematics Institute". Archived from the original on 2008-08-29. Retrieved 2008-09-12. 2. Lawler, Gregory F. (2005), Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, vol. 114, Providence, RI: American Mathematical Society, ISBN 0-8218-3677-3, MR 2129588 3. Chang, Kenneth (September 10, 2008), "Oded Schramm, 46, Mathematician, Is Dead", New York Times 4. "Tselil Schramm". Tselil Schramm. Retrieved 2022-09-04. 5. Gutierrez, Scott (September 3, 2008), "Rescuers recover hiker's body near Cascades' Guye Peak", Seattle Post-Intelligencer 6. "Accomplished Microsoft mathematician died in hiking accident", Seattle Times, September 4, 2008, archived from the original on September 22, 2008 7. Cardy, John (1992), "Critical percolation in finite geometries", J. Phys. A: Math. Gen., 25 (4): L201–L206, arXiv:hep-th/9111026, Bibcode:1992JPhA...25L.201C, doi:10.1088/0305-4470/25/4/009, MR 1151081, S2CID 16037415 8. Cardy, John (2002), "Crossing formulae for critical percolation in an annulus", J. Phys. A: Math. Gen., 35 (41): L565–L572, arXiv:math-ph/0208019, Bibcode:2002math.ph...8019C, doi:10.1088/0305-4470/35/41/102, MR 1946958, S2CID 119655247 9. "Oded Schramm's publications on Google Scholar". 10. Schramm, Oded (2007), Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps, vol. 0709, p. 710, arXiv:0709.0710, Bibcode:2007PhDT.......441S. Modified version of Schramm's Ph.D. thesis from 1990. 11. "Mathematician who made a significant contribution to the study of probability and fractals", Daily Telegraph, September 19, 2008 12. The Anna and Lajos Erdős Prize in Mathematics, Technion. 13. Kehoe, Elaine (2001), "Schramm and Smirnov Awarded 2001 Salem Prize" (PDF), Notices of the American Mathematical Society, 48 (8): 831 14. "Clay Research Award citation: Oded Schramm". Clay Mathematics Institute. Archived from the original on 2008-08-29. Retrieved 2008-09-04. 15. "The Henri Poincaré Prize". International Association of Mathematical Physics. Retrieved 2008-09-04. 16. "Henri Poincaré Prize of the IAMP" (PDF). International Association of Mathematical Physics. 17. "Gregory F. Lawler, Oded Schramm and Wendelin Werner receive George Polya Prize in Boston". Society for Industrial and Applied Mathematics. 2006-04-20. Archived from the original on 2009-05-04. Retrieved 2008-09-04. 18. "2006 Prizes and Awards Luncheon – SIAM Annual Meeting – July 11, 2006". SIAM. 19. "About Microsoft Research: Awards". Microsoft. Retrieved 2008-09-12. External links Wikimedia Commons has media related to Oded Schramm. • Tutorial: SLE, video of MSRI lecture given jointly by Schramm, Lawler and Werner in the special session at the Lawrence Hall of Science, University of California, Berkeley in May 2001. • Conformally Invariant Scaling Limits and SLE, MSRI presentation by Oded Schramm, May 2001. • Terence Tao, "Oded Schramm". • Publication list • Oded Schramm at the Mathematics Genealogy Project • Oded Schramm Memorial page • Oded Schramm Memorial blog • Oded Schramm Memorial Workshop – August 30–31, 2009 at Microsoft Research • Oded Schramm obituary in the IMS Bulletin • O'Connor, John J.; Robertson, Edmund F., "Oded Schramm", MacTutor History of Mathematics Archive, University of St Andrews Authority control International • ISNI • VIAF National • Germany • Israel • United States • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
Saharon Shelah Saharon Shelah (Hebrew: שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Saharon Shelah Shelah in 2005 Born (1945-07-03) July 3, 1945 Jerusalem, British Mandate for Palestine (now Israel) Alma mater • Tel Aviv University (B.Sc) • Hebrew University (M.Sc., Ph.D.) Known forProper Forcing, PCF theory, Sauer–Shelah lemma, Shelah cardinal Awards • Erdős Prize (1977) • Rothschild Prize (1982) • Karp Prize (1983) • George Pólya Prize (1992) • Gödel Lecture (1996) • Bolyai Prize (2000) • Wolf Prize (2001) • Israel Prize (1998) • EMET Prize (2011) • Leroy P. Steele Prize (2013) • Rolf Schock Prize (2018) Scientific career FieldsMathematical logic, model theory, set theory InstitutionsHebrew University, Rutgers University Doctoral advisorMichael O. Rabin Doctoral studentsRami Grossberg[1] Biography Shelah was born in Jerusalem on July 3, 1945. He is the son of the Israeli poet and political activist Yonatan Ratosh.[2] He received his PhD for his work on stable theories in 1969 from the Hebrew University.[1] Shelah is married to Yael,[2] and has three children.[3] His brother, magistrate judge Hamman Shelah was murdered along with his wife and daughter by an Egyptian soldier in the Ras Burqa massacre in 1985. Shelah planned to be a scientist while at primary school, but initially was attracted to physics and biology, not mathematics.[4] Later he found mathematical beauty in studying geometry: He said, "But when I reached the ninth grade I began studying geometry and my eyes opened to that beauty—a system of demonstration and theorems based on a very small number of axioms which impressed me and captivated me." At the age of 15, he decided to become a mathematician, a choice cemented after reading Abraham Halevy Fraenkel's book An Introduction to Mathematics.[4] He received a B.Sc. from Tel Aviv University in 1964, served in the Israel Defense Forces Army between 1964 and 1967, and obtained a M.Sc. from the Hebrew University (under the direction of Haim Gaifman) in 1967.[5] He then worked as a teaching assistant at the Institute of Mathematics of the Hebrew University of Jerusalem while completing a Ph.D. there under the supervision of Michael Oser Rabin,[5] on a study of stable theories. Shelah was a lecturer at Princeton University during 1969–70, and then worked as an assistant professor at the University of California, Los Angeles during 1970–71.[5] He became a professor at Hebrew University in 1974, a position he continues to hold.[5] He has been a visiting professor at the following universities:[5] the University of Wisconsin (1977–78), the University of California, Berkeley (1978 and 1982), the University of Michigan (1984–85), at Simon Fraser University, Burnaby, British Columbia (1985), and Rutgers University, New Jersey (1985). He has been a distinguished visiting professor at Rutgers University since 1986.[5] Academic career Shelah's personal webpage, as of February 2023 lists 1123 published and accepted mathematical papers, as well as more than 100 preprints and papers in preparation, including joint papers with 288 co-authors;[6] the American Mathematical Society's database MathSciNet lists 1147 published books and journal articles with 266 coauthors. His main interests lie in mathematical logic, model theory in particular, and in axiomatic set theory.[7] In model theory, he developed classification theory, which led him to a solution of Morley's problem. In set theory, he discovered the notion of proper forcing, an important tool in iterated forcing arguments. With PCF theory, he showed that in spite of the undecidability of the most basic questions of cardinal arithmetic (such as the continuum hypothesis), there are still highly nontrivial ZFC theorems about cardinal exponentiation. Shelah constructed a Jónsson group, an uncountable group for which every proper subgroup is countable. He showed that Whitehead's problem is independent of ZFC. He gave the first primitive recursive upper bound to van der Waerden's numbers V(C,N).[8] He extended Arrow's impossibility theorem on voting systems.[9] Shelah's work has had a deep impact on model theory and set theory. The tools he developed for his classification theory have been applied to a wide number of topics and problems in model theory and have led to great advances in stability theory and its uses in algebra and algebraic geometry as shown for example by Ehud Hrushovski and many others. Classification theory involves deep work developed in many dozens of papers to completely solve the spectrum problem on classification of first order theories in terms of structure and number of nonisomorphic models, a huge tour de force. Following that he has extended the work far beyond first order theories, for example for abstract elementary classes. This work also has had important applications to algebra by works of Boris Zilber.[10] Awards • Three times speaker at the International Congress of Mathematicians (1974 invited, 1983 plenary, 1986 plenary) • The first recipient of the Erdős Prize, in 1977[11] • The Karp Prize of the Association for Symbolic Logic in 1983[12] • The Israel Prize, for mathematics, in 1998[13] • The Bolyai Prize in 2000[14] • The Wolf Prize in Mathematics in 2001[15] • The EMET Prize for Art, Science and Culture in 2011[16] • The Leroy P. Steele Prize, for Seminal Contribution to Research, in 2013[17] • Honorary member of the Hungarian Academy of Sciences, in 2013[18] • Advanced grant of the European Research Council (2013)[19] • Hausdorff Medal of the European Set Theory Society, joint with Maryanthe Malliaris, 2017[20] • Schock Prize in Logic and Philosophy of the Royal Swedish Academy of Sciences, 2018[21] • Honorary doctorate from the Technische Universität Wien, 2019[22] Selected works • Proper forcing, Springer 1982 • Proper and improper forcing (2nd edition of Proper forcing), Springer 1998 • Around classification theory of models, Springer 1986 • Classification theory and the number of non-isomorphic models, Studies in Logic and the Foundations of Mathematics, 1978,[23] 2nd edition 1990, Elsevier • Classification Theory for Abstract Elementary Classes, College Publications 2009 • Classification Theory for Abstract Elementary Classes, Volume 2, College Publications 2009 • Cardinal Arithmetic, Oxford University Press 1994[24] See also • List of Israel Prize recipients References 1. Saharon Shelah at the Mathematics Genealogy Project 2. (in Hebrew) Shelah, Saharon (April 5, 2001). "זיכרונותיו של בן" [Memoirs of a Son]. Haaretz. Retrieved August 31, 2014. כשעמדתי להציג לפני חברתי יעל (עתה רעייתי) את בני משפחתי...הפרופ' שהרן שלח מן האוניברסיטה העברית בירושלים, בנו של יונתן רטוש... [As I was about to present to friend Yael (now my wife), my family ... Professor Saharon Shelah of the Hebrew University of Jerusalem, son of Yonathan Ratosh ...] 3. (in Hungarian) Réka, Szász (March 2001). "Harc a matematikával és a titkárnőkkel" [Struggle with mathematics and the secretaries]. Magyar Tudományos (in Hungarian). Retrieved August 31, 2014. Hungarian: A gyerekei mivel foglalkoznak? A nagyobbik fiam zeneelméletet tanul, a lányom történelmet, a kisebbik fiam pedig biológiát. (What are your children doing? My elder son is learning the theory of music, my daughter history, my younger son biology.) 4. Moshe Klein. "Interview with Saharon Shelah" (PDF). Gan Adam. Retrieved August 5, 2014. 5. "Saharon Shelah". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved August 5, 2014. 6. "Hyperlinked list of Shelah's papers". Retrieved August 29, 2022. 7. Väänänen, Jouko (April 20, 2020). "An Overview of Saharon Shelah's Contributions to Mathematical Logic, in Particular to Model Theory". Theoria. 87 (2): 349–360. doi:10.1111/theo.12238. eISSN 1755-2567. ISSN 0040-5825. S2CID 216119512. 8. Shelah, Saharon (1988). "Primitive recursive bounds for van der Waerden numbers". Journal of the American Mathematical Society. 1 (3): 683–697. doi:10.2307/1990952. JSTOR 1990952. MR 0929498. 9. On the Arrow property 10. Zilber, Boris (October 2016). "Model theory of special subvarieties and Schanuel-type conjectures". Annals of Pure and Applied Logic. 167 (10): 1000–1028. doi:10.1016/j.apal.2015.02.002. ISSN 0168-0072. S2CID 33799837. 11. "Erdős Prize Website". IMU.org.il. Archived from the original on August 17, 2013. 12. "Karp Prize Recipients". Retrieved September 28, 2019. 13. "Israel Prize Official Site – Recipients in 1998 (in Hebrew)". CMS.education.gov.il. Retrieved August 31, 2014. 14. "Laudation of Shelah on the occasion of winning the Bolyai Prize (in Hungarian)" (PDF). Renyi.hu. Retrieved August 31, 2014. 15. "The Wolf Foundation Prize in Mathematics". Wolf Foundation. 2008. Archived from the original on September 21, 2017. Retrieved August 31, 2014. 16. "EMET Prize". 2011. Retrieved August 31, 2014. 17. "January 2013 Prizes and Awards" (PDF). American Mathematical Society and Mathematical Association of America. January 10, 2013. p. 49. Retrieved August 31, 2014. 18. "New members of the Hungarian Academy of Sciences". Archived from the original on September 3, 2014. Retrieved August 31, 2014. 19. "ERC Grants 2013" (PDF). European Research Council. 2013. Retrieved August 31, 2014. 20. "Hausdorff medal 2017". July 5, 2017. Retrieved September 28, 2019. 21. "Schock Prize 2018". Retrieved September 28, 2019. 22. "Ehrendoktorat der TU Wien für Saharon Shelah". 2019. Retrieved February 2, 2020. 23. Baldwin, John T. (1981). "Review: Classification theory and the number of non-isomorphic models by Saharon Shelah" (PDF). Bull. Amer. Math. Soc. (N.S.). 4 (2): 222–229. doi:10.1090/s0273-0979-1981-14891-6. 24. Baumgartner, James E. (1996). "Review: Cardinal arithmetic by Saharon Shelah" (PDF). Bull. Amer. Math. Soc. (N.S.). 33 (3): 409–411. doi:10.1090/s0273-0979-96-00673-8. External links • Archive of Shelah's mathematical papers, shelah.logic.at • John T. Baldwin (July 7, 2006). "Abstract Elementary Classes: Some Answers, More Questions" (PDF). math.uic.ed. {{cite journal}}: Cite journal requires \|journal= (help) A survey of recent work on AECs. Laureates of the Wolf Prize in Mathematics 1970s • Israel Gelfand / Carl L. Siegel (1978) • Jean Leray / André Weil (1979) 1980s • Henri Cartan / Andrey Kolmogorov (1980) • Lars Ahlfors / Oscar Zariski (1981) • Hassler Whitney / Mark Krein (1982) • Shiing-Shen Chern / Paul Erdős (1983/84) • Kunihiko Kodaira / Hans Lewy (1984/85) • Samuel Eilenberg / Atle Selberg (1986) • Kiyosi Itô / Peter Lax (1987) • Friedrich Hirzebruch / Lars Hörmander (1988) • Alberto Calderón / John Milnor (1989) 1990s • Ennio de Giorgi / Ilya Piatetski-Shapiro (1990) • Lennart Carleson / John G. Thompson (1992) • Mikhail Gromov / Jacques Tits (1993) • Jürgen Moser (1994/95) • Robert Langlands / Andrew Wiles (1995/96) • Joseph Keller / Yakov G. Sinai (1996/97) • László Lovász / Elias M. Stein (1999) 2000s • Raoul Bott / Jean-Pierre Serre (2000) • Vladimir Arnold / Saharon Shelah (2001) • Mikio Sato / John Tate (2002/03) • Grigory Margulis / Sergei Novikov (2005) • Stephen Smale / Hillel Furstenberg (2006/07) • Pierre Deligne / Phillip A. Griffiths / David B. Mumford (2008) 2010s • Dennis Sullivan / Shing-Tung Yau (2010) • Michael Aschbacher / Luis Caffarelli (2012) • George Mostow / Michael Artin (2013) • Peter Sarnak (2014) • James G. Arthur (2015) • Richard Schoen / Charles Fefferman (2017) • Alexander Beilinson / Vladimir Drinfeld (2018) • Jean-François Le Gall / Gregory Lawler (2019) 2020s • Simon K. Donaldson / Yakov Eliashberg (2020) • George Lusztig (2022) • Ingrid Daubechies (2023) Mathematics portal Rolf Schock Prize laureates Logic and philosophy • Willard Van Orman Quine (1993) • Michael Dummett (1995) • Dana Scott (1997) • John Rawls (1999) • Saul Kripke (2001) • Solomon Feferman (2003) • Jaakko Hintikka (2005) • Thomas Nagel (2008) • Hilary Putnam (2011) • Derek Parfit (2014) • Ruth Millikan (2017) • Saharon Shelah (2018) • Dag Prawitz / Per Martin-Löf (2020) • David Kaplan (2022) Mathematics • Elias M. Stein (1993) • Andrew Wiles (1995) • Mikio Sato (1997) • Yuri I. Manin (1999) • Elliott H. Lieb (2001) • Richard P. Stanley (2003) • Luis Caffarelli (2005) • Endre Szemerédi (2008) • Michael Aschbacher (2011) • Yitang Zhang (2014) • Richard Schoen (2017) • Ronald Coifman (2018) • Nikolai G. 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Modern Arabic mathematical notation Modern Arabic mathematical notation is a mathematical notation based on the Arabic script, used especially at pre-university levels of education. Its form is mostly derived from Western notation, but has some notable features that set it apart from its Western counterpart. The most remarkable of those features is the fact that it is written from right to left following the normal direction of the Arabic script. Other differences include the replacement of the Greek and Latin alphabet letters for symbols with Arabic letters and the use of Arabic names for functions and relations. Features • It is written from right to left following the normal direction of the Arabic script. Other differences include the replacement of the Latin alphabet letters for symbols with Arabic letters and the use of Arabic names for functions and relations. • The notation exhibits one of the very few remaining vestiges of non-dotted Arabic scripts, as dots over and under letters (i'jam) are usually omitted. • Letter cursivity (connectedness) of Arabic is also taken advantage of, in a few cases, to define variables using more than one letter. The most widespread example of this kind of usage is the canonical symbol for the radius of a circle نق (Arabic pronunciation: [nɑq]), which is written using the two letters nūn and qāf. When variable names are juxtaposed (as when expressing multiplication) they are written non-cursively. Variations Notation differs slightly from region to another. In tertiary education, most regions use the Western notation. The notation mainly differs in numeral system used, and in mathematical symbol used. Numeral systems There are three numeral systems used in right to left mathematical notation. • "Western Arabic numerals" (sometimes called European) are used in western Arabic regions (e.g. Morocco) • "Eastern Arabic numerals" are used in middle and eastern Arabic regions (e.g. Egypt and Syria) • "Eastern Arabic-Indic numerals" are used in Persian and Urdu speaking regions (e.g. Iran, Pakistan, India) European (descended from Western Arabic) 01234 56789 Arabic-Indic (Eastern Arabic) ٠١٢٣٤ ٥٦٧٨٩ Perso-Arabic variant ۰۱۲۳۴ ۵۶۷۸۹ Urdu variant Devanagari (Hindi) ०१२३४५६७८९ Tamil ௧௨௩௪௫௬௭௮௯ Written numerals are arranged with their lowest-value digit to the right, with higher value positions added to the left. That is identical to the arrangement used by Western texts using Hindu-Arabic numerals even though Arabic script is read from right to left. The symbols "٫" and "٬" may be used as the decimal mark and the thousands separator respectively when writing with Eastern Arabic numerals, e.g. ٣٫١٤١٥٩٢٦٥٣٥٨ 3.14159265358, ١٬٠٠٠٬٠٠٠٬٠٠٠ 1,000,000,000. Negative signs are written to the left of magnitudes, e.g. ٣− −3. In-line fractions are written with the numerator and denominator on the left and right of the fraction slash respectively, e.g. ٢/٧ 2/7. Mirrored Latin symbols Sometimes, symbols used in Arabic mathematical notation differ according to the region: Latin Arabic Persian lim x→∞ x4 س٤ نهــــــــــــا س←∞‏ [a] س۴ حــــــــــــد س←∞‏ [b] • ^a نهــــا nūn-hāʾ-ʾalif is derived from the first three letters of Arabic نهاية nihāya "limit". • ^b حد ḥadd is Persian for "limit". Sometimes, mirrored Latin symbols are used in Arabic mathematical notation (especially in western Arabic regions): Latin Arabic Mirrored Latin n ∑ x=0 3√x ٣‭√‬س ں مجــــــــــــ س=٠ [c] ‪√3‬س ں‭∑‬س=0 • ^c مجــــ is derived from Arabic مجموع maǧmūʿ "sum". However, in Iran, usually Latin symbols are used. Examples Mathematical letters Latin Arabic Notes $a$ ا From the Arabic letter ا ʾalif; a and ا ʾalif are the first letters of the Latin alphabet and the Arabic alphabet's ʾabjadī sequence respectively, and the letters also share a common ancestor and the same sound $b$ ٮ A dotless ب bāʾ; b and ب bāʾ are the second letters of the Latin alphabet and the ʾabjadī sequence respectively $c$ حــــ From the initial form of ح ḥāʾ, or that of a dotless ج jīm; c and ج jīm are the third letters of the Latin alphabet and the ʾabjadī sequence respectively , and the letters also share a common ancestor and the same sound $d$ د From the Arabic letter د dāl; d and د dāl are the fourth letters of the Latin alphabet and the ʾabjadī sequence respectively , and the letters also share a common ancestor and the same sound $x$ س From the Arabic letter س sīn. It is contested that the usage of Latin x in maths is derived from the first letter ش šīn (without its dots) of the Arabic word شيء šayʾ(un) [ʃajʔ(un)], meaning thing.[1] (X was used in old Spanish for the sound /ʃ/). However, according to others there is no historical evidence for this.[2][3] $y$ ص From the Arabic letter ص ṣād $z$ ع From the Arabic letter ع ʿayn Mathematical constants and units Description Latin Arabic Notes Euler's number $e$ ھ Initial form of the Arabic letter ه hāʾ. Both Latin letter e and Arabic letter ه hāʾ are descendants of Phoenician letter hē. imaginary unit $i$ ت From ت tāʾ, which is in turn derived from the first letter of the second word of وحدة تخيلية waḥdaẗun taḫīliyya "imaginary unit" pi $\pi $ ط From ط ṭāʾ; also $\pi $ in some regions radius $r$ نٯ From ن nūn followed by a dotless ق qāf, which is in turn derived from نصف القطر nuṣfu l-quṭr "radius" kilogram kg كجم From كجم kāf-jīm-mīm. In some regions alternative symbols like (كغ kāf-ġayn) or (كلغ kāf-lām-ġayn) are used. All three abbreviations are derived from كيلوغرام kīlūġrām "kilogram" and its variant spellings. gram g جم From جم jīm-mīm, which is in turn derived from جرام jrām, a variant spelling of غرام ġrām "gram" meter m م From م mīm, which is in turn derived from متر mitr "meter" centimeter cm سم From سم sīn-mīm, which is in turn derived from سنتيمتر "centimeter" millimeter mm مم From مم mīm-mīm, which is in turn derived from مليمتر millīmitr "millimeter" kilometer km كم From كم kāf-mīm; also (كلم kāf-lām-mīm) in some regions; both are derived from كيلومتر kīlūmitr "kilometer". second s ث From ث ṯāʾ, which is in turn derived from ثانية ṯāniya "second" minute min د From د dālʾ, which is in turn derived from دقيقة daqīqa "minute"; also (ٯ, i.e. dotless ق qāf) in some regions hour h س From س sīnʾ, which is in turn derived from ساعة sāʿa "hour" kilometer per hour km/h كم/س From the symbols for kilometer and hour degree Celsius °C °س From س sīn, which is in turn derived from the second word of درجة سيلسيوس darajat sīlsīūs "degree Celsius"; also (°م) from م mīmʾ, which is in turn derived from the first letter of the third word of درجة حرارة مئوية "degree centigrade" degree Fahrenheit °F °ف From ف fāʾ, which is in turn derived from the second word of درجة فهرنهايت darajat fahranhāyt "degree Fahrenheit" millimeters of mercury mmHg مم‌ز From مم‌ز mīm-mīm zayn, which is in turn derived from the initial letters of the words مليمتر زئبق "millimeters of mercury" Ångström Å أْ From أْ ʾalif with hamzah and ring above, which is in turn derived from the first letter of "Ångström", variously spelled أنغستروم or أنجستروم Sets and number systems Description Latin Arabic Notes Natural numbers $\mathbb {N} $ ط From ط ṭāʾ, which is in turn derived from the first letter of the second word of عدد طبيعيʿadadun ṭabīʿiyyun "natural number" Integers $\mathbb {Z} $ ص From ص ṣād, which is in turn derived from the first letter of the second word of عدد صحيح ʿadadun ṣaḥīḥun "integer" Rational numbers $\mathbb {Q} $ ن From ن nūn, which is in turn derived from the first letter of نسبة nisba "ratio" Real numbers $\mathbb {R} $ ح From ح ḥāʾ, which is in turn derived from the first letter of the second word of عدد حقيقي ʿadadun ḥaqīqiyyun "real number" Imaginary numbers $\mathbb {I} $ ت From ت tāʾ, which is in turn derived from the first letter of the second word of عدد تخيلي ʿadadun taḫīliyyun "imaginary number" Complex numbers $\mathbb {C} $ م From م mīm, which is in turn derived from the first letter of the second word of عدد مركب ʿadadun murakkabun "complex number" Empty set $\varnothing $ $\varnothing $∅ Is an element of $\in $ $\ni $∈ A mirrored ∈ Subset $\subset $ $\supset $⊂ A mirrored ⊂ Superset $\supset $ $\subset $⊃ A mirrored ⊃ Universal set $\mathbf {S} $ ش From ش šīn, which is in turn derived from the first letter of the second word of مجموعة شاملة majmūʿatun šāmila "universal set" Arithmetic and algebra Description Latin Arabic Notes Percent % ٪ e.g. 100% "٪١٠٠" Permille ‰ ؉ ؊ is an Arabic equivalent of the per ten thousand sign ‱. Is proportional to $\propto $ ∝ A mirrored ∝ n th root ${\sqrt[{n}]{\,\,\,}}$ ں‭√‬ ‏ ں is a dotless ن nūn while √ is a mirrored radical sign √ Logarithm $\log $ لو From لو lām-wāw, which is in turn derived from لوغاريتم lūġārītm "logarithm" Logarithm to base b $\log _{b}$ لوٮ Natural logarithm $\ln $ لوھ From the symbols of logarithm and Euler's number Summation $\sum $ مجــــ مجـــ mīm-medial form of jīm is derived from the first two letters of مجموع majmūʿ "sum"; also (∑, a mirrored summation sign ∑) in some regions Product $\prod $ جــــذ From جذ jīm-ḏāl. The Arabic word for "product" is جداء jadāʾun. Also $\prod $ in some regions. Factorial $n!$ ں Also ( ں! ) in some regions Permutations $^{n}\mathbf {P} _{r}$ ںلر Also ( ل(ں، ر) ) is used in some regions as $\mathbf {P} (n,r)$ Combinations $^{n}\mathbf {C} _{k}$ ںٯك Also ( ٯ(ں، ك) ) is used in some regions as $\mathbf {C} (n,k)$ and ( ⎛⎝ں ك ⎞⎠ ) as the binomial coefficient $n \choose k$ Trigonometric functions Description Latin Arabic Notes Sine $\sin $ حا from حاء ḥāʾ (i.e. dotless ج jīm)-ʾalif; also (جب jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "sine" is جيب jayb Cosine $\cos $ حتا from حتا ḥāʾ (i.e. dotless ج jīm)-tāʾ-ʾalif; also (تجب tāʾ-jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "cosine" is جيب تمام Tangent $\tan $ طا from طا ṭāʾ (i.e. dotless ظ ẓāʾ)-ʾalif; also (ظل ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "tangent" is ظل ẓill Cotangent $\cot $ طتا from طتا ṭāʾ (i.e. dotless ظ ẓāʾ)-tāʾ-ʾalif; also (تظل tāʾ-ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "cotangent" is ظل تمام Secant $\sec $ ٯا from ٯا dotless ق qāf-ʾalif; Arabic for "secant" is قاطع Cosecant $\csc $ ٯتا from ٯتا dotless ق qāf-tāʾ-ʾalif; Arabic for "cosecant" is قاطع تمام Hyperbolic functions The letter (ز zayn, from the first letter of the second word of دالة زائدية "hyperbolic function") is added to the end of trigonometric functions to express hyperbolic functions. This is similar to the way $\operatorname {h} $ is added to the end of trigonometric functions in Latin-based notation. Description Hyperbolic sine Hyperbolic cosine Hyperbolic tangent Hyperbolic cotangent Hyperbolic secant Hyperbolic cosecant Latin $\sinh $$\cosh $$\tanh $$\coth $$\operatorname {sech} $$\operatorname {csch} $ Arabic حاز حتاز طاز طتاز ٯاز ٯتاز Inverse trigonometric functions For inverse trigonometric functions, the superscript −١ in Arabic notation is similar in usage to the superscript $-1$ in Latin-based notation. Description Inverse sine Inverse cosine Inverse tangent Inverse cotangent Inverse secant Inverse cosecant Latin $\sin ^{-1}$$\cos ^{-1}$$\tan ^{-1}$$\cot ^{-1}$$\sec ^{-1}$$\csc ^{-1}$ Arabic حا−١ حتا−١ طا−١ طتا−١ ٯا−١ ٯتا−١ Inverse hyperbolic functions Description Inverse hyperbolic sine Inverse hyperbolic cosine Inverse hyperbolic tangent Inverse hyperbolic cotangent Inverse hyperbolic secant Inverse hyperbolic cosecant Latin $\sinh ^{-1}$$\cosh ^{-1}$$\tanh ^{-1}$$\coth ^{-1}$$\operatorname {sech} ^{-1}$$\operatorname {csch} ^{-1}$ Arabic حاز−١ حتاز−١ طاز−١ طتاز−١ ٯاز−١ ٯتاز−١ Calculus Description Latin Arabic Notes Limit $\lim $ نهــــا نهــــا nūn-hāʾ-ʾalif is derived from the first three letters of Arabic نهاية nihāya "limit" Function $\mathbf {f} (x)$ د(س) د dāl is derived from the first letter of دالة "function". Also called تابع, تا for short, in some regions. Derivatives $\mathbf {f'} (x),{\dfrac {dy}{dx}},{\dfrac {d^{2}y}{dx^{2}}},{\dfrac {\partial {y}}{\partial {x}}}$ ص∂/س∂ ،د٢ص/ د‌س٢ ،د‌ص/ د‌س ،(س)‵د ‵ is a mirrored prime ′ while ، is an Arabic comma. The ∂ signs should be mirrored: ∂. Integrals $\int {},\iint {},\iiint {},\oint {}$ ‪∮ ،∭ ،∬ ،∫‬ Mirrored ∫, ∬, ∭ and ∮ Complex analysis Latin Arabic $z=x+iy=r(\cos {\varphi }+i\sin {\varphi })=re^{i\varphi }=r\angle {\varphi }$ ع = س + ت ص = ل(حتا ى + ت حا ى) = ل ھت‌ى = ل∠ى See also • Mathematical notation • Arabic Mathematical Alphabetic Symbols References 1. Moore, Terry. "Why is X the Unknown". Ted Talk. Archived from the original on 2014-02-22. Retrieved 2012-10-11. 2. Cajori, Florian (1993). A History of Mathematical Notation. Courier Dover Publications. pp. 382–383. ISBN 9780486677668. Retrieved 11 October 2012. Nor is there historical evidence to support the statement found in Noah Webster's Dictionary, under the letter x, to the effect that 'x was used as an abbreviation of Ar. shei (a thing), something, which, in the Middle Ages, was used to designate the unknown, and was then prevailingly transcribed as xei.' 3. Oxford Dictionary, 2nd Edition. There is no evidence in support of the hypothesis that x is derived ultimately from the mediaeval transliteration xei of shei "thing", used by the Arabs to denote the unknown quantity, or from the compendium for L. res "thing" or radix "root" (resembling a loosely-written x), used by mediaeval mathematicians. External links • Multilingual mathematical e-document processing • Arabic mathematical notation - W3C Interest Group Note. • Arabic math editor - by WIRIS.
2 2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures. ← 1 2 3 → −1 0 1 2 3 4 5 6 7 8 9 → • List of numbers • Integers ← 0 10 20 30 40 50 60 70 80 90 → Cardinaltwo Ordinal2nd (second / twoth) Numeral systembinary Factorizationprime Gaussian integer factorization$(1+i)(1-i)$ Prime1st Divisors1, 2 Greek numeralΒ´ Roman numeralII, ii Greek prefixdi- Latin prefixduo-/bi- Old English prefixtwi- Binary102 Ternary23 Senary26 Octal28 Duodecimal212 Hexadecimal216 Greek numeralβ' Arabic, Kurdish, Persian, Sindhi, Urdu٢ Ge'ez፪ Bengali২ Chinese numeral二，弍，貳 Devanāgarī२ Telugu౨ Tamil௨ Kannada೨ Hebrewב Khmer២ Thai๒ Georgian Ⴁ/ⴁ/ბ(Bani) Malayalam൨ Evolution Arabic digit The digit used in the modern Western world to represent the number 2 traces its roots back to the Indic Brahmic script, where "2" was written as two horizontal lines. The modern Chinese and Japanese languages (and Korean Hanja) still use this method. The Gupta script rotated the two lines 45 degrees, making them diagonal. The top line was sometimes also shortened and had its bottom end curve towards the center of the bottom line. In the Nagari script, the top line was written more like a curve connecting to the bottom line. In the Arabic Ghubar writing, the bottom line was completely vertical, and the digit looked like a dotless closing question mark. Restoring the bottom line to its original horizontal position, but keeping the top line as a curve that connects to the bottom line leads to our modern digit.[1] In fonts with text figures, digit 2 usually is of x-height, for example, . As a word Two is most commonly a determiner used with plural countable nouns, as in two days or I'll take these two.[2] Two is a noun when it refers to the number two as in two plus two is four. Etymology of two The word two is derived from the Old English words twā (feminine), tū (neuter), and twēġen (masculine, which survives today in the form twain).[3] The pronunciation /tuː/, like that of who is due to the labialization of the vowel by the w, which then disappeared before the related sound. The successive stages of pronunciation for the Old English twā would thus be /twɑː/, /twɔː/, /twoː/, /twuː/, and finally /tuː/.[3] In mathematics Two is the smallest prime number, and the only even prime number, and for this reason it is sometimes called "the oddest prime".[4] As the smallest prime number, it is also the smallest non-zero pronic number, and the only pronic prime.[5] The next prime is three, which makes two and three the only two consecutive prime numbers. Two is the first prime number that does not have a proper twin prime with a difference two, while three is the first such prime number to have a twin prime, five.[6][7] In consequence, three and five encase four in-between, which is the square of two or $2^{2}$. These are also the two odd prime numbers that lie amongst the only all-Harshad numbers 1, 2, 4, and 6. An integer is called even if it is divisible by 2. For integers written in a numeral system based on an even number such as decimal, divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8.[8] Two is the base of the binary system, the numeral system with the fewest tokens that allows denoting a natural number substantially more concisely (with $\log _{2}$ $n$ tokens) than a direct representation by the corresponding count of a single token (with $n$ tokens). This binary number system is used extensively in computing. The square root of 2 was the first known irrational number. Taking the square root of a number is such a common and essential mathematical operation, that the spot on the root sign where the index would normally be written for cubic and other roots, may simply be left blank for square roots, as it is tacitly understood. Powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent. They are also essential to Fermat primes and Pierpont primes, which have consequences in the constructability of regular polygons using basic tools. In a set-theoretical construction of the natural numbers, two is identified with the set $\{\{\varnothing \},\varnothing \}$. This latter set is important in category theory: it is a subobject classifier in the category of sets. A set that is a field has a minimum of two elements. A Cantor space is a topological space $2^{\mathbb {N} }$ homeomorphic to the Cantor set. The countably infinite product topology of the simplest discrete two-point space, $\{0,1\}$, is the traditional elementary example of a Cantor space. A number is deficient when the sum of its divisors is less than twice the number, whereas an abundant number has a sum of its proper divisors that is larger than the number itself. Primitive abundant numbers are abundant numbers whose proper divisors are all deficient. A number is perfect if it is equal to its aliquot sum, or the sum of all of its positive divisors excluding the number itself. This is equivalent to describing a perfect number $n$ as having a sum of divisors $\sigma (n)$ equal to $2n$. Two is the first Sophie Germain prime,[9] the first factorial prime,[10] the first Lucas prime,[11] and the first Ramanujan prime.[12] It is also a Motzkin number,[13] a Bell number,[14] and the third (or fourth) Fibonacci number.[15] $(3,5)$ are the unique pair of twin primes $(q,q+2)$ that yield the second and only prime quadruplet $(11,13,17,19)$ that is of the form $(d-4,d-2,d+2,d+4)$, where $d$ is the product of said twin primes.[16] Two has the unique property that $2+2=2\times 2=2^{2}=2\uparrow \uparrow 2=2\uparrow \uparrow \uparrow 2={\text{ }}...$ up through any level of hyperoperation, here denoted in Knuth's up-arrow notation, all equivalent to $4.$ Two consecutive twos (as in "22" for "two twos"), or equivalently "2-2", is the only fixed point of John Conway's look-and-say function.[17] Two is the only number $n$ such that the sum of the reciprocals of the natural powers of $n$ equals itself. In symbols, $\sum _{n=0}^{\infty }{\frac {1}{2^{n}}}=1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =2.$ The sum of the reciprocals of all non-zero triangular numbers converges to 2.[18] 2 is the harmonic mean of the divisors of 6, the smallest Ore number greater than 1. Like one, two is a meandric number,[19] a semi-meandric number,[20] and an open meandric number.[21] Euler's number $e$ can be simplified to equal, $e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=2+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots $ A continued fraction for $e=[2;1,2,1,1,4,1,1,8,...]$ repeats a $\{1,2n,1\}$ pattern from the second term onward.[22][23] In a Euclidean space of any dimension greater than zero, two distinct points determine a line. A digon is a polygon with two sides (or edges) and two vertices. On a circle, it is a tessellation with two antipodal points and 180° arc edges. The circumference of a circle of radius $r$ is $2\pi r$. Regarding regular polygons in two dimensions, • The equilateral triangle has the smallest ratio of the circumradius $R$ to the inradius $r$ of any triangle by Euler's inequality, with ${\tfrac {R}{r}}=2.$[24] • The long diagonal of a regular hexagon is of length 2 when its sides are of unit length. • The span of an octagon is in silver ratio $\delta _{s}$ with its sides, which can be computed with the continued fraction $[2;2,2,...]=2.4142\dots $[25] Whereas a square of unit side length has a diagonal equal to ${\sqrt {2}}$, a space diagonal inside a tesseract measures 2 when its side lengths are of unit length. There are no $2\times 2$ magic squares, and as such they are the only null $n$ by $n$ magic square set.[26] Meanwhile, the magic constant of an $n$-pointed normal magic star is $M=4n+2$. For any polyhedron homeomorphic to a sphere, the Euler characteristic is $\chi =V-E+F=2$, where $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces. A double torus has a Euler characteristic of $-2$, on the other hand, and a non-orientable surface of like genus $k$ has a characteristic $\chi =2-k$. The simplest tessellation in two-dimensional space, though an improper tessellation, is that of two $\infty $-sided apeirogons joined along all their edges, coincident about a line that divides the plane in two. This order-2 apeirogonal tiling is the arithmetic limit of the family of dihedra $\{p,2\}$. There are two known sublime numbers, which are numbers with a perfect number of factors, whose sum itself yields a perfect number. 12 is one of the two sublime numbers, with the other being 76 digits long.[27] List of basic calculations Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 2 × x 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 100 200 Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 ÷ x 2 1 0.6 0.5 0.4 0.3 0.285714 0.25 0.2 0.2 0.18 0.16 0.153846 0.142857 0.13 0.125 0.1176470588235294 0.1 0.105263157894736842 0.1 x ÷ 2 0.5 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2x 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 x2 1 9 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 In science • The number of polynucleotide strands in a DNA double helix.[28] • The first magic number.[29] • The atomic number of helium.[30] • The ASCII code of "Start of Text". • 2 Pallas, a large asteroid in the main belt and the second asteroid ever to be discovered.[31] • The Roman numeral II (usually) stands for the second-discovered satellite of a planet or minor planet (e.g. Pluto II or (87) Sylvia II Remus). • A binary star is a stellar system consisting of two stars orbiting around their center of mass.[32] • The number of brain and cerebellar hemispheres.[33] In sports International maritime pennant for 2 International maritime signal flag for 2 • The number of points scored on a safety in American football • A field goal inside the three-point line is worth two points in basketball. • The two in basketball is called the shooting guard. • 2 represents the catcher position in baseball. See also • List of highways numbered 2 • Binary number References 1. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 393, Fig. 24.62 2. Huddleston, Rodney D.; Pullum, Geoffrey K.; Reynolds, Brett (2022). A student's introduction to English grammar (2nd ed.). Cambridge, United Kingdom: Cambridge University Press. p. 117. ISBN 978-1-316-51464-1. OCLC 1255524478. 3. "two, adj., n., and adv.". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.) 4. John Horton Conway & Richard K. Guy, The Book of Numbers. New York: Springer (1996): 25. ISBN 0-387-97993-X. "Two is celebrated as the only even prime, which in some sense makes it the oddest prime of all." 5. "Sloane's A002378: Pronic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 2016-06-09. Retrieved 2020-11-30. 6. Sloane, N. J. A. (ed.). "Sequence A007510 (Single (or isolated or non-twin) primes: Primes p such that neither p-2 nor p+2 is prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-05. 7. Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-05. 8. Sloane, N. J. A. (ed.). "Sequence A005843 (The nonnegative even numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15. 9. Sloane, N. J. A. (ed.). "Sequence A005384 (Sophie Germain primes p: 2p+1 is also prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15. 10. Sloane, N. J. A. (ed.). "Sequence A088054 (Factorial primes: primes which are within 1 of a factorial number.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15. 11. Sloane, N. J. A. (ed.). "Sequence A005479 (Prime Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15. 12. "Sloane's A104272 : Ramanujan primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 2011-04-28. Retrieved 2016-06-01. 13. Sloane, N. J. A. (ed.). "Sequence A001006 (Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15. 14. Sloane, N. J. A. (ed.). "Sequence A000110 (Bell or exponential numbers: number of ways to partition a set of n labeled elements.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15. 15. Sloane, N. J. A. (ed.). "Sequence A000045 (Fibonacci numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15. 16. Sloane, N. J. A. (ed.). "Sequence A136162 (List of prime quadruplets {p, p+2, p+6, p+8}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-09. "{11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3(12k^2-1) and 3(12k^2+1) respectively (k is an integer)." 17. Martin, Oscar (2006). "Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA" (PDF). American Mathematical Monthly. Mathematical association of America. 113 (4): 289–307. doi:10.2307/27641915. ISSN 0002-9890. JSTOR 27641915. Archived from the original (PDF) on 2006-12-24. Retrieved 2022-07-21. 18. Grabowski, Adam (2013). "Polygonal numbers". Formalized Mathematics. Sciendo (De Gruyter). 21 (2): 103–113. doi:10.2478/forma-2013-0012. S2CID 15643540. Zbl 1298.11029. 19. Sloane, N. J. A. (ed.). "Sequence A005315 (Closed meandric numbers (or meanders): number of ways a loop can cross a road 2n times.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15. 20. Sloane, N. J. A. (ed.). "Sequence A000682 (Semi-meanders: number of ways a semi-infinite directed curve can cross a straight line n times.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15. 21. Sloane, N. J. A. (ed.). "Sequence A005316 (Meandric numbers: number of ways a river can cross a road n times.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-15. 22. Cohn, Henry (2006). "A Short Proof of the Simple Continued Fraction Expansion of e". The American Mathematical Monthly. Taylor & Francis, Ltd. 113 (1): 57–62. doi:10.1080/00029890.2006.11920278. JSTOR 27641837. MR 2202921. S2CID 43879696. Zbl 1145.11012. Archived from the original on 2023-04-30. Retrieved 2023-04-30. 23. Sloane, N. J. A. (ed.). "Sequence A005131 (A generalized continued fraction for Euler's number e.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-30. "Only a(1) = 0 prevents this from being a simple continued fraction. The motivation for this alternate representation is that the simple pattern {1, 2*n, 1} (from n=0) may be more mathematically appealing than the pattern in the corresponding simple continued fraction (at A003417)." 24. Svrtan, Dragutin; Veljan, Darko (2012). "Non-Euclidean versions of some classical triangle inequalities" (PDF). Forum Geometricorum. Boca Raton, FL: Department of Mathematical Sciences, Florida Atlantic University. 12: 198. ISSN 1534-1178. MR 2955631. S2CID 29722079. Zbl 1247.51012. Archived (PDF) from the original on 2023-05-03. Retrieved 2023-04-30. 25. Vera W. de Spinadel (1999). "The Family of Metallic Means". Visual Mathematics. Belgrade: Mathematical Institute of the Serbian Academy of Sciences. 1 (3). eISSN 1821-1437. S2CID 125705375. Zbl 1016.11005. Archived from the original on 2023-03-26. Retrieved 2023-02-25. 26. Sloane, N. J. A. (ed.). "Sequence A006052 (Number of magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-21. 27. Sloane, N. J. A. (ed.). "Sequence A081357 (Sublime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-13. 28. "Double-stranded DNA". Scitable. Nature Education. Archived from the original on 2020-07-24. Retrieved 2019-12-22. 29. "The Complete Explanation of the Nuclear Magic Numbers Which Indicate the Filling of Nucleonic Shells and the Revelation of Special Numbers Indicating the Filling of Subshells Within Those Shells". www.sjsu.edu. Archived from the original on 2019-12-02. Retrieved 2019-12-22. 30. Bezdenezhnyi, V. P. (2004). "Nuclear Isotopes and Magic Numbers". Odessa Astronomical Publications. 17: 11. Bibcode:2004OAP....17...11B. 31. "Asteroid Fact Sheet". nssdc.gsfc.nasa.gov. Archived from the original on 2020-02-01. Retrieved 2019-12-22. 32. Staff (2018-01-17). "Binary Star Systems: Classification and Evolution". Space.com. Archived from the original on 2019-12-22. Retrieved 2019-12-22. 33. Lewis, Tanya (2018-09-28). "Human Brain: Facts, Functions & Anatomy". livescience.com. Archived from the original on 2019-12-22. Retrieved 2019-12-22. External links Wikimedia Commons has media related to: 2 (number) (category) • Prime curiosities: 2 Look up two or both in Wiktionary, the free dictionary. Integers 0s • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 100s • 100 • 101 • 102 • 103 • 104 • 105 • 106 • 107 • 108 • 109 • 110 • 111 • 112 • 113 • 114 • 115 • 116 • 117 • 118 • 119 • 120 • 121 • 122 • 123 • 124 • 125 • 126 • 127 • 128 • 129 • 130 • 131 • 132 • 133 • 134 • 135 • 136 • 137 • 138 • 139 • 140 • 141 • 142 • 143 • 144 • 145 • 146 • 147 • 148 • 149 • 150 • 151 • 152 • 153 • 154 • 155 • 156 • 157 • 158 • 159 • 160 • 161 • 162 • 163 • 164 • 165 • 166 • 167 • 168 • 169 • 170 • 171 • 172 • 173 • 174 • 175 • 176 • 177 • 178 • 179 • 180 • 181 • 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• 8000 • 9000 • 10,000 • 20,000 • 30,000 • 40,000 • 50,000 • 60,000 • 70,000 • 80,000 • 90,000 • 100,000 • 1,000,000 • 10,000,000 • 100,000,000 • 1,000,000,000 Authority control: National • Germany • Israel • United States
4 4 (four) is a number, numeral and digit. It is the natural number following 3 and preceding 5. It is a square number, the smallest semiprime and composite number, and is considered unlucky in many East Asian cultures. ← 3 4 5 → −1 0 1 2 3 4 5 6 7 8 9 → • List of numbers • Integers ← 0 10 20 30 40 50 60 70 80 90 → Cardinalfour Ordinal4th (fourth) Numeral systemquaternary Factorization22 Divisors1, 2, 4 Greek numeralΔ´ Roman numeralIV, iv Greek prefixtetra- Latin prefixquadri-/quadr- Binary1002 Ternary113 Senary46 Octal48 Duodecimal412 Hexadecimal416 Arabic, Kurdish٤ Persian, Sindhi۴ Shahmukhi, Urdu۴ Ge'ez፬ Bengali, Assamese৪ Chinese numeral四，亖，肆 Devanagari४ Telugu౪ Malayalam൪ Tamil௪ Hebrewד Khmer៤ Thai๔ Kannada೪ Burmese၄ Evolution of the Hindu-Arabic digit Brahmic numerals represented 1, 2, and 3 with as many lines. 4 was simplified by joining its four lines into a cross that looks like the modern plus sign. The Shunga would add a horizontal line on top of the digit, and the Kshatrapa and Pallava evolved the digit to a point where the speed of writing was a secondary concern. The Arabs' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the digit less cursive, ending up with a digit very close to the original Brahmin cross.[1] While the shape of the character for the digit 4 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in . On the seven-segment displays of pocket calculators and digital watches, as well as certain optical character recognition fonts, 4 is seen with an open top: .[2] Television stations that operate on channel 4 have occasionally made use of another variation of the "open 4", with the open portion being on the side, rather than the top. This version resembles the Canadian Aboriginal syllabics letter ᔦ. The magnetic ink character recognition "CMC-7" font also uses this variety of "4".[3] Mathematics Four is the smallest composite number, its proper divisors being 1 and 2.[4] Four is the sum and product of two with itself: $2+2=4=2\times 2$, the only number $b$ such that $a+a=b=a\times a$, which also makes four the smallest and only even squared prime number $2^{2}$ and hence the first squared prime of the form $p^{2}$, where $p$ is a prime. Four, as the first composite number, has a prime aliquot sum of 3; and as such it is part of the first aliquot sequence with a single composite member, expressly (4, 3, 1, 0). • In Knuth's up-arrow notation, $2+2=2\times 2=2^{2}=2\uparrow \uparrow 2=2\uparrow \uparrow \uparrow 2=\;...\;=4$, and so forth, for any number of up arrows.[5] By consequence, four is the only square one more than a prime number, specifically three. • The sum of the first four prime numbers two + three + five + seven is the only sum of four consecutive prime numbers that yields an odd prime number, seventeen, which is the fourth super-prime. Four lies between the first proper pair of twin primes, three and five, which are the first two Fermat primes, like seventeen, which is the third. On the other hand, the square of four ($4^{2}$), equivalently the fourth power of two ($2^{4}$), is sixteen; the only number that has $a^{b}=b^{a}$ as a form of factorization. Holistically, there are four elementary arithmetic operations in mathematics: addition (+), subtraction (−), multiplication (×), and division (÷); and four basic number systems, the real numbers $\mathbb {R} $, rational numbers $\mathbb {Q} $, integers $\mathbb {Z} $, and natural numbers $\mathbb {N} $. Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. $4x=y^{2}-z^{2}$. A number is a multiple of 4 if its last two digits are a multiple of 4 (for example, 1092 is a multiple of 4 because 92 = 4 × 23).[6] Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers.[7] Three are not always sufficient; 7 for instance cannot be written as the sum of three squares.[8] There are four all-Harshad numbers: 1, 2, 4, and 6. 12, which is divisible by four thrice over, is a Harshad number in all bases except octal. A four-sided plane figure is a quadrilateral or quadrangle, sometimes also called a tetragon. It can be further classified as a rectangle or oblong, kite, rhombus, and square. Four is the highest degree general polynomial equation for which there is a solution in radicals.[9] The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors.[10] Three colors are not, in general, sufficient to guarantee this.[11] The largest planar complete graph has four vertices.[12] A solid figure with four faces as well as four vertices is a tetrahedron, which is the smallest possible number of faces and vertices a polyhedron can have.[13][14] The regular tetrahedron, also called a 3-simplex, is the simplest Platonic solid.[15] It has four regular triangles as faces that are themselves at dual positions with the vertices of another tetrahedron.[16] Tetrahedra can be inscribed inside all other four Platonic solids, and tessellate space alongside the regular octahedron in the alternated cubic honeycomb. The third dimension holds a total of four Coxeter groups that generate convex uniform polyhedra: the tetrahedral group, the octahedral group, the icosahedral group, and a dihedral group (of orders 24, 48, 120, and 4$n$, respectively). There are also four general Coxeter groups of generalized uniform prisms, where two are hosoderal and dihedral groups that form spherical tilings, with another two general prismatic and antiprismatic groups that represent truncated hosohedra (or simply, prisms) and snub antiprisms, respectively. Four-dimensional space is the highest-dimensional space featuring more than three regular convex figures: • Two-dimensional: infinitely many regular polygons. • Three-dimensional: five regular polyhedra; the five Platonic solids which are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. • Four-dimensional: six regular polychora; the 5-cell, 8-cell or tesseract, 16-cell, 24-cell, 120-cell, and 600-cell. The 24-cell, made of regular octahedra, has no analogue in any other dimension; it is self-dual, with its 24-cell honeycomb dual to the 16-cell honeycomb. • Five-dimensional and every higher dimension: three regular convex $n$-polytopes, all within the infinite family of regular $n$-simplexes, $n$-hypercubes, and $n$-orthoplexes. The fourth dimension is also the highest dimension where regular self-intersecting figures exist: • Two-dimensional: infinitely many regular star polygons. • Three-dimensional: four regular star polyhedra, the regular Kepler-Poinsot star polyhedra. • Four-dimensional: ten regular star polychora, the Schläfli–Hess star polychora. They contain cells of Kepler-Poinsot polyhedra alongside regular tetrahedra, icosahedra and dodecahedra. • Five-dimensional and every higher dimension: zero regular star-polytopes; uniform star polytopes in dimensions $n$ > $4$ are the most symmetric, which mainly originate from stellations of regular $n$-polytopes. Altogether, sixteen (or 16 = 42) regular convex and star polychora are generated from symmetries of four (4) Coxeter Weyl groups and point groups in the fourth dimension: the $\mathrm {A} _{4}$ simplex, $\mathrm {B} _{4}$ hypercube, $\mathrm {F} _{4}$ icositetrachoric, and $\mathrm {H} _{4}$ hexacosichoric groups; with the $\mathrm {D} _{4}$ demihypercube group generating two alternative constructions. There are also sixty-four (or 64 = 43) four-dimensional Bravais lattices, alongside sixty-four uniform polychora in the fourth dimension based on the same $\mathrm {A} _{4}$, $\mathrm {B} _{4}$, $\mathrm {F} _{4}$ and $\mathrm {H} _{4}$ Coxeter groups, and extending to prismatic groups of uniform polyhedra, including one special non-Wythoffian form, the grand antiprism. There are also two infinite families of duoprisms and antiprismatic prisms in the fourth dimension. There are only four polytopes with radial equilateral symmetry: the hexagon, the cuboctahedron, the tesseract, and the 24-cell. Four-dimensional differential manifolds have some unique properties. There is only one differential structure on $\mathbb {R} ^{n}$ except when $n$ = $4$, in which case there are uncountably many. The smallest non-cyclic group has four elements; it is the Klein four-group.[17] An alternating groups are not simple for values $n$ ≤ $4$. There are four Hopf fibrations of hyperspheres: ${\begin{aligned}S^{0}&\hookrightarrow S^{1}\to S^{1},\\S^{1}&\hookrightarrow S^{3}\to S^{2},\\S^{3}&\hookrightarrow S^{7}\to S^{4},\\S^{7}&\hookrightarrow S^{15}\to S^{8}.\\\end{aligned}}$ They are defined as locally trivial fibrations that map $f:S^{2n-1}\rightarrow S^{n}$ for values of $n=2,4,8$ (aside from the trivial fibration mapping between two points and a circle).[18] Further extensions of the real numbers under Hurwitz's theorem states that there are four normed division algebras: the real numbers $\mathbb {R} $, the complex numbers $\mathbb {C} $, the quaternions $\mathbb {H} $, and the octonions $\mathbb {O} $. Under Cayley–Dickson constructions, the sedenions $\mathbb {S} $ constitute a further fourth extension over $\mathbb {R} $. The real numbers are ordered, commutative and associative algebras, as well as alternative algebras with power-associativity. The complex numbers $\mathbb {C} $ share all four multiplicative algebraic properties of the reals $\mathbb {R} $, without being ordered. The quaternions loose a further commutative algebraic property, while holding associative, alternative, and power-associative properties. The octonions are alternative and power-associative, while the sedenions are only power-associative. The sedenions and all further extensions of these four normed division algebras are solely power-associative with non-trivial zero divisors, which makes them non-division algebras. $\mathbb {R} $ has a vector space of dimension 1, while $\mathbb {C} $, $\mathbb {H} $, $\mathbb {O} $ and $\mathbb {S} $ work in algebraic number fields of dimensions 2, 4, 8, and 16, respectively. List of basic calculations Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000 4 × x 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 200 400 4000 Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4 ÷ x 4 2 1.3 1 0.8 0.6 0.571428 0.5 0.4 0.4 0.36 0.3 0.307692 0.285714 0.26 0.25 x ÷ 4 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4x 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864 268435456 1073741824 4294967296 x4 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561 38416 50625 65536 In religion Buddhism • Four Noble Truths – Dukkha, Samudaya, Nirodha, Magga[19] • Four sights – observations which affected Prince Siddhartha deeply and made him realize the sufferings of all beings, and compelled him to begin his spiritual journey—an old man, a sick man, a dead man, and an ascetic[20] • Four Great Elements – earth, water, fire, and wind[21] • Four Heavenly Kings[22] • Four Foundations of Mindfulness – contemplation of the body, contemplation of feelings, contemplation of mind, contemplation of mental objects[19] • Four Right Exertions[23] • Four Bases of Power[24] • Four jhānas[25] • Four arūpajhānas[26] • Four Divine Abidings – loving-kindness, compassion, sympathetic joy, and equanimity[27] • Four stages of enlightenment – stream-enterer, once-returner, non-returner, and arahant[28] • Four main pilgrimage sites – Lumbini, Bodh Gaya, Sarnath, and Kusinara[19] Judeo-Christian symbolism • The Tetragrammaton is the four-letter name of God.[29] • Ezekiel has a vision of four living creatures: a man, a lion, an ox, and an eagle.[30] • The four Matriarchs (foremothers) of Judaism are Sarah, Rebekah, Leah, and Rachel.[31] • The Four Species (lulav, hadass, aravah and etrog) are taken as one of the mitzvot on the Jewish holiday of Sukkot. (Judaism)[32] • The Four Cups of Wine to drink on the Jewish holiday of Passover. (Judaism)[33] • The Four Questions to be asked on the Jewish holiday of Passover. (Judaism)[33] • The Four Sons to be dealt with on the Jewish holiday of Passover. (Judaism)[33] • The Four Expressions of Redemption to be said on the Jewish holiday of Passover. (Judaism)[34] • The four Gospels: Matthew, Mark, Luke, and John. (Christianity)[35] • The Four Horsemen of the Apocalypse ride in the Book of Revelation. (Christianity)[36] • The four holy cities of Judaism: Jerusalem, Hebron, Safed, and Tiberius[37] Hinduism • There are four Vedas: Rigveda, Samaveda, Yajurveda and Atharvaveda.[38] • In Puruṣārtha, there are four aims of human life: Dharma, Artha, Kāma, Moksha.[39] • The four stages of life Brahmacharya (student life), Grihastha (household life), Vanaprastha (retired life) and Sannyasa (renunciation).[40] • The four primary castes or strata of society: Brahmana (priest/teacher), Kshatriya (warrior/politician), Vaishya (landowner/entrepreneur) and Shudra (servant/manual laborer).[41] • The swastika symbol is traditionally used in Hindu religions as a sign of good luck and signifies good from all four directions.[42] • The god Brahma has four faces.[43] • There are four yugas: Satya, Dvapara, Treta and Kali[44] Islam • Eid al-Adha lasts for four days, from the 10th to the 14th of Dhul Hijja.[45] • The four holy cities of Islam: Mecca, Medina, Jerusalem and Damascus. • The four tombs in the Green Dome: Muhammad, Abu Bakr, Umar ibn Khattab and Isa ibn Maryam (Jesus). • There are four Rashidun or Rightly Guided Caliphs: Abu Bakr, Umar ibn al-Khattab, Uthman ibn Affan and Ali ibn Abi Talib.[46] • The Four Arch Angels in Islam are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael)[47] • There are four months in which war is not permitted: Muharram, Rajab, Dhu al-Qi'dah and Dhu al-Hijjah.[48] • There are four Sunni schools of fiqh: Hanafi, Shafi`i, Maliki and Hanbali. • There are four major Sunni Imams: Abū Ḥanīfa, Muhammad ibn Idris ash-Shafi`i, Malik ibn Anas and Ahmad ibn Hanbal. • There are four books in Islam: Taurāt, Zābūr, Injīl, Qur'ān.[49] • Waiting for four months is ordained for those who take an oath for abstention from their wives.[50] • The waiting period of the woman whose husband dies is four months and ten days.[51] • When Abraham said: "My Lord, show me how You give life to the dead," Allah said: "Why! Do you have no faith?" Abraham replied: "Yes, but in order that my heart be at rest." He said: "Then take four birds, and tame them to yourself, then put a part of them on every hill, and summon them; they will come to you flying. [Al-Baqara 2:260][52] • The respite of four months was granted to give time to the mushriks in Surah At-Tawba so that they should consider their position carefully and decide whether to make preparation for war or to emigrate from the country or to accept Islam.[53] • Those who accuse honorable women (of unchastity) but do not produce four witnesses, flog them with eighty lashes, and do not admit their testimony ever after. They are indeed transgressors. [An-Noor 24:4][54] Taoism • Four Symbols of I Ching[55] Other • In a more general sense, numerous mythological and cosmogonical systems consider Four corners of the world as essentially corresponding to the four points of the compass.[56] • Four is the sacred number of the Zia, an indigenous tribe located in the U.S. state of New Mexico.[57] • The Chinese, the Koreans, and the Japanese are superstitious about the number four because it is a homonym for "death" in their languages.[58] • In Slavic mythology, the god Svetovid has four heads.[59] In politics • Four Freedoms: four fundamental freedoms that Franklin D. Roosevelt declared ought to be enjoyed by everyone in the world: Freedom of Speech, Freedom of Religion, Freedom from Want, Freedom from Fear.[60] • Gang of Four: Popular name for four Chinese Communist Party leaders who rose to prominence during China's Cultural Revolution, but were ousted in 1976 following the death of Chairman Mao Zedong. Among the four was Mao's widow, Jiang Qing. Since then, many other political factions headed by four people have been called "Gangs of Four".[61] In computing • Four bits (half a byte) are sometimes called a nibble.[62] In science • A tetramer is an oligomer formed out of four sub-units.[63] In astronomy • Four terrestrial (or rocky) planets in the Solar System: Mercury, Venus, Earth, and Mars.[64] • Four giant gas/ice planets in the Solar System: Jupiter, Saturn, Uranus, and Neptune.[65] • Four of Jupiter's moons (the Galilean moons) are readily visible from Earth with a hobby telescope.[66] • Messier object M4, a magnitude 7.5 globular cluster in the constellation Scorpius.[67] • The Roman numeral IV stands for subgiant in the Yerkes spectral classification scheme.[68] In biology • Four is the number of nucleobase types in DNA and RNA – adenine, guanine, cytosine, thymine (uracil in RNA).[69] • Many chordates have four feet, legs or leglike appendages (tetrapods). • The mammalian heart consists of four chambers.[70] • Many mammals (Carnivora, Ungulata) use four fingers for movement. • All insects with wings except flies and some others have four wings.[71] • Insects of the superorder Endopterygota, also known as Holometabola, such as butterflies, ants, bees, beetles, fleas, flies, moths, and wasps, undergo holometabolism—complete metamorphism in four stages—from (1) embryo (ovum, egg), to (2) larva (such as grub, caterpillar), then (3) pupa (such as the chrysalis), and finally (4) the imago.[72] • In the common ABO blood group system, there are four blood types (A, B, O, AB).[73] • Humans have four canines and four wisdom teeth.[74] • The cow's stomach is divided in four digestive compartments: reticulum, rumen, omasum and abomasum.[75] In chemistry • Valency of carbon (that is basis of life on the Earth) is four. Also because of its tetrahedral crystal bond structure, diamond (one of the natural allotropes of carbon) is the hardest known naturally occurring material. It is also the valence of silicon, whose compounds form the majority of the mass of the Earth's crust.[76] • The atomic number of beryllium[77] • There are four basic states of matter: solid, liquid, gas, and plasma.[78] In physics • Special relativity and general relativity treat nature as four-dimensional: 3D regular space and one-dimensional time are treated together and called spacetime.[79] Also, any event E has a light cone composed of four zones of possible communication and cause and effect (outside the light cone is strictly incommunicado). • There are four fundamental forces (electromagnetism, gravitation, the weak nuclear force, and the strong nuclear force).[80] • In statistical mechanics, the four functions inequality is an inequality for four functions on a finite distributive lattice.[81] In logic and philosophy • The symbolic meanings of the number four are linked to those of the cross and the square. "Almost from prehistoric times, the number four was employed to signify what was solid, what could be touched and felt. Its relationship to the cross (four points) made it an outstanding symbol of wholeness and universality, a symbol which drew all to itself". Where lines of latitude and longitude intersect, they divide the earth into four proportions. Throughout the world kings and chieftains have been called "lord of the four suns" or "lord of the four quarters of the earth",[82] which is understood to refer to the extent of their powers both territorially and in terms of total control of their subjects' doings. • The Square of Opposition, in both its Aristotelian version and its Boolean version, consists of four forms: A ("All S is R"), I ("Some S is R"), E ("No S is R"), and O ("Some S is not R"). • In regard to whether two given propositions can have the same truth value, there are four separate logical possibilities: the propositions are subalterns (possibly both are true, and possibly both are false); subcontraries (both may be true, but not that both are false); contraries (both may be false, but not that both are true); or contradictories (it is not possible that both are true, and it is not possible that both are false). • Aristotle held that there are basically four causes in nature: the material, the formal, the efficient, and the final.[83] • The Stoics held with four basic categories, all viewed as bodies (substantial and insubstantial): (1) substance in the sense of substrate, primary formless matter; (2) quality, matter's organization to differentiate and individualize something, and coming down to a physical ingredient such as pneuma, breath; (3) somehow holding (or disposed), as in a posture, state, shape, size, action, and (4) somehow holding (or disposed) toward something, as in relative location, familial relation, and so forth. • Immanuel Kant expounded a table of judgments involving four three-way alternatives, in regard to (1) Quantity, (2) Quality, (3) Relation, (4) Modality, and, based thereupon, a table of four categories, named by the terms just listed, and each with three subcategories. • Arthur Schopenhauer's doctoral thesis was On the Fourfold Root of the Principle of Sufficient Reason. • Franz Brentano held that any major philosophical period has four phases: (1) Creative and rapidly progressing with scientific interest and results; then declining through the remaining phases, (2) practical, (3) increasingly skeptical, and (4) literary, mystical, and scientifically worthless—until philosophy is renewed through a new period's first phase. (See Brentano's essay "The Four Phases of Philosophy and Its Current State" 1895, tr. by Mezei and Smith 1998.) • C. S. Peirce, usually a trichotomist, discussed four methods for overcoming troublesome uncertainties and achieving secure beliefs: (1) the method of tenacity (policy of sticking to initial belief), (2) the method of authority, (3) the method of congruity (following a fashionable paradigm), and (4) the fallibilistic, self-correcting method of science (see "The Fixation of Belief", 1877); and four barriers to inquiry, barriers refused by the fallibilist: (1) assertion of absolute certainty; (2) maintaining that something is unknowable; (3) maintaining that something is inexplicable because absolutely basic or ultimate; (4) holding that perfect exactitude is possible, especially such as to quite preclude unusual and anomalous phenomena (see "F.R.L." [First Rule of Logic], 1899). • Paul Weiss built a system involving four modes of being: Actualities (substances in the sense of substantial, spatiotemporally finite beings), Ideality or Possibility (pure normative form), Existence (the dynamic field), and God (unity). (See Weiss's Modes of Being, 1958). • Karl Popper outlined a tetradic schema to describe the growth of theories and, via generalization, also the emergence of new behaviors and living organisms: (1) problem, (2) tentative theory, (3) (attempted) error-elimination (especially by way of critical discussion), and (4) new problem(s). (See Popper's Objective Knowledge, 1972, revised 1979.) • John Boyd (military strategist) made his key concept the decision cycle or OODA loop, consisting of four stages: (1) observation (data intake through the senses), (2) orientation (analysis and synthesis of data), (3) decision, and (4) action.[84] Boyd held that his decision cycle has philosophical generality, though for strategists the point remains that, through swift decisions, one can disrupt an opponent's decision cycle. • Richard McKeon outlined four classes (each with four subclasses) of modes of philosophical inquiry: (1) Modes of Being (Being); (2) Modes of Thought (That which is); (3) Modes of Fact (Existence); (4) Modes of Simplicity (Experience)—and, corresponding to them, four classes (each with four subclasses) of philosophical semantics: Principles, Methods, Interpretations, and Selections. (See McKeon's "Philosophic Semantics and Philosophic Inquiry" in Freedom and History and Other Essays, 1989.) • Jonathan Lowe (E.J. Lowe) argues in The Four-Category Ontology, 2006, for four categories: kinds (substantial universals), attributes (relational universals and property-universals), objects (substantial particulars), and modes (relational particulars and property-particulars, also known as "tropes"). (See Lowe's "Recent Advances in Metaphysics," 2001, Eprint) • Four opposed camps of the morality and nature of evil: moral absolutism, amoralism, moral relativism, and moral universalism. In technology • The resin identification code used in recycling to identify low-density polyethylene.[85] • Most furniture has four legs – tables, chairs, etc. • The four color process (CMYK) is used for printing.[86] • Wide use of rectangles (with four angles and four sides) because they have effective form and capability for close adjacency to each other (houses, rooms, tables, bricks, sheets of paper, screens, film frames). • In the Rich Text Format specification, language code 4 is for the Chinese language. Codes for regional variants of Chinese are congruent to 4 mod 256. • Credit card machines have four-twelve function keys. • On most phones, the 4 key is associated with the letters G, H, and I,[87] but on the BlackBerry Pearl, it is the key for D and F. • On many computer keyboards, the "4" key may also be used to type the dollar sign ($) if the shift key is held down. • It is the number of bits in a nibble, equivalent to half a byte[88] • In internet slang, "4" can replace the word "for" (as "four" and "for" are pronounced similarly). For example, typing "4u" instead of "for you". • In Leetspeak, "4" may be used to replace the letter "A". • The TCP/IP stack consists of four layers.[89] In transport • Many internal combustion engines are called four-stroke engines because they complete one thermodynamic cycle in four distinct steps: Intake, compression, power, and exhaust. • Most vehicles, including motor vehicles, and particularly cars/automobiles and light commercial vehicles have four road wheels. • "Quattro", meaning four in the Italian language, is used by Audi as a trademark to indicate that all-wheel drive (AWD) technologies are used on Audi-branded cars.[90] The word "Quattro" was initially used by Audi in 1980 in its original 4WD coupé, the Audi Quattro. Audi also has a privately held subsidiary company called quattro GmbH. • List of highways numbered 4 In sports • In the Australian Football League, the top level of Australian rules football, each team is allowed 4 "interchanges" (substitute players), who can be freely substituted at any time, subject to a limit on the total number of substitutions. • In baseball: • There are four bases in the game: first base, second base, third base, and home plate; to score a run, an offensive player must complete, in the sequence shown, a circuit of those four bases. • When a batter receives four pitches that the umpire declares to be "balls" in a single at-bat, a base on balls, informally known as a "walk", is awarded, with the batter sent to first base. • For scoring, number 4 is assigned to the second baseman. • Four is the most runs that can be scored on any single at bat, whereby all three baserunners and the batter score (the most common being via a grand slam). • The fourth batter in the batting lineup is called the cleanup hitter. • In basketball, the number four is used to designate the power forward position, often referred to as "the four spot" or "the four".[91] • In cricket, a four is a specific type of scoring event, whereby the ball crosses the boundary after touching the ground at least one time, scoring four runs. Taking four wickets in four consecutive balls is typically referred to as a double hat trick (two consecutive, overlapping hat tricks). • In American Football teams get four downs to reach the line of gain. • In rowing, a four refers to a boat for four rowers, with or without coxswain. In rowing nomenclature, 4− represents a coxless four and 4+ represents a coxed four. • In rugby league: • A try is worth 4 points. • One of the two starting centres wears the jersey number 4. (An exception to this rule is the Super League, which uses static squad numbering.) • In rugby union: • One of the two starting locks wears the jersey number 4. • In the standard bonus points system, a point is awarded in the league standings to a team that scores at least 4 tries in a match, regardless of the match result. In other fields • The phrase "four-letter word" is used to describe many swear words in the English language.[92] • Four is the only number whose name in English has the same number of letters as its value. • Four (四, formal writing: 肆, pinyin sì) is considered an unlucky number in Chinese, Korean, Vietnamese and Japanese cultures mostly in Eastern Asia because it sounds like the word "death" (死, pinyin sǐ). To avoid complaints from people with tetraphobia, many numbered product lines skip the "four": e.g. Nokia cell phones (there was no series beginning with a 4 until the Nokia 4.2), Palm PDAs, etc. Some buildings skip floor 4 or replace the number with the letter "F", particularly in heavily Asian areas. See tetraphobia and Numbers in Chinese culture. • In Pythagorean numerology (a pseudocience) the number 4 represents security and stability. • The number of characters in a canonical four-character idiom. • In the NATO phonetic alphabet, the digit 4 is called "fower".[93] • In astrology, Cancer is the 4th astrological sign of the Zodiac.[94] • In Tarot, The Emperor is the fourth trump or Major Arcana card.[95] • In Tetris, a game named for the Greek word for 4, every shape in the game is formed of 4 blocks each.[96] • 4 represents the number of Justices on the Supreme Court of the United States necessary to grant a writ of certiorari (i.e., agree to hear a case; it is one less than the number necessary to render a majority decision) at the court's current size.[97] • Number Four is a character in the book series Lorien Legacies.[98] • In the performing arts, the fourth wall is an imaginary barrier which separates the audience from the performers, and is "broken" when performers communicate directly to the audience.[99] In music • In written music, common time is constructed of four beats per measure and a quarter note receives one beat.[100] • In popular or modern music, the most common time signature is also founded on four beats, i.e., 4/4 having four quarter note beats. • The common major scale is built on two sets of four notes (e.g., CDEF, GABC), where the first and last notes create an octave interval (a pair-of-four relationship). • The interval of a perfect fourth is a foundational element of many genres of music, represented in music theory as the tonic and subdominant relationship. Four is also embodied within the circle of fifths (also known as circle of fourths), which reveals the interval of four in more active harmonic contexts. • The typical number of movements in a symphony.[101] • The number of completed, numbered symphonies by Johannes Brahms.[102] • The number of strings on a violin, a viola, a cello, double bass, a cuatro, a typical bass guitar, and a ukulele, and the number of string pairs on a mandolin. • "Four calling birds" is the gift on the fourth day of Christmas in the carol "The Twelve Days of Christmas".[103] Groups of four • Big Four (disambiguation) • Four basic operations of arithmetic: addition, subtraction, multiplication, division.[104] • Greek classical elements (fire, air, water, earth).[105] • Four seasons: spring, summer, autumn, winter. • The Four Seasons (disambiguation) • A leap year generally occurs every four years.[106] • Approximately four weeks (4 times 7 days) to a lunar month (synodic month = 29.53 days). Thus the number four is universally an integral part of primitive sacred calendars. • Four weeks of Advent (and four Advent candles on the Advent wreath). • Four cardinal directions: north, south, east, west.[107] • Four Temperaments: sanguine, choleric, melancholic, phlegmatic. • Four Humors: blood, yellow bile, black bile, phlegm.[108] • Four Great Ancient Capitals of China. • Four-corner method. • Four Asian Tigers, referring to the economies of Hong Kong, Taiwan, South Korea, and Singapore • Cardinal principles. • Four cardinal virtues: justice, prudence, temperance, fortitude. • Four suits of playing cards: hearts, diamonds, clubs, spades.[109] • Four nations of the United Kingdom: England, Wales, Scotland, Northern Ireland. • Four provinces of Ireland: Munster, Ulster, Leinster, Connacht. • Four estates: politics, administration, judiciary, journalism. Especially in the expression "Fourth Estate", which means journalism. • Four Corners is the only location in the United States where four states come together at a single point: Colorado, Utah, New Mexico, and Arizona. • Four Evangelists – Matthew, Mark, Luke, and John • Four Doctors of Western Church – Saint Gregory the Great, Saint Ambrose, Saint Augustine, and Saint Jerome • Four Doctors of Eastern Church – Saint John Chrysostom, Saint Basil the Great, and Gregory of Nazianzus and Saint Athanasius • Four Galilean moons of Jupiter – Io, Europa, Ganymede, and Callisto • The Gang of Four was a Chinese communist political faction. • The Fantastic Four: Mr. Fantastic, The Invisible Woman, The Human Torch, The Thing. • The Teenage Mutant Ninja Turtles: Leonardo, Michelangelo, Donatello, Raphael • The Beatles were also known as the "Fab Four": John Lennon, Paul McCartney, George Harrison, Ringo Starr. • Gang of Four is a British post-punk rock band formed in the late 1970s. • Four rivers in the Garden of Eden (Genesis 2:10–14): Pishon (perhaps the Jaxartes or Syr Darya), Gihon (perhaps the Oxus or Amu Darya), Hiddekel (Tigris), and P'rat (Euphrates). • There are also four years in a single Olympiad (duration between the Olympic Games). Many major international sports competitions follow this cycle, among them the FIFA World Cup and its women's version, the FIBA World Championships for men and women, and the Rugby World Cup. • There are four limbs on the human body. • Four Houses of Hogwarts in the Harry Potter series: Gryffindor, Hufflepuff, Ravenclaw, Slytherin.[110] • Four known continents of the world in the A Song of Ice and Fire series: Westeros, Essos, Sothoryos, Ulthos. • Each Grand Prix in Nintendo's Mario Kart series is divided into four cups and each cup is divided into four courses. The Mushroom Cup, Flower Cup, Star Cup, and Special Cup make up the Nitro Grand Prix, while the Shell Cup, Banana Cup, Leaf Cup, and the Lightning Cup make up the Retro Grand Prix. See also • List of highways numbered 4 References 1. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.64 2. "Seven Segment Displays (7-Segment) \| Pinout, Types and Applications". Electronics Hub. 22 April 2019. Archived from the original on 28 July 2020. Retrieved 28 July 2020. 3. "Battle of the MICR Fonts: Which Is Better, E13B or CMC7? - Digital Check". Digital Check. 2 February 2017. Archived from the original on 3 August 2020. Retrieved 28 July 2020. 4. Fiore, Gregory (1 August 1993). Basic mathematics for college students: concepts and applications. HarperCollins College. p. 162. ISBN 978-0-06-042046-8. The smallest composite number is 4. 5. Hodges, Andrew (17 May 2008). One to Nine: The Inner Life of Numbers. W. W. Norton & Company. p. 249. ISBN 978-0-393-06863-4. 2 ↑↑ ... ↑↑ 2 is always 4 6. Kaplan Test Prep (3 January 2017). SAT Subject Test Mathematics Level 1. Simon and Schuster. p. 289. ISBN 978-1-5062-0922-7. An integer is divisible by 4 if the last two digits form a multiple of 4. 7. Spencer, Joel (1996), Chudnovsky, David V.; Chudnovsky, Gregory V.; Nathanson, Melvyn B. (eds.), "Four Squares with Few Squares", Number Theory: New York Seminar 1991–1995, New York, NY: Springer US, pp. 295–297, doi:10.1007/978-1-4612-2418-1_22, ISBN 978-1-4612-2418-1 8. Peterson, Ivars (2002). Mathematical Treks: From Surreal Numbers to Magic Circles. MAA. p. 95. ISBN 978-0-88385-537-9. 7 is an example of an integer that can't be written as the sum of three squares. 9. Bajnok, Béla (13 May 2013). An Invitation to Abstract Mathematics. Springer Science & Business Media. ISBN 978-1-4614-6636-9. There is no algebraic formula for the roots of the general polynomial of degrees 5 or higher. 10. Bunch, Bryan (2000). The Kingdom of Infinite Number. New York: W. H. Freeman & Company. p. 48. 11. Ben-Menahem, Ari (6 March 2009). Historical Encyclopedia of Natural and Mathematical Sciences. Springer Science & Business Media. p. 2147. ISBN 978-3-540-68831-0. (i.e. That there are maps for which three colors are not sufficient) 12. Molitierno, Jason J. (19 April 2016). Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs. CRC Press. p. 197. ISBN 978-1-4398-6339-8. ... The complete graph on the largest number of vertices that is planar is K4 and that a(K4) equals 4. 13. Weisstein, Eric W. "Tetrahedron". mathworld.wolfram.com. Archived from the original on 20 August 2020. Retrieved 28 July 2020. 14. Grossnickle, Foster Earl; Reckzeh, John (1968). Discovering Meanings in Elementary School Mathematics. Holt, Rinehart and Winston. p. 337. ISBN 9780030676451. ...the smallest possible number of faces that a polyhedron may have is four 15. Grossnickle, Foster Earl; Reckzeh, John (1968). Discovering Meanings in Elementary School Mathematics. Holt, Rinehart and Winston. p. 337. ISBN 9780030676451. ...face of the platonic solid. The simplest of these shapes is the tetrahedron... 16. Hilbert, David; Cohn-Vossen, Stephan (1999). Geometry and the Imagination. American Mathematical Soc. p. 143. ISBN 978-0-8218-1998-2. ...the tetrahedron plays an anomalous role in that it is self-dual, whereas the four remaining polyhedra are mutually dual in pairs... 17. Horne, Jeremy (19 May 2017). Philosophical Perceptions on Logic and Order. IGI Global. p. 299. ISBN 978-1-5225-2444-1. Archived from the original on 31 October 2022. Retrieved 31 October 2022. The Klein four-group is the smallest noncyclic group,... 18. Michiel Hazewinkel, ed. (2002). Hopf fibration. ISBN 1402006098. OCLC 1013220521. Archived from the original on 1 May 2023. Retrieved 30 April 2023. {{cite book}}: \|website= ignored (help) 19. Chwalkowski, Farrin (14 December 2016). Symbols in Arts, Religion and Culture: The Soul of Nature. Cambridge Scholars Publishing. p. 22. ISBN 978-1-4438-5728-4. The four main pilgrimages sites are: Lumbini, Bodh Gaya, Sarnath and Kusinara....four Noble Truths of Buddhism 20. Van Voorst, Robert (1 January 2012). RELG: World. Cengage Learning. p. 108. ISBN 978-1-111-72620-1. He first observed the suffering of the world in the Four Passing Sites 21. Yun, Hsing; Xingyun (2010). The Great Realizations: A Commentary on the Eight Realizations of a Bodhisattva Sutra. Buddha's Light Publishing. p. 27. ISBN 978-1-932293-44-9. The four great elements, earth, water, fire and wind... 22. Chaudhuri, Saroj Kumar (2003). Hindu Gods and Goddesses in Japan. Vedams eBooks (P) Ltd. p. 20. ISBN 978-81-7936-009-5. The Buddhists adopted him as one of the four Devarajas or Heavenly Kings 23. Bronkhorst, Johannes (22 December 2009). Buddhist Teaching in India. Simon and Schuster. p. 66. ISBN 978-0-86171-566-4. The four right exertions are... 24. Mistry, Freny (2 May 2011). Nietzsche and Buddhism: Prolegomenon to a Comparative Study. Walter de Gruyter. p. 69. ISBN 978-3-11-083724-7. these four bases of psychic power 25. Arbel, Keren (16 March 2017). Early Buddhist Meditation: The Four Jhanas as the Actualization of Insight. Taylor & Francis. p. 1. ISBN 978-1-317-38399-4. This book is about the four jhanas 26. Jayatilleke, K. N. (16 October 2013). Early Buddhist Theory of Knowledge. Routledge. ISBN 978-1-134-54294-9. ...the states of the four arupajhanas. 27. van Gorkom, Nina. The Perfections Leading to Enlightenment. Рипол Классик. p. 171. ISBN 978-5-88139-786-9. There are four of them: loving-kindness, metta, compassion, karuna, sympathetic joy, mudita and equanimity, upekkha. 28. Rinpoche, Khenchen Konchog Gyaltshen; Milarepa; Sumgon, Jigten (8 October 2013). Opening the Treasure of the Profound: Teachings on the Songs of Jigten Sumgon and Milarepa. Shambhala Publications. ISBN 978-0-8348-2896-4. ...four types of shravaka (stream enterer, oncereturner, nonreturner, and arhat) 29. Fahlbusch, Erwin; Bromiley, Geoffrey William; Lochman, Jan Milic; Mbiti, John; Pelikan, Jaroslav (14 February 2008). The Encyclodedia of Christianity, Vol. 5. Wm. B. Eerdmans Publishing. p. 823. ISBN 978-0-8028-2417-2. 30. Stevenson, Kenneth; Glerup, Michael (19 March 2014). Ezekiel, Daniel. InterVarsity Press. pp. xlv. ISBN 978-0-8308-9738-4. We have already mentioned the four living creatures—the man, the lion, the ox and the eagle 31. Butnick, Stephanie; Leibovitz, Liel; Oppenheimer, Mark (1 October 2019). The Newish Jewish Encyclopedia: From Abraham to Zabar's and Everything in Between. Artisan Books. ISBN 978-1-57965-893-9. ...be like Sarah, Rachel, Rebecca, and Leah, the foremothers of Judaism 32. Kaplan, Aryeh (1990). Innerspace: Introduction to Kabbalah, Meditation and Prophecy. Moznaim. p. 109. ISBN 9780940118560. ...as well as to the palm ( lulav ), myrtle ( hadas ), willow ( aravah ) and citron ( etrog ), the four species of plants 33. Dennis, Geoffrey W. (2007). The Encyclopedia of Jewish Myth, Magic and Mysticism. Llewellyn Worldwide. p. 188. ISBN 978-0-7387-0905-5. The Passover Seder is particularly structured around fours: the Four Questions, the Four Sons, and four cups of wine. 34. "Four Expressions of Redemption to be said on the Jewish holiday of Passover - Google Search". p. 46. Archived from the original on 25 October 2022. Retrieved 29 July 2020. There are four expressions of redemption in the Torah 35. Templeton, Charles (1973). Jesus: the four Gospels, Matthew, Mark, Luke, and John, combined in one narrative and rendered in modern English. Simon and Schuster. ISBN 9780671217150. 36. Wagner, Richard; Helyer, Larry R. (31 January 2011). The Book of Revelation For Dummies. John Wiley & Sons. p. 308. ISBN 978-1-118-05086-6. The four horsemen of the Apocalypse are one of the most familiar images of Revelation 37. Turfe, Tallal Alie (19 July 2013). Children of Abraham: United We Prevail, Divided We Fail. iUniverse. p. 91. ISBN 978-1-4759-9047-8. The four holy cities of Judaism are Jerusalem, Hebron, Safed, and Tiberius. 38. Frawley, David (7 October 2014). Vedic Yoga: The Path of the Rishi. Lotus Press. ISBN 978-0-940676-25-1. There are four Vedas 39. Fritz, Stephen Martin (14 May 2019). Our Human Herds: The Theory of Dual Morality (Second Edition, Unabridged). Dog Ear Publishing. p. 491. ISBN 978-1-4575-6755-1. that these four proper aims and objects 40. Maanas - Individual and Society. Rapid Publications. ISBN 978-1-937192-06-8. The Four Stages of Life 41. Chwalkowski, Farrin (14 December 2016). Symbols in Arts, Religion and Culture: The Soul of Nature. Cambridge Scholars Publishing. p. 23. ISBN 978-1-4438-5728-4. The four primary castes or strata of society:... 42. Kulendiren, Pon (11 October 2012). Hinduism a Scientific Religion: & Some Temples in Sri Lanka. iUniverse. p. 32. ISBN 978-1-4759-3675-9. 43. Jansen, Eva Rudy (1993). The Book of Hindu Imagery: Gods, Manifestations and Their Meaning. Binkey Kok Publications. p. 87. ISBN 978-90-74597-07-4. Brahma has four faces,... 44. "Definition of yuga". Dictionary.com. Archived from the original on 28 July 2020. Retrieved 28 July 2020. 45. Çakmak, Cenap (18 May 2017). Islam: A Worldwide Encyclopedia [4 volumes]. ABC-CLIO. p. 397. ISBN 978-1-61069-217-5. ...Eid al-Adha (Feast of Sacrifice) lasts four days ... 46. Leonard, Timothy; Willis, Peter (11 June 2008). Pedagogies of the Imagination: Mythopoetic Curriculum in Educational Practice. Springer Science & Business Media. p. 144. ISBN 978-1-4020-8350-1. ... four Rightly Guided Caliphs, Abu-Bakr, Umar ibn al-Khattab, Uthman ibn Affan and Ali ibn Abi Talib,... 47. Chwalkowski, Farrin (14 December 2016). Symbols in Arts, Religion and Culture: The Soul of Nature. Cambridge Scholars Publishing. p. 23. ISBN 978-1-4438-5728-4. According to Islam, the Four Arch Angels are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael). 48. Busool, Assad Nimer (28 December 2010). The Wise Qur'an: These are the Verses of the Wise Book: These are the verses of the Wise Book. Xlibris Corporation. p. 50. ISBN 978-1-4535-2526-5. The sacred months are four, Rajab, Dhu al-Qi'dah, Dhu al-Hijjah, and al-Muharram. During those four sacred months there were no war... 49. Shabazz, Hassan (6 January 2020). Al Islaam, and the Transformation of Society. Lulu.com. p. 15. ISBN 978-1-7948-3337-1. There are four books in Islam: Torah, Zaboor, Injeel and Holy Qur'an... 50. Bukhari, Muohammad Ben Ismail Al (1 January 2007). THE CORRECT TRADITIONS OF AL'BUKHARI 1-4 VOL 3: صحيح البخاري 1/4 [عربي/انكليزي] ج3. Dar Al Kotob Al Ilmiyah دار الكتب العلمية. p. 840. For those who take an oath for abstention from their wives, awaiting for four months is ordained; 51. Ahmad, Yusuf Al-Hajj. The Book Of Nikkah: Encyclopaedia of Islamic Law. Darussalam Publishers. ...for four months and ten days. 52. Mawdudi, Sayyid Abul A'la (15 December 2016). Towards Understanding the Qur'an: English Only Edition. Kube Publishing Ltd. p. 59. ISBN 978-0-86037-613-2. Then take four birds, ... 53. Maudoodi, Syed Abul ʻAla (2000). Sūrah al-Aʻarāf to Sūrah bani Isrāel. Islamic Publications. p. 177. The respite of four months... 54. Barazangi, Nimat Hafez (9 March 2016). Woman's Identity and Rethinking the Hadith. Routledge. p. 138. ISBN 978-1-134-77065-6. And those who launch a charge against chaste women and do not produce four witnesses... 55. SK, Lim. Origins of Chinese Auspicious Symbols. Asiapac Books Pte Ltd. p. 16. ISBN 978-981-317-026-1. Taoism later incorporated the four symbols into its immortality system... 56. Terry, Milton Spenser (1883). Biblical Hermenutics: A Treatise on the Interpretation of the Old and New Testaments. Phillips & Hunt. p. 382. the four corners or extremities of the earth (Isa. xi, 12; Ezek. vii, 2.; Rev. vii, 1 ; xx, 8), corresponding, doubtless, with the four points of the compass 57. Bulletin - State Department of Education. Department of Education. 1955. p. 151. Four was a sacred number of Zia 58. Lachenmeyer, Nathaniel (2005). 13: The Story of the World's Most Notorious Superstition. Penguin Group (USA) Incorporated. p. 187. ISBN 978-0-452-28496-8. In Chinese , Japanese , and Korean , the word for four is , unfortunately , an exact homonym for death 59. Maberry, Jonathan; Kramer, David F. (2007). The Cryptopedia: A Dictionary of the Weird, Strange & Downright Bizarre. Citadel Press. p. 211. ISBN 978-0-8065-2819-9. Svetovid is portrayed as having four heads ... 60. "FDR, 'The Four Freedoms,' Speech Text". Voices of Democracy. Archived from the original on 1 August 2020. Retrieved 30 July 2020. 61. "Yao Wenyuan". The Economist. ISSN 0013-0613. Archived from the original on 1 May 2018. Retrieved 30 July 2020. 62. Raphael, Howard A., ed. (November 1974). "The Functions Of A Computer: Instruction Register And Decoder" (PDF). MCS-40 User's Manual For Logic Designers. Santa Clara, California, USA: Intel Corporation. p. viii. Archived (PDF) from the original on 3 March 2020. Retrieved 3 March 2020. [...] The characteristic eight bit field is sometimes referred to as a byte, a four bit field can be referred to as a nibble. [...] 63. Petsko, Gregory A.; Ringe, Dagmar (2004). Protein Structure and Function. New Science Press. p. 40. ISBN 978-0-87893-663-2. Oligomers containing two, three, four, five, six or even more subunits are known as dimers, trimers, tetramers, pentamers, hexamers, and so on. 64. Yaqoob, Tahir (2011). Exoplanets and Alien Solar Systems. New Earth Labs. p. 12. ISBN 978-0-9741689-2-0. The four inner planets (known as terrestrial, or rocky planets 65. Encrenaz, Therese; Bibring, Jean-Pierre; Blanc, M.; Barucci, Maria-Antonietta; Roques, Francoise; Zarka, Philippe (26 January 2004). The Solar System. Springer Science & Business Media. p. 283. ISBN 978-3-540-00241-3. Archived from the original on 12 March 2022. Retrieved 4 November 2020. ...the gas giants (Jupiter and Saturn), and the icy giants (Uranus and Neptune) 66. Pidwirny, Michael (7 May 2020). Chapter 3: Matter, Energy and the Universe: Single chapter from the eBook Understanding Physical Geography. Our Planet Earth Publishing. p. 10. including the four large Galilean moons that are easily visible from a hobby telescope 67. Pugh, Philip (2 November 2011). Observing the Messier Objects with a Small Telescope: In the Footsteps of a Great Observer. Springer Science & Business Media. p. 41. ISBN 978-0-387-85357-4. M4 is a globular star cluster near Antares in Scorpius. 68. Bok, Bart Jan; Bok, Priscilla Fairfield (1981). The Milky Way. Harvard University Press. p. 66. ISBN 978-0-674-57503-5. IV , subgiants 69. Encyclopedia of Cell Biology. Academic Press. 7 August 2015. p. 25. ISBN 978-0-12-394796-3. 70. Chien, Shu; Chen, Peter C. Y.; Fung, Yuan-cheng (2008). An Introductory Text to Bioengineering. World Scientific. p. 54. ISBN 978-981-270-793-2. The mammalian heart consists of four chambers,... 71. Creation Research Society Textbook Committee (1970). Biology: a search for order in complexity. Zondervan Pub. House. p. 209. ISBN 978-0-310-29490-0. Except for the flies, mosquitoes, and some others, insects with wings have four wings. 72. Pittenger, Dennis (15 December 2014). California Master Gardener Handbook, 2nd Edition. UCANR Publications. p. 180. ISBN 978-1-60107-857-5. metamorphosis is marked by four distinct stages 73. Darpan, Pratiyogita (2008). Pratiyogita Darpan. Pratiyogita Darpan. p. 85. In the 'ABO' system, all blood belongs one of four major groups — A, B, AB or O 74. Daniels, Patricia; Stein, Lisa (2009). Body: The Complete Human : how it Grows, how it Works, and how to Keep it Healthy and Strong. National Geographic Books. p. 94. ISBN 978-1-4262-0449-4. Four canines for tearing + Eight premolars for crushing +Twelve molars (including four wisdom teeth) 75. Woodward, Thompson Elwyn; Nystrom, Amer Benjamin (1930). Feeding Dairy Cows. U.S. Department of Agriculture. p. 4. The cow's stomach is divided into four compartments. 76. Lucas, Jerry (1993). Great unsolved mysteries of science. F & W Pubns Inc. p. 168. ISBN 978-1-55870-291-2. Of course, carbon is not the only chemical element with a valence of +4 or -4 77. Walsh, Kenneth A. (1 January 2009). Beryllium Chemistry and Processing. ASM International. p. 93. ISBN 978-0-87170-721-5. Beryllium has an atomic number of four 78. Ebeling, Werner; Fortov, Vladimir E.; Filinov, Vladimir (27 November 2017). Quantum Statistics of Dense Gases and Nonideal Plasmas. Springer. p. 39. ISBN 978-3-319-66637-2. Plasma is one of the four fundamental states of matter, the others being solid, liquid, and gas. 79. Petkov, Vesselin (23 June 2009). Relativity and the Nature of Spacetime. Springer Science & Business Media. p. 124. ISBN 978-3-642-01962-3. should be regarded as a four-dimensional world 80. Giordano, Nicholas (13 February 2009). College Physics: Reasoning and Relationships. Cengage Learning. p. 1073. ISBN 978-0-534-42471-8. We have referred to the four fundamental forces in nature,... 81. Alon, Noga; Spencer, Joel H. (20 September 2011). The Probabilistic Method. John Wiley & Sons. p. 6.1. ISBN 978-1-118-21044-4. The Four Functions Theorem of Ahlswede Daykin 82. Chevalier, Jean and Gheerbrant, Alain (1994), The Dictionary of Symbols. The quote beginning "Almost from prehistoric times..." is on p. 402. 83. Hennig, Boris (5 December 2018). Aristotle's Four Causes. Peter Lang. ISBN 978-1-4331-5929-9. This book examines Aristotle's four causes (material, formal, efficient, and final) 84. Wilkinson, Amy (17 February 2015). The Creator's Code: The Six Essential Skills of Extraordinary Entrepreneurs. Simon and Schuster. p. 79. ISBN 978-1-4516-6609-0. The OODA loop consists of four steps. 85. Howard, Brian Clark; Abdelrahman, Amina Lake; Good Housekeeping Institute (26 February 2020). "You Might Be Recycling Wrong — Here's Everything You Need to Know About Recycling Symbols". Good Housekeeping. Archived from the original on 13 March 2015. Retrieved 28 July 2020. Plastic Recycling Symbol #4: LDPE 86. Conover, Charles (8 November 2011). Designing for Print. John Wiley & Sons. p. 62. ISBN 978-1-118-13088-9. CMYK is the standard four-color model used for all full-color print jobs that will be output on an offset printing press 87. Vermaat, Misty E.; Sebok, Susan L.; Freund, Steven M.; Campbell, Jennifer T.; Frydenberg, Mark (1 January 2015). Discovering Computers, Essentials. Cengage Learning. p. 123. ISBN 978-1-305-53402-5. ...the 4 key (labeled with the letters g,h and i)... 88. Bunting, Steve; Wei, William (6 March 2006). EnCase Computer Forensics: The Official EnCE: EnCase?Certified Examiner Study Guide. John Wiley & Sons. p. 246. ISBN 978-0-7821-4435-2. A byte also contains two 4-bit nibbles... 89. Braden, R. (1989). Braden, R (ed.). "Requirements for Internet Hosts - Communication Layers". tools.ietf.org: 9–10. doi:10.17487/RFC1122. Archived from the original on 28 July 2020. Retrieved 28 July 2020. 90. Assenza, Tony (June 1982). "Audi Quattro: Germany's 4x4 Cruise Missile". Popular Mechanics. Hearst Magazines. 91. Schaller, Bob; Harnish, Dave (18 September 2009). The Everything Kids' Basketball Book: The all-time greats, legendary teams, today's superstars - and tips on playing like a pro. Simon and Schuster. ISBN 978-1-4405-0177-7. Power forward Referred to as the number 4 spot 92. "Definition of FOUR-LETTER WORD". merriam-webster.com. Archived from the original on 22 August 2016. Retrieved 28 July 2020. 93. Wells, J. C. (25 September 2014). Sounds Interesting: Observations on English and General Phonetics. Cambridge University Press. p. 33. ISBN 978-1-316-12385-0. But one confused re-spelling is fower for 'four'. 94. Guttman, Ariel; Guttman, Gail; Johnson, Kenneth (1993). Mythic Astrology: Archetypal Powers in the Horoscope. Llewellyn Worldwide. p. 263. ISBN 978-0-87542-248-0. Sign: Cancer, the fourth Zodiacal Sign 95. Curtiss, Harriette A. (1996). The Key to the Universe. Health Research Books. p. 161. ISBN 978-0-7873-1233-6. The 4th Tarot Card is called "The Emperor." 96. Weller, David; Lobao, Alexandre Santos; Hatton, Ellen (20 September 2004). Beginning .NET Game Programming in VB .NET. Apress. p. 383. ISBN 978-1-4302-0724-5. ...tetraminos (the shapes used in Tetris) are all just a collection of four blocks 97. Bardes, Barbara; Shelley, Mack; Schmidt, Steffen (16 December 2008). American Government and Politics Today: The Essentials 2009 - 2010 Edition. Cengage Learning. p. 453. ISBN 978-0-495-57170-4. The court will not issue a writ unless at least four justices approve of it. This is called the rule of four. 98. "Movie Projector: 'I Am Number Four' to be No. 1 at holiday weekend box office [Updated]". LA Times Blogs - Company Town. 17 February 2011. Archived from the original on 20 August 2020. Retrieved 28 July 2020. 99. "fourth wall". dictionary.cambridge.org. Retrieved 29 November 2021. 100. Roberts, Gareth E. (15 February 2016). From Music to Mathematics: Exploring the Connections. JHU Press. p. 3. ISBN 978-1-4214-1918-3. ... called common time and denoted by C, which has four beats per measure 101. Bonds, Mark Evan (10 January 2009). Music as Thought: Listening to the Symphony in the Age of Beethoven. Princeton University Press. p. 1. ISBN 978-1-4008-2739-8. The number, character and sequence of movements in the symphony, moreover, did not stabilize until the 1770s when the familiar format of four movements... 102. Frisch, Walter (2003). Brahms: The Four Symphonies. Yale University Press. ISBN 978-0-300-09965-2. 103. Brech, Lewis (2010). Storybook Advent Carols Collection Songbook. Couples Company, Inc. p. 26. ISBN 978-1-4524-7763-3. 104. Wright, Robert J.; Ellemor-Collins, David; Tabor, Pamela D. (4 November 2011). Developing Number Knowledge: Assessment,Teaching and Intervention with 7-11 year olds. SAGE. ISBN 978-1-4462-5368-7. 105. Macauley, David (29 September 2010). Elemental Philosophy: Earth, Air, Fire, and Water as Environmental Ideas. SUNY Press. ISBN 978-1-4384-3246-5. 106. Brooks, Edward (1876). Normal Higher Arithmetic Designed for Advanced Classes in Common Schools, Normal Schools, and High Schools, Academics, Etc. Sower. p. 227. Every year that is divisible by four, except the Centennial years, and every Centennial year divisible by 400, is a leap year... 107. Touche, Fred; Price, Anne (2005). Wilderness Navigation Handbook. Touche Publishing. p. 48. ISBN 978-0-9732527-0-5. Each of the familiar cardinal directions is equivalent to a particular true bearing: north (0°), east (90°), south (180°), and west (270°) 108. Roeckelein, J. E. (19 January 2006). Elsevier's Dictionary of Psychological Theories. Elsevier. p. 235. ISBN 978-0-08-046064-2. ...four substances or humors: blood, yellow bile, black bile and phlegm 109. Medley, H. Anthony (1997). Bridge. Penguin. p. 6. ISBN 978-0-02-861735-0. The four playing card suits, as you probably already know, are spades, hearts, diamonds, and clubs 110. Baker, Felicity (2017). Houses of Hogwarts: Cinematic Guide. Scholastic Incorporated. ISBN 978-1-338-12861-1. ...the four houses of Hogwarts School of Witchcraft and Wizardry: Gryffindor, Ravenclaw, Hufflepuff, and Slytherin • Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 55–58 External links Look up four in Wiktionary, the free dictionary. Wikimedia Commons has media related to 4 (number). • Marijn.Org on Why is everything four? • A few thoughts on the number four, by Penelope Merritt at samuel-beckett.net • The Number 4 • The Positive Integer 4 • Prime curiosities: 4 Integers 0s • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 100s • 100 • 101 • 102 • 103 • 104 • 105 • 106 • 107 • 108 • 109 • 110 • 111 • 112 • 113 • 114 • 115 • 116 • 117 • 118 • 119 • 120 • 121 • 122 • 123 • 124 • 125 • 126 • 127 • 128 • 129 • 130 • 131 • 132 • 133 • 134 • 135 • 136 • 137 • 138 • 139 • 140 • 141 • 142 • 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5 5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits. ← 4 5 6 → −1 0 1 2 3 4 5 6 7 8 9 → • List of numbers • Integers ← 0 10 20 30 40 50 60 70 80 90 → Cardinalfive Ordinal5th (fifth) Numeral systemquinary Factorizationprime Prime3rd Divisors1,5 Greek numeralΕ´ Roman numeralV, v Greek prefixpenta-/pent- Latin prefixquinque-/quinqu-/quint- Binary1012 Ternary123 Senary56 Octal58 Duodecimal512 Hexadecimal516 Greekε (or Ε) Arabic, Kurdish٥ Persian, Sindhi, Urdu۵ Ge'ez፭ Bengali৫ Kannada೫ Punjabi੫ Chinese numeral五 Devanāgarī५ Hebrewה Khmer៥ Telugu౫ Malayalam൫ Tamil௫ Thai๕ Evolution of the Arabic digit The evolution of the modern Western digit for the numeral 5 cannot be traced back to the Indian system, as for the digits 1 to 4. The Kushana and Gupta empires in what is now India had among themselves several forms that bear no resemblance to the modern digit. The Nagari and Punjabi took these digits and all came up with forms that were similar to a lowercase "h" rotated 180°. The Ghubar Arabs transformed the digit in several ways, producing from that were more similar to the digits 4 or 3 than to 5.[1] It was from those digits that Europeans finally came up with the modern 5. While the shape of the character for the digit 5 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in . On the seven-segment display of a calculator and digital clock, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa. It is one of three numbers, along with 4 and 6, where the number of segments matches the number. Mathematics Five is the third smallest prime number, and the second super-prime.[2] It is the first safe prime,[3] the first good prime,[4] the first balanced prime,[5] and the first of three known Wilson primes.[6] Five is the second Fermat prime,[2] the second Proth prime,[7] and the third Mersenne prime exponent,[8] as well as the third Catalan number[9] and the third Sophie Germain prime.[2] Notably, 5 is equal to the sum of the only consecutive primes 2 + 3 and it is the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7).[10][11] It also forms the first pair of sexy primes with 11,[12] which is the fifth prime number and Heegner number,[13] as well as the first repunit prime in decimal; a base in-which five is also the first non-trivial 1-automorphic number.[14] Five is the third factorial prime,[15] and an alternating factorial.[16] It is also an Eisenstein prime (like 11) with no imaginary part and real part of the form $3p-1$.[2] In particular, five is the first congruent number, since it is the length of the hypotenuse of the smallest integer-sided right triangle.[17] Number theory 5 is the fifth Fibonacci number, being 2 plus 3.[2] It is the only Fibonacci number that is equal to its position aside from 1, which is both the first and second Fibonacci numbers. Five is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13, 194), (5, 29, 433), ... (OEIS: A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5). Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers.[18] 5 is the second Fermat prime of the form $2^{2^{n}}+1$, and more generally the second Sierpiński number of the first kind, $n^{n}+1$.[19] There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537.[20] The sum of the first three Fermat primes, 3, 5 and 17, yields 25 or 52, while 257 is the 55th prime number. Combinations from these five Fermat primes generate thirty-one polygons with an odd number of sides that can be constructed purely with a compass and straight-edge, which includes the five-sided regular pentagon.[21][22]: pp.137–142 Apropos, thirty-one is also equal to the sum of the maximum number of areas inside a circle that are formed from the sides and diagonals of the first five $n$-sided polygons, which is equal to the maximum number of areas formed by a six-sided polygon; per Moser's circle problem.[23][22]: pp.76-78 The first prime centered pentagonal number is 31,[24] which is also the fifth centered triangular number.[25] 5 is also the third Mersenne prime exponent of the form $2^{n}-1$, which yields $31$, the eleventh prime number and fifth super-prime.[26][2] This is the prime index of the third Mersenne prime and second double Mersenne prime 127,[27] as well as the third double Mersenne prime exponent for the number 2,147,483,647,[27] which is the largest value that a signed 32-bit integer field can hold. There are only four known double Mersenne prime numbers, with a fifth candidate double Mersenne prime $M_{M_{61}}$ = 223058...93951 − 1 too large to compute with current computers. In a related sequence, the first five terms in the sequence of Catalan–Mersenne numbers $M_{c_{n}}$ are the only known prime terms, with a sixth possible candidate in the order of 101037.7094. These prime sequences are conjectured to be prime up to a certain limit. There are a total of five known unitary perfect numbers, which are numbers that are the sums of their positive proper unitary divisors.[28][29] The smallest such number is 6, and the largest of these is equivalent to the sum of 4095 divisors, where 4095 is the largest of five Ramanujan–Nagell numbers that are both triangular numbers and Mersenne numbers of the general form.[30][31] The sums of the first five non-primes greater than zero 1 + 4 + 6 + 8 + 9 and the first five prime numbers 2 + 3 + 5 + 7 + 11 both equal 28; the seventh triangular number and like 6 a perfect number, which also includes 496, the thirty-first triangular number and perfect number of the form $2^{p-1}$($2^{p}-1$) with a $p$ of $5$, by the Euclid–Euler theorem.[32][33][34] Within the larger family of Ore numbers, 140 and 496, respectively the fourth and sixth indexed members, both contain a set of divisors that produce integer harmonic means equal to 5.[35][36] The fifth Mersenne prime, 8191,[26] splits into 4095 and 4096, with the latter being the fifth superperfect number[37] and the sixth power of four, 46. Figurate numbers and magic figures In figurate numbers, 5 is a pentagonal number, with the sequence of pentagonal numbers starting: 1, 5, 12, 22, 35, ...[38] • 5 is a centered tetrahedral number: 1, 5, 15, 35, 69, ...[39] Every centered tetrahedral number with an index of 2, 3 or 4 modulo 5 is divisible by 5. • 5 is a square pyramidal number: 1, 5, 14, 30, 55, ...[40] The first four members add to 50 while the fifth indexed member in the sequence is 55. • 5 is a centered square number: 1, 5, 13, 25, 41, ...[41] The fifth square number or 52 is 25, which features in the proportions of the two smallest (3, 4, 5) and (5, 12, 13) primitive Pythagorean triples.[42] The factorial of five $5!=120$ is multiply perfect like 28 and 496.[43] It is the sum of the first fifteen non-zero positive integers and 15th triangular number, which in-turn is the sum of the first five non-zero positive integers and 5th triangular number. Furthermore, $120+5=125=5^{3}$, where 125 is the second number to have an aliquot sum of 31 (after the fifth power of two, 32).[44] On its own, 31 is the first prime centered pentagonal number,[45] and the fifth centered triangular number.[46] Collectively, five and thirty-one generate a sum of 36 (the square of 6) and a difference of 26, which is the only number to lie between a square $a^{2}$ and a cube $b^{3}$ (respectively, 25 and 27).[47] The fifth pentagonal and tetrahedral number is 35, which is equal to the sum of the first five triangular numbers: 1, 3, 6, 10, 15.[48] In the sequence of pentatope numbers that start from the first (or fifth) cell of the fifth row of Pascal's triangle (left to right or from right to left), the first few terms are: 1, 5, 15, 35, 70, 126, 210, 330, 495, ...[49] The first five members in this sequence add to 126, which is the fifth non-trivial pentagonal pyramidal number[50] as well as the fifth ${\mathcal {S}}$-perfect Granville number.[51] This is the third Granville number not to be perfect, and the only known such number with three distinct prime factors.[52] 5 is the value of the central cell of the first non-trivial normal magic square, called the Luoshu square. Its $3$ x $3$ array has a magic constant $M$ of $15$, where the sums of its rows, columns, and diagonals are all equal to fifteen.[53] 5 is also the value of the central cell the only non-trivial normal magic hexagon made of nineteen cells.[54] Collatz conjecture In the Collatz 3x + 1 problem, 5 requires five steps to reach one by multiplying terms by three and adding one if the term is odd (starting with five itself), and dividing by two if they are even: {5 ➙ 16 ➙ 8 ➙ 4 ➙ 2 ➙ 1}; the only other number to require five steps is 32 (since 16 must be part of such path).[55][56] When generalizing the Collatz conjecture to all positive or negative integers, −5 becomes one of only four known possible cycle starting points and endpoints, and in its case in five steps too: {−5 ➙ −14 ➙ −7 ➙ −20 ➙ −10 ➙ −5 ➙ ...}. The other possible cycles begin and end at −17 in eighteen steps, −1 in two steps, and 1 in three steps. This behavior is analogous to the path cycle of five in the 3x − 1 problem, where 5 takes five steps to return cyclically, in this instance by multiplying terms by three and subtracting 1 if the terms are odd, and also halving if even.[57] It is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...).[58] Generalizations Five is conjectured to be the only odd untouchable number, and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.[59] Meanwhile: • Every odd number greater than $1$ is the sum of at most five prime numbers,[60] and • Every odd number greater than $5$ is conjectured to be expressible as the sum of three prime numbers.[61] Helfgott has provided a proof of this, also known as the odd Goldbach conjecture, that is already widely acknowledged by mathematicians as it still undergoes peer-review. Polynomial equations of degree 4 and below can be solved with radicals, while quintic equations of degree 5 and higher cannot generally be so solved (see, Abel–Ruffini theorem). This is related to the fact that the symmetric group $\mathrm {S} _{n}$ is a solvable group for $n$ ⩽ $4$, and not for $n$ ⩾ $5$. There are five countably infinite Ramsey classes of permutations, where the age of each countable homogeneous permutation forms an individual Ramsey class $K$ of objects such that, for each natural number $r$ and each choice of objects $A,B\in K$, there is no object $C\in K$ where in any $r$-coloring of all subobjects of $C$ isomorphic to $A$ there is a monochromatic subobject isomorphic to $B$.[62]: pp.1, 2 Aside from $\{1\}$, the five classes of Ramsey permutations are the class of identity permutations, the class of reversals, the class of increasing sequences of decreasing sequences, the class of decreasing sequences of increasing sequences, and the class of all permutations.[62]: p.4 In general, the Fraïssé limit of a class $K$ of finite relational structure is the age of a countable homogeneous relational structure $U$ if and only if five conditions hold for $K$: it is closed under isomorphism, it has only countably many isomorphism classes, it is hereditary, it is joint-embedded, and it holds the amalgamation property.[62]: p.3 Inside the classification of number systems, the real numbers $\mathbb {R} $ and its three subsequent Cayley-Dickson constructions of algebras over the field of the real numbers (i.e. the complex numbers $\mathbb {C} $, the quaternions $\mathbb {H} $, and the octonions $\mathbb {O} $) are normed division algebras that hold up to five different principal algebraic properties of interest: whether the algebras are ordered, and whether they hold commutative, associative, alternative, and power-associative properties.[63] Whereas the real numbers contain all five properties, the octonions are only alternative and power-associative. On the other hand, the sedenions $\mathbb {S} $, which represent a fifth algebra in this series, is not a composition algebra unlike $\mathbb {H} $ and $\mathbb {O} $, is only power-associative, and is the first algebra to contain non-trivial zero divisors as with all further algebras over larger fields.[64] Altogether, these five algebras operate, respectively, over fields of dimension 1, 2, 4, 8, and 16. Geometry A pentagram, or five-pointed polygram, is the first proper star polygon constructed from the diagonals of a regular pentagon as self-intersecting edges that are proportioned in golden ratio, $\varphi $. Its internal geometry appears prominently in Penrose tilings, and is a facet inside Kepler-Poinsot star polyhedra and Schläfli–Hess star polychora, represented by its Schläfli symbol {5/2}. A similar figure to the pentagram is a five-pointed simple isotoxal star ☆ without self-intersecting edges. It is often found as a facet inside Islamic Girih tiles, of which there are five different rudimentary types.[65] Generally, star polytopes that are regular only exist in dimensions $2$ ⩽ $n$ < $5$, and can be constructed using five Miller rules for stellating polyhedra or higher-dimensional polytopes.[66] Graphs theory, and planar geometry In graph theory, all graphs with four or fewer vertices are planar, however, there is a graph with five vertices that is not: K5, the complete graph with five vertices, where every pair of distinct vertices in a pentagon is joined by unique edges belonging to a pentagram. By Kuratowski's theorem, a finite graph is planar iff it does not contain a subgraph that is a subdivision of K5, or the complete bipartite utility graph K3,3.[67] A similar graph is the Petersen graph, which is strongly connected and also nonplanar. It is most easily described as graph of a pentagram embedded inside a pentagon, with a total of 5 crossings, a girth of 5, and a Thue number of 5.[68][69] The Petersen graph, which is also a distance-regular graph, is one of only 5 known connected vertex-transitive graphs with no Hamiltonian cycles.[70] The automorphism group of the Petersen graph is the symmetric group $\mathrm {S} _{5}$ of order 120 = 5!. The chromatic number of the plane is at least five, depending on the choice of set-theoretical axioms: the minimum number of colors required to color the plane such that no pair of points at a distance of 1 has the same color.[71][72] Whereas the hexagonal Golomb graph and the regular hexagonal tiling generate chromatic numbers of 4 and 7, respectively, a chromatic coloring of 5 can be attained under a more complicated graph where multiple four-coloring Moser spindles are linked so that no monochromatic triples exist in any coloring of the overall graph, as that would generate an equilateral arrangement that tends toward a purely hexagonal structure. The plane also contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations: hexagonal, oblique, rectangular, centered rectangular, and square lattices. Uniform tilings of the plane, furthermore, are generated from combinations of only five regular polygons: the triangle, square, hexagon, octagon, and the dodecagon.[73] The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.[74] Polyhedra There are five Platonic solids in three-dimensional space: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.[75] The dodecahedron in particular contains pentagonal faces, while the icosahedron, its dual polyhedron, has a vertex figure that is a regular pentagon. There are also five: • Regular polyhedron compounds: the stella octangula, compound of five tetrahedra, compound of five cubes, compound of five octahedra, and compound of ten tetrahedra.[76] Icosahedral symmetry $\mathrm {I} _{h}$ is isomorphic to the alternating group on five letters $\mathrm {A} _{5}$ of order 120, realized by actions on these uniform polyhedron compounds. • Space-filling convex polyhedra with regular faces: the triangular prism, hexagonal prism, cube, truncated octahedron, and gyrobifastigium.[77] The cube is the only Platonic solid that can tessellate space on its own, and the truncated octahedron and gyrobifastigium are the only Archimedean and Johnson solids, respectively, that can tessellate space with their own copies. • Cell-transitive parallelohedra: any parallelepiped, as well as the rhombic dodecahedron, the elongated dodecahedron, the hexagonal prism and the truncated octahedron.[78] The cube is a special case of a parallelepiped, and the rhombic dodecahedron (with five stellations per Miller's rules) is the only Catalan solid to tessellate space on its own.[79] • Regular abstract polyhedra, which include the excavated dodecahedron and the dodecadodecahedron.[80] They have combinatorial symmetries transitive on flags of their elements, with topologies equivalent to that of toroids and the ability to tile the hyperbolic plane. There are also five semiregular prisms that are facets inside non-prismatic uniform four-dimensional figures: the triangular, pentagonal, hexagonal, octagonal, and decagonal prisms. Five uniform prisms and antiprisms contain pentagons or pentagrams: the pentagonal prism and antiprism, and the pentagrammic prism, antiprism, and crossed-antirprism.[81] Fourth dimension The pentatope, or 5-cell, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry $\mathrm {A} _{4}$ of order 120 = 5! and $\mathrm {S} _{5}$ group structure. Made of five tetrahedra, its Petrie polygon is a regular pentagon and its orthographic projection is equivalent to the complete graph K5. It is one of six regular 4-polytopes, made of thirty-one elements: five vertices, ten edges, ten faces, five tetrahedral cells and one 4-face.[82] • A regular 120-cell, the dual polychoron to the regular 600-cell, can fit one hundred and twenty 5-cells. Also, five 24-cells fit inside a small stellated 120-cell, the first stellation of the 120-cell. • A subset of the vertices of the small stellated 120-cell are matched by the great duoantiprism star, which is the only uniform nonconvex duoantiprismatic solution in the fourth dimension, constructed from the polytope cartesian product $\{5\}\otimes \{5/3\}$ and made of fifty tetrahedra, ten pentagrammic crossed antiprisms, ten pentagonal antiprisms, and fifty vertices.[83] • The grand antiprism, which is the only known non-Wythoffian construction of a uniform polychoron, is made of twenty pentagonal antiprisms and three hundred tetrahedra, with a total of one hundred vertices and five hundred edges.[84] • The abstract four-dimensional 57-cell is made of fifty-seven hemi-icosahedral cells, in-which five surround each edge.[85] The 11-cell, another abstract 4-polytope with eleven vertices and fifty-five edges, is made of eleven hemi-dodecahedral cells each with fifteen edges.[86] The skeleton of the hemi-dodecahedron is the Petersen graph. Overall, the fourth dimension contains five fundamental Weyl groups that form a finite number of uniform polychora: $\mathrm {A} _{4}$, $\mathrm {B} _{4}$, $\mathrm {D} _{4}$, $\mathrm {F} _{4}$, and $\mathrm {H} _{4}$, accompanied by a fifth or sixth general group of unique 4-prisms of Platonic and Archimedean solids. All of these uniform 4-polytopes are generated from twenty-five uniform polyhedra, which include the five Platonic solids, fifteen Archimedean solids counting two enantiomorphic forms, and five prisms. There are also a total of five Coxeter groups that generate non-prismatic Euclidean honeycombs in 4-space, alongside five compact hyperbolic Coxeter groups that generate five regular compact hyperbolic honeycombs with finite facets, as with the order-5 5-cell honeycomb and the order-5 120-cell honeycomb, both of which have five cells around each face. Compact hyperbolic honeycombs only exist through the fourth dimension, or rank 5, with paracompact hyperbolic solutions existing through rank 10. Likewise, analogues of four-dimensional $\mathrm {H} _{4}$ hexadecachoric or $\mathrm {F} _{4}$ icositetrachoric symmetry do not exist in dimensions $n$ ⩾ $5$; however, there are prismatic groups in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have $\mathrm {H} _{4}$ and $\mathrm {F} _{4}$ symmetry. There are also five regular projective 4-polytopes in the fourth dimension, all of which are hemi-polytopes of the regular 4-polytopes, with the exception of the 5-cell.[87] Only two regular projective polytopes exist in each higher dimensional space. In particular, Bring's surface is the curve in the projective plane $\mathbb {P} ^{4}$ that is represented by the homogeneous equations:[88] $v+w+x+y+z=v^{2}+w^{2}+x^{2}+y^{2}+z^{2}=v^{3}+w^{3}+x^{3}+y^{3}+z^{3}=0.$ It holds the largest possible automorphism group of a genus four complex curve, with group structure $\mathrm {S} _{5}$. This is the Riemann surface associated with the small stellated dodecahedron, whose fundamental polygon is a regular hyperbolic icosagon, with an area of $12\pi $ (by the Gauss-Bonnet theorem). Including reflections, its full group of symmetries is $\mathrm {S} _{5}\times \mathbb {Z} _{2}$, of order 240; which is also the number of (2,4,5) hyperbolic triangles that tessellate its fundamental polygon. Bring quintic $x^{5}+ax+b=0$ holds roots $x_{i}$ that satisfy Bring's curve. Fifth dimension The 5-simplex or hexateron is the five-dimensional analogue of the 5-cell, or 4-simplex. It has Coxeter group $\mathrm {A} _{5}$ as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group $\mathrm {S} _{6}$, the only finite symmetric group which has an outer automorphism. The 5-cube, made of ten tesseracts and the 5-cell as its vertex figure, is also regular and one of thirty-one uniform 5-polytopes under the Coxeter $\mathrm {B} _{5}$ hypercubic group. The demipenteract, with one hundred and twenty cells, is the only fifth-dimensional semiregular polytope, and has the rectified 5-cell as its vertex figure, which is one of only three semiregular 4-polytopes alongside the rectified 600-cell and the snub 24-cell. In the fifth dimension, there are five regular paracompact honeycombs, all with infinite facets and vertex figures; no other regular paracompact honeycombs exist in higher dimensions.[89] There are also exclusively twelve complex aperiotopes in $\mathbb {C} ^{n}$ complex spaces of dimensions $n$ ⩾ $5$; alongside complex polytopes in $\mathbb {C} ^{5}$ and higher under simplex, hypercubic and orthoplex groups (with van Oss polytopes).[90] A Veronese surface in the projective plane $\mathbb {P} ^{5}$ generalizes a linear condition $\nu :\mathbb {P} ^{2}\to \mathbb {P} ^{5}$ :\mathbb {P} ^{2}\to \mathbb {P} ^{5}} for a point to be contained inside a conic, which requires five points in the same way that two points are needed to determine a line.[91] Finite simple groups There are five exceptional Lie algebras: ${\mathfrak {g}}_{2}$, ${\mathfrak {f}}_{4}$, ${\mathfrak {e}}_{6}$, ${\mathfrak {e}}_{7}$, and ${\mathfrak {e}}_{8}$. The smallest of these, ${\mathfrak {g}}_{2}$, can be represented in five-dimensional complex space and projected as a ball rolling on top of another ball, whose motion is described in two-dimensional space.[92] ${\mathfrak {e}}_{8}$ is the largest of all five exceptional groups, with the other four as subgroups, and an associated lattice that is constructed with one hundred and twenty quaternionic unit icosians that make up the vertices of the 600-cell, whose Euclidean norms define a quadratic form on a lattice structure isomorphic to the optimal configuration of spheres in eight dimensions.[93] This sphere packing $\mathrm {E} _{8}$ lattice structure in 8-space is held by the vertex arrangement of the 521 honeycomb, one of five Euclidean honeycombs that admit Gosset's original definition of a semiregular honeycomb, which includes the three-dimensional alternated cubic honeycomb.[94][95] While there are specifically five solvable groups that are excluded from finite simple groups of Lie type, the smallest duplicate found inside finite simple Lie groups is $\mathrm {A_{5}} \cong A_{1}(4)\cong A_{1}(5)$, where $\mathrm {A_{n}} $ represents alternating groups and $A_{n}(q)$ classical Chevalley groups. The smallest alternating group that is simple is the alternating group on five letters. The five Mathieu groups constitute the first generation in the happy family of sporadic groups. These are also the first five sporadic groups to have been described, defined as $\mathrm {M} _{n}$ multiply transitive permutation groups on $n$ objects, with $n$ ∈ {11, 12, 22, 23, 24}.[96]: p.54 In particular, $\mathrm {M} _{11}$, the smallest of all sporadic groups, has a rank 3 action on fifty-five points from an induced action on unordered pairs, as well as two five-dimensional faithful complex irreducible representations over the field with three elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with $n$ elements.[97] Of precisely five different conjugacy classes of maximal subgroups of $\mathrm {M} _{11}$, one is the almost simple symmetric group $\mathrm {S} _{5}$ (of order 5!), and another is $\mathrm {M} _{10}$, also almost simple, that functions as a point stabilizer which contains five as its largest prime factor in its group order: 24·32·5 = 2·3·4·5·6 = 8·9·10 = 720. On the other hand, whereas $\mathrm {M} _{11}$ is sharply 4-transitive, $\mathrm {M} _{12}$ is sharply 5-transitive and $\mathrm {M} _{24}$ is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups.[98] $\mathrm {M} _{22}$ has the first five prime numbers as its distinct prime factors in its order of 27·32·5·7·11, and is the smallest of five sporadic groups with five distinct prime factors in their order.[96]: p.17 All Mathieu groups are subgroups of $\mathrm {M} _{24}$, which under the Witt design $\mathrm {W} _{24}$ of Steiner system $\operatorname {S(5,8,24)} $ emerges a construction of the extended binary Golay code $\mathrm {B} _{24}$ that has $\mathrm {M} _{24}$ as its automorphism group.[96]: pp.39, 47, 55 $\mathrm {W} _{24}$ generates octads from code words of Hamming weight 8 from the extended binary Golay code, one of five different Hamming weights the extended binary Golay code uses: 0, 8, 12, 16, and 24.[96]: p.38 The Witt design and the extended binary Golay code in turn can be used to generate a faithful construction of the 24-dimensional Leech lattice Λ24, which is the subject of the second generation of seven sporadic groups that are subquotients of the automorphism of the Leech lattice, Conway group $\mathrm {Co} _{0}$.[96]: pp.99, 125 There are five non-supersingular prime numbers — 37, 43, 53, 61, and 67 — less than 71, which is the largest of fifteen supersingular primes that divide the order of the friendly giant, itself the largest sporadic group.[99] In particular, a centralizer of an element of order 5 inside this group arises from the product between Harada–Norton sporadic group $\mathrm {HN} $ and a group of order 5.[100][101] On its own, $\mathrm {HN} $ can be represented using standard generators $(a,b,ab)$ that further dictate a condition where $o([a,b])=5$.[102][103] This condition is also held by other generators that belong to the Tits group $\mathrm {T} $,[104] the only finite simple group that is a non-strict group of Lie type that can also classify as sporadic. Furthermore, over the field with five elements, $\mathrm {HN} $ holds a 133-dimensional representation where 5 acts on a commutative yet non-associative product as a 5-modular analogue of the Griess algebra $V_{2}$♮,[105] which holds the friendly giant as its automorphism group. Euler's identity Euler's identity, $e^{i\pi }$+ $1$ = $0$, contains five essential numbers used widely in mathematics: Archimedes' constant $\pi $, Euler's number $e$, the imaginary number $i$, unity $1$, and zero $0$.[106][107][108] List of basic calculations Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 5 × x 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5 ÷ x 5 2.5 1.6 1.25 1 0.83 0.714285 0.625 0.5 0.5 0.45 0.416 0.384615 0.3571428 0.3 x ÷ 5 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5x 5 25 125 625 3125 15625 78125 390625 1953125 9765625 48828125 244140625 1220703125 6103515625 30517578125 x5 1 32 243 1024 7776 16807 32768 59049 100000 161051 248832 371293 537824 759375 In decimal All multiples of 5 will end in either 5 or 0, and vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions because they are prime factors of 10, the base. In the powers of 5, every power ends with the number five, and from 53 onward, if the exponent is odd, then the hundreds digit is 1, and if it is even, the hundreds digit is 6. A number $n$ raised to the fifth power always ends in the same digit as $n$. Science • The atomic number of boron.[109] • The number of appendages on most starfish, which exhibit pentamerism.[110] • The most destructive known hurricanes rate as Category 5 on the Saffir–Simpson hurricane wind scale.[111] • The most destructive known tornadoes rate an F-5 on the Fujita scale or EF-5 on the Enhanced Fujita scale.[112] Astronomy • There are five Lagrangian points in a two-body system. • There are currently five dwarf planets in the Solar System: Ceres, Pluto, Haumea, Makemake, and Eris.[113] • The New General Catalogue object NGC 5, a magnitude 13 spiral galaxy in the constellation Andromeda.[114] • Messier object M5, a magnitude 7.0 globular cluster in the constellation Serpens.[115] Biology • There are usually considered to be five senses (in general terms). • The five basic tastes are sweet, salty, sour, bitter, and umami.[116] • Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.[117] Computing • 5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.[118] Religion and culture Hinduism • The god Shiva has five faces[119] and his mantra is also called panchakshari (five-worded) mantra. • The goddess Saraswati, goddess of knowledge and intellectual is associated with panchami or the number 5. • There are five elements in the universe according to Hindu cosmology: dharti, agni, jal, vayu evam akash (earth, fire, water, air and space respectively). • The most sacred tree in Hinduism has 5 leaves in every leaf stunt. • Most of the flowers have 5 petals in them. • The epic Mahabharata revolves around the battle between Duryodhana and his 99 other brothers and the 5 pandava princes—Dharma, Arjuna, Bhima, Nakula and Sahadeva. Christianity • There are traditionally five wounds of Jesus Christ in Christianity: the Scourging at the Pillar, the Crowning with Thorns, the wounds in Christ's hands, the wounds in Christ's feet, and the Side Wound of Christ.[120] Gnosticism • The number five was an important symbolic number in Manichaeism, with heavenly beings, concepts, and others often grouped in sets of five. • Five Seals in Sethianism • Five Trees in the Gospel of Thomas Islam • The Five Pillars of Islam[121] • Muslims pray to Allah five times a day[122] • According to Shia Muslims, the Panjetan or the Five Holy Purified Ones are the members of Muhammad's family: Muhammad, Ali, Fatimah, Hasan, and Husayn and are often symbolically represented by an image of the Khamsa.[123] Judaism • The Torah contains five books—Genesis, Exodus, Leviticus, Numbers, and Deuteronomy—which are collectively called the Five Books of Moses, the Pentateuch (Greek for "five containers", referring to the scroll cases in which the books were kept), or Humash (חומש, Hebrew for "fifth").[124] • The book of Psalms is arranged into five books, paralleling the Five Books of Moses.[125] • The Khamsa, an ancient symbol shaped like a hand with four fingers and one thumb, is used as a protective amulet by Jews; that same symbol is also very popular in Arabic culture, known to protect from envy and the evil eye.[126] Sikhism • The five sacred Sikh symbols prescribed by Guru Gobind Singh are commonly known as panj kakars or the "Five Ks" because they start with letter K representing kakka (ਕ) in the Punjabi language's Gurmukhi script. They are: kesh (unshorn hair), kangha (the comb), kara (the steel bracelet), kachhehra (the soldier's shorts), and kirpan (the sword) (in Gurmukhi: ਕੇਸ, ਕੰਘਾ, ਕੜਾ, ਕਛਹਰਾ, ਕਿਰਪਾਨ).[127] Also, there are five deadly evils: kam (lust), krodh (anger), moh (attachment), lobh (greed), and ankhar (ego). Daoism • 5 Elements[128] • 5 Emperors[129] Other religions and cultures • According to ancient Greek philosophers such as Aristotle, the universe is made up of five classical elements: water, earth, air, fire, and ether. This concept was later adopted by medieval alchemists and more recently by practitioners of Neo-Pagan religions such as Wicca. • The pentagram, or five-pointed star, bears religious significance in various faiths including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca. • In Cantonese, "five" sounds like the word "not" (character: 唔). When five appears in front of a lucky number, e.g. "58", the result is considered unlucky. • In East Asian tradition, there are five elements: (water, fire, earth, wood, and metal).[130] The Japanese names for the days of the week, Tuesday through Saturday, come from these elements via the identification of the elements with the five planets visible with the naked eye.[131] Also, the traditional Japanese calendar has a five-day weekly cycle that can be still observed in printed mixed calendars combining Western, Chinese-Buddhist, and Japanese names for each weekday. • In numerology, 5 or a series of 555, is often associated with change, evolution, love and abundance. • Members of The Nation of Gods and Earths, a primarily African American religious organization, call themselves the "Five-Percenters" because they believe that only 5% of mankind is truly enlightened.[132] Art, entertainment, and media Fictional entities • James the Red Engine, a fictional character numbered 5.[133] • Johnny 5 is the lead character in the film Short Circuit (1986)[134] • Number Five is a character in Lorien Legacies[135] • Numbuh 5, real name Abigail Lincoln, from Codename: Kids Next Door • Sankara Stones, five magical rocks in Indiana Jones and the Temple of Doom that are sought by the Thuggees for evil purposes[136] • The Mach Five Mahha-gō? (マッハ号), the racing car Speed Racer (Go Mifune in the Japanese version) drives in the anime series of the same name (known as "Mach Go! Go! Go!" in Japan) • In the works of J. R. R. Tolkien, five wizards (Saruman, Gandalf, Radagast, Alatar and Pallando) are sent to Middle-earth to aid against the threat of the Dark Lord Sauron[137] • In the A Song of Ice and Fire series, the War of the Five Kings is fought between different claimants to the Iron Throne of Westeros, as well as to the thrones of the individual regions of Westeros (Joffrey Baratheon, Stannis Baratheon, Renly Baratheon, Robb Stark and Balon Greyjoy)[138] • In The Wheel of Time series, the "Emond's Field Five" are a group of five of the series' main characters who all come from the village of Emond's Field (Rand al'Thor, Matrim Cauthon, Perrin Aybara, Egwene al'Vere and Nynaeve al'Meara) • Myst uses the number 5 as a unique base counting system. In The Myst Reader series, it is further explained that the number 5 is considered a holy number in the fictional D'ni society. • Number Five is also a character in The Umbrella Academy comic book and TV series adaptation[139] Films • Towards the end of the film Monty Python and the Holy Grail (1975), the character of King Arthur repeatedly confuses the number five with the number three. • Five Go Mad in Dorset (1982) was the first of the long-running series of The Comic Strip Presents... television comedy films[140] • The Fifth Element (1997), a science fiction film[141] • Fast Five (2011), the fifth installment of the Fast and Furious film series.[142] • V for Vendetta (2005), produced by Warner Bros., directed by James McTeigue, and adapted from Alan Moore's graphic novel V for Vendetta prominently features number 5 and Roman Numeral V; the story is based on the historical event in which a group of men attempted to destroy Parliament on November 5, 1605[143] Music • Modern musical notation uses a musical staff made of five horizontal lines.[144] • A scale with five notes per octave is called a pentatonic scale.[145] • A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.[146] • In harmonics, the fifth partial (or 4th overtone) of a fundamental has a frequency ratio of 5:1 to the frequency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure third. A major triad chord when played in just intonation (most often the case in a cappella vocal ensemble singing), will contain such a pure major third. • Using the Latin root, five musicians are called a quintet.[147] • Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter. Groups • Five (group), a UK Boy band[148] • The Five (composers), 19th-century Russian composers[149] • 5 Seconds of Summer, pop band that originated in Sydney, Australia • Five Americans, American rock band active 1965–1969[150] • Five Finger Death Punch, American heavy metal band from Las Vegas, Nevada. Active 2005–present • Five Man Electrical Band, Canadian rock group billed (and active) as the Five Man Electrical Band, 1969–1975[151] • Maroon 5, American pop rock band that originated in Los Angeles, California[152] • MC5, American punk rock band[153] • Pentatonix, a Grammy-winning a cappella group originated in Arlington, Texas[154] • The 5th Dimension, American pop vocal group, active 1977–present[155] • The Dave Clark Five, a.k.a. DC5, an English pop rock group comprising Dave Clark, Lenny Davidson, Rick Huxley, Denis Payton, and Mike Smith; active 1958–1970[156] • The Jackson 5, American pop rock group featuring various members of the Jackson family; they were billed (and active) as The Jackson 5, 1966–1975[157] • Hi-5, Australian pop kids group, where it has several international adaptations, and several members throughout the history of the band. It was also a TV show. • We Five: American folk rock group active 1965–1967 and 1968–1977 • Grandmaster Flash and the Furious Five: American rap group, 1970–80's[158] • Fifth Harmony, an American girl group.[159] • Ben Folds Five, an American alternative rock trio, 1993–2000, 2008 and 2011–2013[160] • R5 (band), an American pop and alternative rock group, 2009–2018[161] Other • The number of completed, numbered piano concertos of Ludwig van Beethoven, Sergei Prokofiev, and Camille Saint-Saëns Television Stations • Channel 5 (UK), a television channel that broadcasts in the United Kingdom[162] • 5 (TV channel) (formerly known as ABC 5 and TV5) (DWET-TV channel 5 In Metro Manila) a television network in the Philippines.[163] Series • Babylon 5, a science fiction television series[164] • The number 5 features in the television series Battlestar Galactica in regards to the Final Five cylons and the Temple of Five • Hi-5 (Australian TV series), a television series from Australia[165] • Hi-5 (UK TV series), a television show from the United Kingdom • Hi-5 Philippines a television show from the Philippines • Odyssey 5, a 2002 science fiction television series[166] • Tillbaka till Vintergatan, a Swedish children's television series featuring a character named "Femman" (meaning five), who can only utter the word 'five'. • The Five (talk show): Fox News Channel roundtable current events television show, premiered 2011, so-named for its panel of five commentators. • Yes! PreCure 5 is a 2007 anime series which follows the adventures of Nozomi and her friends. It is also followed by the 2008 sequel Yes! Pretty Cure 5 GoGo! • The Quintessential Quintuplets is a 2019 slice of life romance anime series which follows the everyday life of five identical quintuplets and their interactions with their tutor. It has two seasons, and a final movie is scheduled in summer 2022. • Hawaii Five-0, CBS American TV series.[167] Literature • The Famous Five is a series of children's books by British writer Enid Blyton • The Power of Five is a series of children's books by British writer and screenwriter Anthony Horowitz • The Fall of Five is a book written under the collective pseudonym Pittacus Lore in the series Lorien Legacies • The Book of Five Rings is a text on kenjutsu and the martial arts in general, written by the swordsman Miyamoto Musashi circa 1645 • Slaughterhouse-Five is a book by Kurt Vonnegut about World War II[168] Sports • The Olympic Games have five interlocked rings as their symbol, representing the number of inhabited continents represented by the Olympians (Europe, Asia, Africa, Australia and Oceania, and the Americas).[169] • In AFL Women's, the top level of women's Australian rules football, each team is allowed 5 "interchanges" (substitute players), who can be freely substituted at any time. • In baseball scorekeeping, the number 5 represents the third baseman's position. • In basketball: • The number 5 is used to represent the position of center. • Each team has five players on the court at a given time. Thus, the phrase "five on five" is commonly used to describe standard competitive basketball.[170] • The "5-second rule" refers to several related rules designed to promote continuous play. In all cases, violation of the rule results in a turnover. • Under the FIBA (used for all international play, and most non-US leagues) and NCAA women's rule sets, a team begins shooting bonus free throws once its opponent has committed five personal fouls in a quarter. • Under the FIBA rules, A player fouls out and must leave the game after committing five fouls • Five-a-side football is a variation of association football in which each team fields five players.[171] • In ice hockey: • A major penalty lasts five minutes.[172] • There are five different ways that a player can score a goal (teams at even strength, team on the power play, team playing shorthanded, penalty shot, and empty net).[173] • The area between the goaltender's legs is known as the five-hole.[174] • In most rugby league competitions, the starting left wing wears this number. An exception is the Super League, which uses static squad numbering. • In rugby union: • A try is worth 5 points.[175] • One of the two starting lock forwards wears number 5, and usually jumps at number 4 in the line-out. • In the French variation of the bonus points system, a bonus point in the league standings is awarded to a team that loses by 5 or fewer points. Technology • 5 is the most common number of gears for automobiles with manual transmission.[176] • In radio communication, the term "Five by five" is used to indicate perfect signal strength and clarity.[177] • On almost all devices with a numeric keypad such as telephones, computers, etc., the 5 key has a raised dot or raised bar to make dialing easier. Persons who are blind or have low vision find it useful to be able to feel the keys of a telephone. All other numbers can be found with their relative position around the 5 button (on computer keyboards, the 5 key of the numpad has the raised dot or bar, but the 5 key that shifts with % does not).[178] • On most telephones, the 5 key is associated with the letters J, K, and L,[179] but on some of the BlackBerry phones, it is the key for G and H. • The Pentium, coined by Intel Corporation, is a fifth-generation x86 architecture microprocessor.[180] • The resin identification code used in recycling to identify polypropylene.[181] Miscellaneous fields Five can refer to: • "Give me five" is a common phrase used preceding a high five. • An informal term for the British Security Service, MI5. • Five babies born at one time are quintuplets. The most famous set of quintuplets were the Dionne quintuplets born in the 1930s.[182] • In the United States legal system, the Fifth Amendment to the United States Constitution can be referred to in court as "pleading the fifth", absolving the defendant from self-incrimination.[183] • Pentameter is verse with five repeating feet per line; iambic pentameter was the most popular form in Shakespeare.[184] • Quintessence, meaning "fifth element", refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth)[185] • The designation of an Interstate Highway (Interstate 5) that runs from San Diego, California to Blaine, Washington.[186] In addition, all major north-south Interstate Highways in the United States end in 5.[187] • In the computer game Riven, 5 is considered a holy number, and is a recurring theme throughout the game, appearing in hundreds of places, from the number of islands in the game to the number of bolts on pieces of machinery. • The Garden of Cyrus (1658) by Sir Thomas Browne is a Pythagorean discourse based upon the number 5. • The holy number of Discordianism, as dictated by the Law of Fives.[188] • The number of Justices on the Supreme Court of the United States necessary to render a majority decision.[189] • The number of dots in a quincunx.[190] • The number of permanent members with veto power on the United Nations Security Council.[191] • The number of Korotkoff sounds when measuring blood pressure[192] • The drink Five Alive is named for its five ingredients. The drink punch derives its name after the Sanskrit पञ्च (pañc) for having five ingredients.[193] • The Keating Five were five United States Senators accused of corruption in 1989.[194] • The Inferior Five: Merryman, Awkwardman, The Blimp, White Feather, and Dumb Bunny. DC Comics parody superhero team.[195] • No. 5 is the name of the iconic fragrance created by Coco Chanel.[196] • The Committee of Five was delegated to draft the United States Declaration of Independence.[197] • The five-second rule is a commonly used rule of thumb for dropped food.[198] • 555 95472, usually referred to simply as 5, is a minor male character in the comic strip Peanuts.[199] See also • List of highways numbered 5 References 1. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.65 2. Weisstein, Eric W. "5". mathworld.wolfram.com. Retrieved 2020-07-30. 3. Sloane, N. J. A. (ed.). "Sequence A005385 (Safe primes p: (p-1)/2 is also prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14. 4. Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 5. Sloane, N. J. A. (ed.). "Sequence A006562 (Balanced primes (of order one): primes which are the average of the previous prime and the following prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14. 6. Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 7. Sloane, N. J. A. (ed.). "Sequence A080076 (Proth primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21. 8. Weisstein, Eric W. "Mersenne Prime". mathworld.wolfram.com. Retrieved 2020-07-30. 9. Weisstein, Eric W. "Catalan Number". mathworld.wolfram.com. Retrieved 2020-07-30. 10. Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14. 11. Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14. 12. Sloane, N. J. A. (ed.). "Sequence A023201 (Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-14. 13. Sloane, N. J. A. (ed.). "Sequence A003173 (Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-20. 14. Sloane, N. J. A. (ed.). "Sequence A003226 (Automorphic numbers: m^2 ends with m.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-26. 15. Sloane, N. J. A. (ed.). "Sequence A088054 (Factorial primes: primes which are within 1 of a factorial number.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14. 16. Weisstein, Eric W. "Twin Primes". mathworld.wolfram.com. Retrieved 2020-07-30. 17. Sloane, N. J. A. (ed.). "Sequence A003273 (Congruent numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 18. Weisstein, Eric W. "Perrin Sequence". mathworld.wolfram.com. Retrieved 2020-07-30. 19. Weisstein, Eric W. "Sierpiński Number of the First Kind". mathworld.wolfram.com. Retrieved 2020-07-30. 20. Sloane, N. J. A. (ed.). "Sequence A019434 (Fermat primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-21. 21. Sloane, N. J. A. (ed.). "Sequence A004729 (... the 31 regular polygons with an odd number of sides constructible with ruler and compass)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-26. 22. Conway, John H.; Guy, Richard K. (1996). The Book of Numbers. New York, NY: Copernicus (Springer). pp. ix, 1–310. doi:10.1007/978-1-4612-4072-3. ISBN 978-1-4612-8488-8. OCLC 32854557. S2CID 115239655. 23. Sloane, N. J. A. (ed.). "Sequence A000127 (Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-31. 24. Sloane, N. J. A. (ed.). "Sequence A005891 (Centered pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21. 25. Sloane, N. J. A. (ed.). "Sequence A005448 (Centered triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21. 26. Sloane, N. J. A. (ed.). "Sequence A000668 (Mersenne primes (primes of the form 2^n - 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-03. 27. Sloane, N. J. A. (ed.). "Sequence A103901 (Mersenne primes p such that M(p) equal to 2^p - 1 is also a (Mersenne) prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-03. 28. Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. pp. 84–86. ISBN 0-387-20860-7. 29. Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10. 30. Sloane, N. J. A. (ed.). "Sequence A076046 (Ramanujan-Nagell numbers: the triangular numbers...which are also of the form 2^b - 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10. 31. Sloane, N. J. A. (ed.). "Sequence A000225 (... (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-13. 32. Bourcereau (2015-08-19). "28". Prime Curios!. PrimePages. Retrieved 2022-10-13. The only known number which can be expressed as the sum of the first non-negative integers (1 + 2 + 3 + 4 + 5 + 6 + 7), the first primes (2 + 3 + 5 + 7 + 11) and the first non-primes (1 + 4 + 6 + 8 + 9). There is probably no other number with this property. 33. Sloane, N. J. A. (ed.). "Sequence A000396 (Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13. 34. Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13. 35. Sloane, N. J. A. (ed.). "Sequence A001599 (Harmonic or Ore numbers: numbers n such that the harmonic mean of the divisors of n is an integer.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-26. 36. Sloane, N. J. A. (ed.). "Sequence A001600 (Harmonic means of divisors of harmonic numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-26. 37. Sloane, N. J. A. (ed.). "Sequence A019279 (Superperfect numbers: numbers k such that sigma(sigma(k)) equals 2*k where sigma is the sum-of-divisors function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-26. 38. Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08. 39. Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08. 40. Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08. 41. Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers...Sum of two squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08. 42. Sloane, N. J. A. (ed.). "Sequence A103606 (Primitive Pythagorean triples in nondecreasing order of perimeter, with each triple in increasing order, and if perimeters coincide then increasing order of the even members.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-26. 43. Sloane, N. J. A. (ed.). "Sequence A007691 (Multiply-perfect numbers: n divides sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-28. 44. Sloane, N. J. A. (ed.). "Sequence A001065". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-11. 45. Sloane, N. J. A. (ed.). "Sequence A005891 (Centered pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21. 46. Sloane, N. J. A. (ed.). "Sequence A005448 (Centered triangular numbers)". 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Retrieved 2020-08-03. 166. Odyssey 5, retrieved 2020-08-03 167. Hawaii Five-0, retrieved 2020-08-03 168. Powers, Kevin (2019-03-06). "The Moral Clarity of 'Slaughterhouse-Five' at 50". The New York Times. ISSN 0362-4331. Retrieved 2020-08-03. 169. "Olympic Rings – Symbol of the Olympic Movement". International Olympic Committee. 2020-06-23. Retrieved 2020-08-02. 170. "Rules of the Game". FIBA.basketball. Retrieved 2020-08-02. 171. Macalister, Terry (2007-09-04). "Popularity of five-a-side kicks off profits". The Guardian. ISSN 0261-3077. Retrieved 2020-08-02. 172. Sharp, Anne Wallace (2010-11-08). Ice Hockey. Greenhaven Publishing LLC. p. 18. ISBN 978-1-4205-0589-4. Major penalties of five minutes 173. Blevins, David (2012). The Sports Hall of Fame Encyclopedia: Baseball, Basketball, Football, Hockey, Soccer. Rowman & Littlefield. p. 585. ISBN 978-0-8108-6130-5. scoring five goals in five different ways: an even-strength goal, a power-play goal, a shorthanded goal, a penalty shot goal... 174. Times, The New York (2004-11-05). The New York Times Guide to Essential Knowledge: A Desk Reference for the Curious Mind. Macmillan. p. 713. ISBN 978-0-312-31367-8. five-hole the space between a goaltender's legs 175. McNeely, Scott (2012-09-14). Ultimate Book of Sports: The Essential Collection of Rules, Stats, and Trivia for Over 250 Sports. Chronicle Books. p. 189. ISBN 978-1-4521-2187-1. a "try," worth 5 points; 176. Poulton, Mark L. (1997). Fuel Efficient Car Technology. Computational Mechanics Publications. p. 65. ISBN 978-1-85312-447-1. The 5 – speed manual gearbox is likely to remain the most common type 177. "What Does "Five by Five" mean? \| Five by Five Definition Brand Evolution". Five by Five. 2019-07-16. Retrieved 2020-08-02. 178. Gaskin, Shelley (2009-01-31). Go! with 2007. CRC PRESS. p. 615. ISBN 978-0-13-239020-0. the number 5 key has a raised bar or dot that helps you identify it by touch 179. Stewart, George (1985). The C-64 Program Factory. Osborn McGraw-Hill. p. 278. ISBN 978-0-88134-150-8. ...digit in the phone number is a 5 , which corresponds to the triplet J , K , L 180. Atlantic (2007-06-13). Encyclopedia Of Information Technology. Atlantic Publishers & Dist. p. 659. ISBN 978-81-269-0752-6. The Pentium is a fifth-generation x86 architecture... 181. Stevens, E. S. (2020-06-16). Green Plastics: An Introduction to the New Science of Biodegradable Plastics. Princeton University Press. p. 45. ISBN 978-0-691-21417-7. polypropylene 5 182. Corporation, Bonnier (1937). Popular Science. Bonnier Corporation. p. 32. ...another picture of one of the world's most famous babies was made. Fred Davis is official photographer of the Dionne quintuplets... 183. Smith, Rich (2010-09-01). Fifth Amendment: The Right to Fairness. ABDO Publishing Company. p. 20. ISBN 978-1-61784-256-6. Someone who stands on his or her right to avoid self incrimination is said in street language to be "taking the Fifth," or "pleading the Fifth." 184. Veith (Jr.), Gene Edward; Wilson, Douglas (2009). Omnibus IV: The Ancient World. Veritas Press. p. 52. ISBN 978-1-932168-86-0. The most common accentual-syllabic lines are five-foot iambic lines (iambic pentameter) 185. Kronland-Martinet, Richard; Ystad, Sølvi; Jensen, Kristoffer (2008-07-19). Computer Music Modeling and Retrieval. Sense of Sounds: 4th International Symposium, CMMR 2007, Copenhagen, Denmark, August 2007, Revised Papers. Springer. p. 502. ISBN 978-3-540-85035-9. Plato and Aristotle postulated a fifth state of matter, which they called "idea" or quintessence" (from "quint" which means "fifth") 186. Roads, United States Congress Senate Committee on Public Works Subcommittee on (1970). Designating Highway U.S. 50 as Part of the Interstate System, Nevada: Hearings, Ninety-first Congress, First Session; Carson City, Nevada, October 6, 1969; [and] Ely, Nevada, October 7, 1969. U.S. Government Printing Office. p. 78. 187. Sonderman, Joe (2010). Route 66 in New Mexico. Arcadia Publishing. p. 7. ISBN 978-0-7385-8029-6. North – south highways got odd numbers , the most important ending in 5 188. Cusack, Professor Carole M. (2013-06-28). Invented Religions: Imagination, Fiction and Faith. Ashgate Publishing, Ltd. p. 31. ISBN 978-1-4094-8103-4. Law of Fives is never wrong'. This law is the reason 23 is a significant number for Discordians... 189. Lazarus, Richard J. (2020-03-10). The Rule of Five: Making Climate History at the Supreme Court. Harvard University Press. p. 252. ISBN 978-0-674-24515-0. ...Justice Brennan's infamous "Rule of Five," 190. Laplante, Philip A. (2018-10-03). Comprehensive Dictionary of Electrical Engineering. CRC Press. p. 562. ISBN 978-1-4200-3780-7. quincunx five points 191. Hargrove, Julia (2000-03-01). John F. Kennedy's Inaugural Address. Lorenz Educational Press. p. 24. ISBN 978-1-57310-222-3. The five permanent members have a veto power over actions proposed by members of the United Nations. 192. McGee, Steven R. (2012-01-01). Evidence-based Physical Diagnosis. Elsevier Health Sciences. p. 120. ISBN 978-1-4377-2207-9. There are five Korotkoff phases... 193. "punch \| Origin and meaning of punch by Online Etymology Dictionary". www.etymonline.com. Retrieved 2020-08-01. ...said to derive from Hindi panch "five," in reference to the number of original ingredients 194. Berke, Richard L.; Times, Special To the New York (1990-10-15). "G.O.P. Senators See Politics In Pace of Keating 5 Inquiry". The New York Times. ISSN 0362-4331. Retrieved 2020-08-01. 195. "Keith Giffen Revives Inferior Five for DC Comics in September – What to Do With Woody Allen?". bleedingcool.com. 14 June 2019. Retrieved 2020-08-01. 196. "For the first time". Inside Chanel. Archived from the original on 2020-09-18. Retrieved 2020-08-01. 197. Beeman, Richard R. (2013-05-07). Our Lives, Our Fortunes and Our Sacred Honor: The Forging of American Independence, 1774–1776. Basic Books. p. 407. ISBN 978-0-465-03782-7. On Friday, June 28, the Committee of Five delivered its revised draft of Jefferson's draft of the Declaration of Independence 198. Skarnulis, Leanna. "5 Second Rule For Food". WebMD. Retrieved 2020-08-01. 199. Newsweek. Newsweek. 1963. p. 71. His newest characters: a boy named 555 95472, or 5 for short, Further reading • Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 58–67 External links • Media related to 5 (number) at Wikimedia Commons • The dictionary definition of five at Wiktionary • Prime curiosities: 5 Integers 0s • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 100s • 100 • 101 • 102 • 103 • 104 • 105 • 106 • 107 • 108 • 109 • 110 • 111 • 112 • 113 • 114 • 115 • 116 • 117 • 118 • 119 • 120 • 121 • 122 • 123 • 124 • 125 • 126 • 127 • 128 • 129 • 130 • 131 • 132 • 133 • 134 • 135 • 136 • 137 • 138 • 139 • 140 • 141 • 142 • 143 • 144 • 145 • 146 • 147 • 148 • 149 • 150 • 151 • 152 • 153 • 154 • 155 • 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983 • 984 • 985 • 986 • 987 • 988 • 989 • 990 • 991 • 992 • 993 • 994 • 995 • 996 • 997 • 998 • 999 ≥1000 • 1000 • 2000 • 3000 • 4000 • 5000 • 6000 • 7000 • 8000 • 9000 • 10,000 • 20,000 • 30,000 • 40,000 • 50,000 • 60,000 • 70,000 • 80,000 • 90,000 • 100,000 • 1,000,000 • 10,000,000 • 100,000,000 • 1,000,000,000 Authority control: National • France • BnF data • Germany • Israel • United States
−1 In mathematics, −1 (negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0. ← −2 −1 0 → −1 0 1 2 3 4 5 6 7 8 9 → • List of numbers • Integers ← 0 10 20 30 40 50 60 70 80 90 → Cardinal−1, minus one, negative one Ordinal−1st (negative first) Arabic−١ Chinese numeral负一，负弌，负壹 Bengali−১ Binary (byte) S&M: 1000000012 2sC: 111111112 Hex (byte) S&M: 0x10116 2sC: 0xFF16 Algebraic properties Multiplication Further information: Additive inverse Multiplying a number by −1 is equivalent to changing the sign of the number – that is, for any x we have (−1) ⋅ x = −x. This can be proved using the distributive law and the axiom that 1 is the multiplicative identity: x + (−1) ⋅ x = 1 ⋅ x + (−1) ⋅ x = (1 + (−1)) ⋅ x = 0 ⋅ x = 0. Here we have used the fact that any number x times 0 equals 0, which follows by cancellation from the equation 0 ⋅ x = (0 + 0) ⋅ x = 0 ⋅ x + 0 ⋅ x. In other words, x + (−1) ⋅ x = 0, so (−1) ⋅ x is the additive inverse of x, i.e. (−1) ⋅ x = −x, as was to be shown. Square of −1 The square of −1, i.e. −1 multiplied by −1, equals 1. As a consequence, a product of two negative numbers is positive. For an algebraic proof of this result, start with the equation 0 = −1 ⋅ 0 = −1 ⋅ [1 + (−1)]. The first equality follows from the above result, and the second follows from the definition of −1 as additive inverse of 1: it is precisely that number which when added to 1 gives 0. Now, using the distributive law, it can be seen that 0 = −1 ⋅ [1 + (−1)] = −1 ⋅ 1 + (−1) ⋅ (−1) = −1 + (−1) ⋅ (−1). The third equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies (−1) ⋅ (−1) = 1. The above arguments hold in any ring, a concept of abstract algebra generalizing integers and real numbers. Square roots of −1 Although there are no real square roots of −1, the complex number i satisfies i2 = −1, and as such can be considered as a square root of −1.[1][2] The only other complex number whose square is −1 is −i because there are exactly two square roots of any non‐zero complex number, which follows from the fundamental theorem of algebra. In the algebra of quaternions – where the fundamental theorem does not apply – which contains the complex numbers, the equation x2 = −1 has infinitely many solutions. Exponentiation to negative integers Exponentiation of a non‐zero real number can be extended to negative integers. We make the definition that x−1 = 1/x, meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This definition is then extended to negative integers, preserving the exponential law xaxb = x(a + b) for real numbers a and b. Exponentiation to negative integers can be extended to invertible elements of a ring, by defining x−1 as the multiplicative inverse of x. A −1 that appears as a superscript of a function does not mean taking the (pointwise) reciprocal of that function, but rather the inverse function of the function. For example, sin−1(x) is a notation for the arcsine function, and in general f −1(x) denotes the inverse function of f(x),. When a subset of the codomain is specified inside the function, it instead denotes the preimage of that subset under the function. Uses • In software development, −1 is a common initial value for integers and is also used to show that a variable contains no useful information. • −1 bears relation to Euler's identity since eiπ = −1. See also • Balanced ternary • Menelaus's theorem References 1. "Imaginary Numbers". Math is Fun. Retrieved 15 February 2021. 2. Weisstein, Eric W. "Imaginary Number". MathWorld. Retrieved 15 February 2021.
Fifth power (algebra) In arithmetic and algebra, the fifth power or sursolid[1] of a number n is the result of multiplying five instances of n together: n5 = n × n × n × n × n. Fifth powers are also formed by multiplying a number by its fourth power, or the square of a number by its cube. The sequence of fifth powers of integers is: 0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 9765625, ... (sequence A000584 in the OEIS) Properties For any integer n, the last decimal digit of n5 is the same as the last (decimal) digit of n, i.e. $n\equiv n^{5}{\pmod {10}}$ By the Abel–Ruffini theorem, there is no general algebraic formula (formula expressed in terms of radical expressions) for the solution of polynomial equations containing a fifth power of the unknown as their highest power. This is the lowest power for which this is true. See quintic equation, sextic equation, and septic equation. Along with the fourth power, the fifth power is one of two powers k that can be expressed as the sum of k − 1 other k-th powers, providing counterexamples to Euler's sum of powers conjecture. Specifically, 275 + 845 + 1105 + 1335 = 1445 (Lander & Parkin, 1966)[2] See also • Eighth power • Seventh power • Sixth power • Fourth power • Cube (algebra) • Square (algebra) • Perfect power Footnotes 1. "Webster's 1913". 2. Lander, L. J.; Parkin, T. R. (1966). "Counterexample to Euler's conjecture on sums of like powers". Bull. Amer. Math. Soc. 72 (6): 1079. doi:10.1090/S0002-9904-1966-11654-3. References • Råde, Lennart; Westergren, Bertil (2000). Springers mathematische Formeln: Taschenbuch für Ingenieure, Naturwissenschaftler, Informatiker, Wirtschaftswissenschaftler (in German) (3 ed.). Springer-Verlag. p. 44. ISBN 3-540-67505-1. • Vega, Georg (1783). Logarithmische, trigonometrische, und andere zum Gebrauche der Mathematik eingerichtete Tafeln und Formeln (in German). Vienna: Gedruckt bey Johann Thomas Edlen von Trattnern, kaiferl. königl. Hofbuchdruckern und Buchhändlern. p. 358. 1 32 243 1024. • Jahn, Gustav Adolph (1839). Tafeln der Quadrat- und Kubikwurzeln aller Zahlen von 1 bis 25500, der Quadratzahlen aller Zahlen von 1 bis 27000 und der Kubikzahlen aller Zahlen von 1 bis 24000 (in German). Leipzig: Verlag von Johann Ambrosius Barth. p. 241. • Deza, Elena; Deza, Michel (2012). Figurate Numbers. Singapore: World Scientific Publishing. p. 173. ISBN 978-981-4355-48-3. • Rosen, Kenneth H.; Michaels, John G. (2000). Handbook of Discrete and Combinatorial Mathematics. Boca Raton, Florida: CRC Press. p. 159. ISBN 0-8493-0149-1. • Prändel, Johann Georg (1815). Arithmetik in weiterer Bedeutung, oder Zahlen- und Buchstabenrechnung in einem Lehrkurse - mit Tabellen über verschiedene Münzsorten, Gewichte und Ellenmaaße und einer kleinen Erdglobuslehre (in German). Munich. p. 264. Classes of natural numbers Powers and related numbers • Achilles • Power of 2 • Power of 3 • Power of 10 • Square • Cube • Fourth power • Fifth power • Sixth power • Seventh power • Eighth power • Perfect power • Powerful • Prime power Of the form a × 2b ± 1 • Cullen • Double Mersenne • Fermat • Mersenne • Proth • Thabit • Woodall Other polynomial numbers • Hilbert • Idoneal • Leyland • Loeschian • Lucky numbers of Euler Recursively defined numbers • Fibonacci • Jacobsthal • Leonardo • Lucas • Padovan • Pell • Perrin Possessing a specific set of other numbers • Amenable • Congruent • Knödel • Riesel • Sierpiński Expressible via specific sums • Nonhypotenuse • Polite • Practical • Primary pseudoperfect • Ulam • Wolstenholme Figurate numbers 2-dimensional centered • Centered triangular • Centered square • Centered pentagonal • Centered hexagonal • Centered heptagonal • Centered octagonal • Centered nonagonal • Centered decagonal • Star non-centered • Triangular • Square • Square triangular • Pentagonal • Hexagonal • Heptagonal • Octagonal • Nonagonal • Decagonal • Dodecagonal 3-dimensional centered • Centered tetrahedral • Centered cube • Centered octahedral • Centered dodecahedral • Centered icosahedral non-centered • Tetrahedral • Cubic • Octahedral • Dodecahedral • Icosahedral • Stella octangula pyramidal • Square pyramidal 4-dimensional non-centered • Pentatope • Squared triangular • Tesseractic Combinatorial numbers • Bell • Cake • Catalan • Dedekind • Delannoy • Euler • Eulerian • Fuss–Catalan • Lah • Lazy caterer's sequence • Lobb • Motzkin • Narayana • Ordered Bell • Schröder • Schröder–Hipparchus • Stirling first • Stirling second • Telephone number • Wedderburn–Etherington Primes • Wieferich • Wall–Sun–Sun • Wolstenholme prime • Wilson Pseudoprimes • Carmichael number • Catalan pseudoprime • Elliptic pseudoprime • Euler pseudoprime • Euler–Jacobi pseudoprime • Fermat pseudoprime • Frobenius pseudoprime • Lucas pseudoprime • Lucas–Carmichael number • Somer–Lucas pseudoprime • Strong pseudoprime Arithmetic functions and dynamics Divisor functions • Abundant • Almost perfect • Arithmetic • Betrothed • Colossally abundant • Deficient • Descartes • Hemiperfect • Highly abundant • Highly composite • Hyperperfect • Multiply perfect • Perfect • Practical • Primitive abundant • Quasiperfect • Refactorable • Semiperfect • Sublime • Superabundant • Superior highly composite • Superperfect Prime omega functions • Almost prime • Semiprime Euler's totient function • Highly cototient • Highly totient • Noncototient • Nontotient • Perfect totient • Sparsely totient Aliquot sequences • Amicable • Perfect • Sociable • Untouchable Primorial • Euclid • Fortunate Other prime factor or divisor related numbers • Blum • Cyclic • Erdős–Nicolas • Erdős–Woods • Friendly • Giuga • Harmonic divisor • Jordan–Pólya • Lucas–Carmichael • Pronic • Regular • Rough • Smooth • Sphenic • Størmer • Super-Poulet • Zeisel Numeral system-dependent numbers Arithmetic functions and dynamics • Persistence • Additive • Multiplicative Digit sum • Digit sum • Digital root • Self • Sum-product Digit product • Multiplicative digital root • Sum-product Coding-related • Meertens Other • Dudeney • Factorion • Kaprekar • Kaprekar's constant • Keith • Lychrel • Narcissistic • Perfect digit-to-digit invariant • Perfect digital invariant • Happy P-adic numbers-related • Automorphic • Trimorphic Digit-composition related • Palindromic • Pandigital • Repdigit • Repunit • Self-descriptive • Smarandache–Wellin • Undulating Digit-permutation related • Cyclic • Digit-reassembly • Parasitic • Primeval • Transposable Divisor-related • Equidigital • Extravagant • Frugal • Harshad • Polydivisible • Smith • Vampire Other • Friedman Binary numbers • Evil • Odious • Pernicious Generated via a sieve • Lucky • Prime Sorting related • Pancake number • Sorting number Natural language related • Aronson's sequence • Ban Graphemics related • Strobogrammatic • Mathematics portal
Sixth power In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So: n6 = n × n × n × n × n × n. Sixth powers can be formed by multiplying a number by its fifth power, multiplying the square of a number by its fourth power, by cubing a square, or by squaring a cube. The sequence of sixth powers of integers is: 0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 244140625, 308915776, 387420489, 481890304, ... (sequence A001014 in the OEIS) They include the significant decimal numbers 106 (a million), 1006 (a short-scale trillion and long-scale billion), 10006 (a Quintillion and a long-scale trillion) and so on. Squares and cubes The sixth powers of integers can be characterized as the numbers that are simultaneously squares and cubes.[1] In this way, they are analogous to two other classes of figurate numbers: the square triangular numbers, which are simultaneously square and triangular, and the solutions to the cannonball problem, which are simultaneously square and square-pyramidal. Because of their connection to squares and cubes, sixth powers play an important role in the study of the Mordell curves, which are elliptic curves of the form $y^{2}=x^{3}+k.$ When $k$ is divisible by a sixth power, this equation can be reduced by dividing by that power to give a simpler equation of the same form. A well-known result in number theory, proven by Rudolf Fueter and Louis J. Mordell, states that, when $k$ is an integer that is not divisible by a sixth power (other than the exceptional cases $k=1$ and $k=-432$), this equation either has no rational solutions with both $x$ and $y$ nonzero or infinitely many of them.[2] In the archaic notation of Robert Recorde, the sixth power of a number was called the "zenzicube", meaning the square of a cube. Similarly, the notation for sixth powers used in 12th century Indian mathematics by Bhāskara II also called them either the square of a cube or the cube of a square.[3] Sums There are numerous known examples of sixth powers that can be expressed as the sum of seven other sixth powers, but no examples are yet known of a sixth power expressible as the sum of just six sixth powers.[4] This makes it unique among the powers with exponent k = 1, 2, ... , 8, the others of which can each be expressed as the sum of k other k-th powers, and some of which (in violation of Euler's sum of powers conjecture) can be expressed as a sum of even fewer k-th powers. In connection with Waring's problem, every sufficiently large integer can be represented as a sum of at most 24 sixth powers of integers.[5] There are infinitely many different nontrivial solutions to the Diophantine equation[6] $a^{6}+b^{6}+c^{6}=d^{6}+e^{6}+f^{6}.$ It has not been proven whether the equation $a^{6}+b^{6}=c^{6}+d^{6}$ has a nontrivial solution,[7] but the Lander, Parkin, and Selfridge conjecture would imply that it does not. Other properties • $n^{6}-1$ is divisible by 7 iff n isn't divisible by 7. See also • Sextic equation • Eighth power • Seventh power • Fifth power (algebra) • Fourth power • Cube (algebra) • Square (algebra) References 1. Dowden, Richard (April 30, 1825), "(untitled)", Mechanics' Magazine and Journal of Science, Arts, and Manufactures, Knight and Lacey, vol. 4, no. 88, p. 54 2. Ireland, Kenneth F.; Rosen, Michael I. (1982), A classical introduction to modern number theory, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York-Berlin, p. 289, ISBN 0-387-90625-8, MR 0661047. 3. Cajori, Florian (2013), A History of Mathematical Notations, Dover Books on Mathematics, Courier Corporation, p. 80, ISBN 9780486161167 4. Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions". Retrieved 17 July 2017. 5. Vaughan, R. C.; Wooley, T. D. (1994), "Further improvements in Waring's problem. II. Sixth powers", Duke Mathematical Journal, 76 (3): 683–710, doi:10.1215/S0012-7094-94-07626-6, MR 1309326 6. Brudno, Simcha (1976), "Triples of sixth powers with equal sums", Mathematics of Computation, 30 (135): 646–648, doi:10.1090/s0025-5718-1976-0406923-6, MR 0406923 7. Bremner, Andrew; Guy, Richard K. (1988), "Unsolved Problems: A Dozen Difficult Diophantine Dilemmas", American Mathematical Monthly, 95 (1): 31–36, doi:10.2307/2323442, JSTOR 2323442, MR 1541235 External links • Weisstein, Eric W. "Diophantine Equation—6th Powers". MathWorld. Classes of natural numbers Powers and related numbers • Achilles • Power of 2 • Power of 3 • Power of 10 • Square • Cube • Fourth power • Fifth power • Sixth power • Seventh power • Eighth power • Perfect power • Powerful • Prime power Of the form a × 2b ± 1 • Cullen • Double Mersenne • Fermat • Mersenne • Proth • Thabit • Woodall Other polynomial numbers • Hilbert • Idoneal • Leyland • Loeschian • Lucky numbers of Euler Recursively defined numbers • Fibonacci • Jacobsthal • Leonardo • Lucas • Padovan • Pell • Perrin Possessing a specific set of other numbers • Amenable • Congruent • Knödel • Riesel • Sierpiński Expressible via specific sums • Nonhypotenuse • Polite • Practical • Primary pseudoperfect • Ulam • Wolstenholme Figurate numbers 2-dimensional centered • Centered triangular • Centered square • Centered pentagonal • Centered hexagonal • Centered heptagonal • Centered octagonal • Centered nonagonal • Centered decagonal • Star non-centered • Triangular • Square • Square triangular • Pentagonal • Hexagonal • Heptagonal • Octagonal • Nonagonal • Decagonal • Dodecagonal 3-dimensional centered • Centered tetrahedral • Centered cube • Centered octahedral • Centered dodecahedral • Centered icosahedral non-centered • Tetrahedral • Cubic • Octahedral • Dodecahedral • Icosahedral • Stella octangula pyramidal • Square pyramidal 4-dimensional non-centered • Pentatope • Squared triangular • Tesseractic Combinatorial numbers • Bell • Cake • Catalan • Dedekind • Delannoy • Euler • Eulerian • Fuss–Catalan • Lah • Lazy caterer's sequence • Lobb • Motzkin • Narayana • Ordered Bell • Schröder • Schröder–Hipparchus • Stirling first • Stirling second • Telephone number • Wedderburn–Etherington Primes • Wieferich • Wall–Sun–Sun • Wolstenholme prime • Wilson Pseudoprimes • Carmichael number • Catalan pseudoprime • Elliptic pseudoprime • Euler pseudoprime • Euler–Jacobi pseudoprime • Fermat pseudoprime • Frobenius pseudoprime • Lucas pseudoprime • Lucas–Carmichael number • Somer–Lucas pseudoprime • Strong pseudoprime Arithmetic functions and dynamics Divisor functions • Abundant • Almost perfect • Arithmetic • Betrothed • Colossally abundant • Deficient • Descartes • Hemiperfect • Highly abundant • Highly composite • Hyperperfect • Multiply perfect • Perfect • Practical • Primitive abundant • Quasiperfect • Refactorable • Semiperfect • Sublime • Superabundant • Superior highly composite • Superperfect Prime omega functions • Almost prime • Semiprime Euler's totient function • Highly cototient • Highly totient • Noncototient • Nontotient • Perfect totient • Sparsely totient Aliquot sequences • Amicable • Perfect • Sociable • Untouchable Primorial • Euclid • Fortunate Other prime factor or divisor related numbers • Blum • Cyclic • Erdős–Nicolas • Erdős–Woods • Friendly • Giuga • Harmonic divisor • Jordan–Pólya • Lucas–Carmichael • Pronic • Regular • Rough • Smooth • Sphenic • Størmer • Super-Poulet • Zeisel Numeral system-dependent numbers Arithmetic functions and dynamics • Persistence • Additive • Multiplicative Digit sum • Digit sum • Digital root • Self • Sum-product Digit product • Multiplicative digital root • Sum-product Coding-related • Meertens Other • Dudeney • Factorion • Kaprekar • Kaprekar's constant • Keith • Lychrel • Narcissistic • Perfect digit-to-digit invariant • Perfect digital invariant • Happy P-adic numbers-related • Automorphic • Trimorphic Digit-composition related • Palindromic • Pandigital • Repdigit • Repunit • Self-descriptive • Smarandache–Wellin • Undulating Digit-permutation related • Cyclic • Digit-reassembly • Parasitic • Primeval • Transposable Divisor-related • Equidigital • Extravagant • Frugal • Harshad • Polydivisible • Smith • Vampire Other • Friedman Binary numbers • Evil • Odious • Pernicious Generated via a sieve • Lucky • Prime Sorting related • Pancake number • Sorting number Natural language related • Aronson's sequence • Ban Graphemics related • Strobogrammatic • Mathematics portal
Seventh power In arithmetic and algebra the seventh power of a number n is the result of multiplying seven instances of n together. So: n7 = n × n × n × n × n × n × n. Seventh powers are also formed by multiplying a number by its sixth power, the square of a number by its fifth power, or the cube of a number by its fourth power. The sequence of seventh powers of integers is: 0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, 410338673, 612220032, 893871739, 1280000000, 1801088541, 2494357888, 3404825447, 4586471424, 6103515625, 8031810176, ... (sequence A001015 in the OEIS) In the archaic notation of Robert Recorde, the seventh power of a number was called the "second sursolid".[1] Properties Leonard Eugene Dickson studied generalizations of Waring's problem for seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers[2] (17 is 1, and 27 is 128). All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers.[3] If powers of negative integers are allowed, only 12 powers are required.[4] The smallest number that can be represented in two different ways as a sum of four positive seventh powers is 2056364173794800.[5] The smallest seventh power that can be represented as a sum of eight distinct seventh powers is:[6] $102^{7}=12^{7}+35^{7}+53^{7}+58^{7}+64^{7}+83^{7}+85^{7}+90^{7}.$ The two known examples of a seventh power expressible as the sum of seven seventh powers are $568^{7}=127^{7}+258^{7}+266^{7}+413^{7}+430^{7}+439^{7}+525^{7}$ (M. Dodrill, 1999);[7] and $626^{7}=625^{7}+309^{7}+258^{7}+255^{7}+158^{7}+148^{7}+91^{7}$ (Maurice Blondot, 11/14/2000);[7] any example with fewer terms in the sum would be a counterexample to Euler's sum of powers conjecture, which is currently only known to be false for the powers 4 and 5. See also • Eighth power • Sixth power • Fifth power (algebra) • Fourth power • Cube (algebra) • Square (algebra) References 1. Womack, D. (2015), "Beyond tetration operations: their past, present and future", Mathematics in School, 44 (1): 23–26 2. Dickson, L. E. (1934), "A new method for universal Waring theorems with details for seventh powers", American Mathematical Monthly, 41 (9): 547–555, doi:10.2307/2301430, JSTOR 2301430, MR 1523212 3. Kumchev, Angel V. (2005), "On the Waring-Goldbach problem for seventh powers", Proceedings of the American Mathematical Society, 133 (10): 2927–2937, doi:10.1090/S0002-9939-05-07908-6, MR 2159771 4. Choudhry, Ajai (2000), "On sums of seventh powers", Journal of Number Theory, 81 (2): 266–269, doi:10.1006/jnth.1999.2465, MR 1752254 5. Ekl, Randy L. (1996), "Equal sums of four seventh powers", Mathematics of Computation, 65 (216): 1755–1756, Bibcode:1996MaCom..65.1755E, doi:10.1090/S0025-5718-96-00768-5, MR 1361807 6. Stewart, Ian (1989), Game, set, and math: Enigmas and conundrums, Basil Blackwell, Oxford, p. 123, ISBN 0-631-17114-2, MR 1253983 7. Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions". Retrieved 17 July 2017. Classes of natural numbers Powers and related numbers • Achilles • Power of 2 • Power of 3 • Power of 10 • Square • Cube • Fourth power • Fifth power • Sixth power • Seventh power • Eighth power • Perfect power • Powerful • Prime power Of the form a × 2b ± 1 • Cullen • Double Mersenne • Fermat • Mersenne • Proth • Thabit • Woodall Other polynomial numbers • Hilbert • Idoneal • Leyland • Loeschian • Lucky numbers of Euler Recursively defined numbers • Fibonacci • Jacobsthal • Leonardo • Lucas • Padovan • Pell • Perrin Possessing a specific set of other numbers • Amenable • Congruent • Knödel • Riesel • Sierpiński Expressible via specific sums • Nonhypotenuse • Polite • Practical • Primary pseudoperfect • Ulam • Wolstenholme Figurate numbers 2-dimensional centered • Centered triangular • Centered square • Centered pentagonal • Centered hexagonal • Centered heptagonal • Centered octagonal • Centered nonagonal • Centered decagonal • Star non-centered • Triangular • Square • Square triangular • Pentagonal • Hexagonal • Heptagonal • Octagonal • Nonagonal • Decagonal • Dodecagonal 3-dimensional centered • Centered tetrahedral • Centered cube • Centered octahedral • Centered dodecahedral • Centered icosahedral non-centered • Tetrahedral • Cubic • Octahedral • Dodecahedral • Icosahedral • Stella octangula pyramidal • Square pyramidal 4-dimensional non-centered • Pentatope • Squared triangular • Tesseractic Combinatorial numbers • Bell • Cake • Catalan • Dedekind • Delannoy • Euler • Eulerian • Fuss–Catalan • Lah • Lazy caterer's sequence • Lobb • Motzkin • Narayana • Ordered Bell • Schröder • Schröder–Hipparchus • Stirling first • Stirling second • Telephone number • Wedderburn–Etherington Primes • Wieferich • Wall–Sun–Sun • Wolstenholme prime • Wilson Pseudoprimes • Carmichael number • Catalan pseudoprime • Elliptic pseudoprime • Euler pseudoprime • Euler–Jacobi pseudoprime • Fermat pseudoprime • Frobenius pseudoprime • Lucas pseudoprime • Lucas–Carmichael number • Somer–Lucas pseudoprime • Strong pseudoprime Arithmetic functions and dynamics Divisor functions • Abundant • Almost perfect • Arithmetic • Betrothed • Colossally abundant • Deficient • Descartes • Hemiperfect • Highly abundant • Highly composite • Hyperperfect • Multiply perfect • Perfect • Practical • Primitive abundant • Quasiperfect • Refactorable • Semiperfect • Sublime • Superabundant • Superior highly composite • Superperfect Prime omega functions • Almost prime • Semiprime Euler's totient function • Highly cototient • Highly totient • Noncototient • Nontotient • Perfect totient • Sparsely totient Aliquot sequences • Amicable • Perfect • Sociable • Untouchable Primorial • Euclid • Fortunate Other prime factor or divisor related numbers • Blum • Cyclic • Erdős–Nicolas • Erdős–Woods • Friendly • Giuga • Harmonic divisor • Jordan–Pólya • Lucas–Carmichael • Pronic • Regular • Rough • Smooth • Sphenic • Størmer • Super-Poulet • Zeisel Numeral system-dependent numbers Arithmetic functions and dynamics • Persistence • Additive • Multiplicative Digit sum • Digit sum • Digital root • Self • Sum-product Digit product • Multiplicative digital root • Sum-product Coding-related • Meertens Other • Dudeney • Factorion • Kaprekar • Kaprekar's constant • Keith • Lychrel • Narcissistic • Perfect digit-to-digit invariant • Perfect digital invariant • Happy P-adic numbers-related • Automorphic • Trimorphic Digit-composition related • Palindromic • Pandigital • Repdigit • Repunit • Self-descriptive • Smarandache–Wellin • Undulating Digit-permutation related • Cyclic • Digit-reassembly • Parasitic • Primeval • Transposable Divisor-related • Equidigital • Extravagant • Frugal • Harshad • Polydivisible • Smith • Vampire Other • Friedman Binary numbers • Evil • Odious • Pernicious Generated via a sieve • Lucky • Prime Sorting related • Pancake number • Sorting number Natural language related • Aronson's sequence • Ban Graphemics related • Strobogrammatic • Mathematics portal
7 7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube. ← 6 7 8 → −1 0 1 2 3 4 5 6 7 8 9 → • List of numbers • Integers ← 0 10 20 30 40 50 60 70 80 90 → Cardinalseven Ordinal7th (seventh) Numeral systemseptenary Factorizationprime Prime4th Divisors1, 7 Greek numeralΖ´ Roman numeralVII, vii Greek prefixhepta-/hept- Latin prefixseptua- Binary1112 Ternary213 Senary116 Octal78 Duodecimal712 Hexadecimal716 Greek numeralZ, ζ Amharic፯ Arabic, Kurdish, Persian٧ Sindhi, Urdu۷ Bengali৭ Chinese numeral七, 柒 Devanāgarī७ Telugu౭ Tamil௭ Hebrewז Khmer៧ Thai๗ Kannada೭ Malayalam൭ As an early prime number in the series of positive integers, the number seven has greatly symbolic associations in religion, mythology, superstition and philosophy. The seven Classical planets resulted in seven being the number of days in a week. It is often considered lucky in Western culture and is often seen as highly symbolic. Unlike Western culture, in Vietnamese culture, the number seven is sometimes considered unlucky. Evolution of the Arabic digit In the beginning, Indians wrote 7 more or less in one stroke as a curve that looks like an uppercase ⟨J⟩ vertically inverted (ᒉ). The western Ghubar Arabs' main contribution was to make the longer line diagonal rather than straight, though they showed some tendencies to making the digit more rectilinear. The eastern Arabs developed the digit from a form that looked something like 6 to one that looked like an uppercase V. Both modern Arab forms influenced the European form, a two-stroke form consisting of a horizontal upper stroke joined at its right to a stroke going down to the bottom left corner, a line that is slightly curved in some font variants. As is the case with the European digit, the Cham and Khmer digit for 7 also evolved to look like their digit 1, though in a different way, so they were also concerned with making their 7 more different. For the Khmer this often involved adding a horizontal line to the top of the digit.[1] This is analogous to the horizontal stroke through the middle that is sometimes used in handwriting in the Western world but which is almost never used in computer fonts. This horizontal stroke is, however, important to distinguish the glyph for seven from the glyph for one in writing that uses a long upstroke in the glyph for 1. In some Greek dialects of the early 12th century the longer line diagonal was drawn in a rather semicircular transverse line. On the seven-segment displays of pocket calculators and digital watches, 7 is the digit with the most common graphic variation (1, 6 and 9 also have variant glyphs). Most calculators use three line segments, but on Sharp, Casio, and a few other brands of calculators, 7 is written with four line segments because in Japan, Korea and Taiwan 7 is written with a "hook" on the left, as ① in the following illustration. While the shape of the character for the digit 7 has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender (⁊), as, for example, in . Most people in Continental Europe,[2] Indonesia,[3] and some in Britain, Ireland, and Canada, as well as Latin America, write 7 with a line in the middle ("7"), sometimes with the top line crooked. The line through the middle is useful to clearly differentiate the digit from the digit one, as the two can appear similar when written in certain styles of handwriting. This form is used in official handwriting rules for primary school in Russia, Ukraine, Bulgaria, Poland, other Slavic countries,[4] France,[5] Italy, Belgium, the Netherlands, Finland,[6] Romania, Germany, Greece,[7] and Hungary. Mathematics Seven, the fourth prime number, is not only a Mersenne prime (since 23 − 1 = 7) but also a double Mersenne prime since the exponent, 3, is itself a Mersenne prime.[8] It is also a Newman–Shanks–Williams prime,[9] a Woodall prime,[10] a factorial prime,[11] a Harshad number, a lucky prime,[12] a happy number (happy prime),[13] a safe prime (the only Mersenne safe prime), a Leyland prime of the second kind and the fourth Heegner number.[14] • Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers. (See Lagrange's four-square theorem#Historical development.) • Seven is the aliquot sum of one number, the cubic number 8 and is the base of the 7-aliquot tree. • 7 is the only number D for which the equation 2n − D = x2 has more than two solutions for n and x natural. In particular, the equation 2n − 7 = x2 is known as the Ramanujan–Nagell equation. • There are 7 frieze groups in two dimensions, consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers.[15] These are related to the 17 wallpaper groups whose transformations and isometries repeat two-dimensional patterns in the plane.[16][17] The seventh indexed prime number is seventeen.[18] • A seven-sided shape is a heptagon.[19] The regular n-gons for n ⩽ 6 can be constructed by compass and straightedge alone, which makes the heptagon the first regular polygon that cannot be directly constructed with these simple tools.[20] Figurate numbers representing heptagons are called heptagonal numbers.[21] 7 is also a centered hexagonal number.[22] A heptagon in Euclidean space is unable to generate uniform tilings alongside other polygons, like the regular pentagon. However, it is one of fourteen polygons that can fill a plane-vertex tiling, in its case only alongside a regular triangle and a 42-sided polygon (3.7.42).[23][24] This is also one of twenty-one such configurations from seventeen combinations of polygons, that features the largest and smallest polygons possible.[25][26] • In Wythoff's kaleidoscopic constructions, seven distinct generator points that lie on mirror edges of a three-sided Schwarz triangle are used to create most uniform tilings and polyhedra; an eighth point lying on all three mirrors is technically degenerate, reserved to represent snub forms only.[27] Seven of eight semiregular tilings are Wythoffian, the only exception is the elongated triangular tiling.[28] Seven of nine uniform colorings of the square tiling are also Wythoffian, and between the triangular tiling and square tiling, there are seven non-Wythoffian uniform colorings of a total twenty-one that belong to regular tilings (all hexagonal tiling uniform colorings are Wythoffian).[29] In two dimensions, there are precisely seven 7-uniform Krotenheerdt tilings, with no other such k-uniform tilings for k > 7, and it is also the only k for which the count of Krotenheerdt tilings agrees with k.[30][31] • The Fano plane is the smallest possible finite projective plane with 7 points and 7 lines such that every line contains 3 points and 3 lines cross every point.[32] With group order 168 = 23·3·7, this plane holds 35 total triples of points where 7 are collinear and another 28 are non-collinear, whose incidence graph is the 3-regular bipartate Heawood graph with 14 vertices and 21 edges.[33] This graph embeds in three dimensions as the Szilassi polyhedron, the simplest toroidal polyhedron alongside its dual with 7 vertices, the Császár polyhedron.[34][35] • In three-dimensional space there are seven crystal systems and fourteen Bravais lattices which classify under seven lattice systems, six of which are shared with the seven crystal systems.[36][37][38] There are also collectively seventy-seven Wythoff symbols that represent all uniform figures in three dimensions.[39] • The seventh dimension is the only dimension aside from the familiar three where a vector cross product can be defined.[40] This is related to the octonions over the imaginary subspace Im(O) in 7-space whose commutator between two octonions defines this vector product, wherein the Fano plane describes the multiplicative algebraic structure of the unit octonions {e0, e1, e2, ..., e7}, with e0 an identity element.[41] Also, the lowest known dimension for an exotic sphere is the seventh dimension, with a total of 28 differentiable structures; there may exist exotic smooth structures on the four-dimensional sphere.[42][43] In hyperbolic space, 7 is the highest dimension for non-simplex hypercompact Vinberg polytopes of rank n + 4 mirrors, where there is one unique figure with eleven facets.[44] On the other hand, such figures with rank n + 3 mirrors exist in dimensions 4, 5, 6 and 8; not in 7.[45] Hypercompact polytopes with lowest possible rank of n + 2 mirrors exist up through the 17th dimension, where there is a single solution as well.[46] • There are seven fundamental types of catastrophes.[47] • When rolling two standard six-sided dice, seven has a 6 in 62 (or 1/6) probability of being rolled (1–6, 6–1, 2–5, 5–2, 3–4, or 4–3), the greatest of any number.[48] The opposite sides of a standard six-sided dice always add to 7. • The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.[49] Currently, six of the problems remain unsolved.[50] Basic calculations Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000 7 × x 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 161 168 175 350 700 7000 Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 7 ÷ x 7 3.5 2.3 1.75 1.4 1.16 1 0.875 0.7 0.7 0.63 0.583 0.538461 0.5 0.46 x ÷ 7 0.142857 0.285714 0.428571 0.571428 0.714285 0.857142 1.142857 1.285714 1.428571 1.571428 1.714285 1.857142 2 2.142857 Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 7x 7 49 343 2401 16807 117649 823543 5764801 40353607 282475249 1977326743 13841287201 96889010407 x7 1 128 2187 16384 78125 279936 823543 2097152 4782969 10000000 19487171 35831808 62748517 Radix 1 5 10 15 20 25 50 75 100 125 150 200 250 500 1000 10000 100000 1000000 x7 1 5 137 217 267 347 1017 1357 2027 2367 3037 4047 5057 13137 26267 411047 5643557 113333117 In decimal 999,999 divided by 7 is exactly 142,857. Therefore, when a vulgar fraction with 7 in the denominator is converted to a decimal expansion, the result has the same six-digit repeating sequence after the decimal point, but the sequence can start with any of those six digits.[51] For example, 1/7 = 0.142857 142857... and 2/7 = 0.285714 285714.... In fact, if one sorts the digits in the number 142,857 in ascending order, 124578, it is possible to know from which of the digits the decimal part of the number is going to begin with. The remainder of dividing any number by 7 will give the position in the sequence 124578 that the decimal part of the resulting number will start. For example, 628 ÷ 7 = 89+5/7; here 5 is the remainder, and would correspond to number 7 in the ranking of the ascending sequence. So in this case, 628 ÷ 7 = 89.714285. Another example, 5238 ÷ 7 = 748+2/7, hence the remainder is 2, and this corresponds to number 2 in the sequence. In this case, 5238 ÷ 7 = 748.285714. In science • Seven colors in a rainbow: ROYGBIV • Seven Continents • Seven Seas • Seven climes • The neutral pH balance • Number of music notes in a scale • Number of spots most commonly found on ladybugs • Atomic number for nitrogen In psychology • Seven, plus or minus two as a model of working memory. • Seven psychological types called the Seven Rays in the teachings of Alice A. Bailey • In Western culture, Seven is consistently listed as people's favorite number.[52][53] • When guessing numbers 1–10, the number 7 is most likely to be picked.[54] • Seven-year itch: happiness in marriage said to decline after 7 years Classical antiquity The Pythagoreans invested particular numbers with unique spiritual properties. The number seven was considered to be particularly interesting because it consisted of the union of the physical (number 4) with the spiritual (number 3).[55] In Pythagorean numerology the number 7 means spirituality. References from classical antiquity to the number seven include: • Seven Classical planets and the derivative Seven Heavens • Seven Wonders of the Ancient World • Seven metals of antiquity • Seven days in the week • Seven Seas • Seven Sages • Seven champions that fought Thebes • Seven hills of Rome and Seven Kings of Rome • Seven Sisters, the daughters of Atlas also known as the Pleiades Religion and mythology Judaism The number seven forms a widespread typological pattern within Hebrew scripture, including: • Seven days (more precisely yom) of Creation, leading to the seventh day or Sabbath (Genesis 1) • Seven-fold vengeance visited on upon Cain for the killing of Abel (Genesis 4:15) • Seven pairs of every clean animal loaded onto the ark by Noah (Genesis 7:2) • Seven years of plenty and seven years of famine in Pharaoh's dream (Genesis 41) • Seventh son of Jacob, Gad, whose name means good luck (Genesis 46:16) • Seven times bullock's blood is sprinkled before God (Leviticus 4:6) • Seven nations God told the Israelites they would displace when they entered the land of Israel (Deuteronomy 7:1) • Seven days (de jure, but de facto eight days) of the Passover feast (Exodus 13:3–10) • Seven-branched candelabrum or Menorah (Exodus 25) • Seven trumpets played by seven priests for seven days to bring down the walls of Jericho (Joshua 6:8) • Seven things that are detestable to God (Proverbs 6:16–19) • Seven Pillars of the House of Wisdom (Proverbs 9:1) • Seven archangels in the deuterocanonical Book of Tobit (12:15) References to the number seven in Jewish knowledge and practice include: • Seven divisions of the weekly readings or aliyah of the Torah • Seven Jewish men (over the age of 13) called to read aliyahs in Shabbat morning services • Seven blessings recited under the chuppah during a Jewish wedding ceremony • Seven days of festive meals for a Jewish bride and groom after their wedding, known as Sheva Berachot or Seven Blessings • Seven Ushpizzin prayers to the Jewish patriarchs during the holiday of Sukkot Christianity Following the tradition of the Hebrew Bible, the New Testament likewise uses the number seven as part of a typological pattern: • Seven loaves multiplied into seven basketfuls of surplus (Matthew 15:32–37) • Seven demons were driven out of Mary Magdalene (Luke 8:2) • Seven last sayings of Jesus on the cross • Seven men of honest report, full of the Holy Ghost and wisdom (Acts 6:3) • Seven Spirits of God, Seven Churches and Seven Seals in the Book of Revelation References to the number seven in Christian knowledge and practice include: • Seven Gifts of the Holy Spirit • Seven Corporal Acts of Mercy and Seven Spiritual Acts of Mercy • Seven deadly sins: lust, gluttony, greed, sloth, wrath, envy, and pride, and seven terraces of Mount Purgatory • Seven Virtues: chastity, temperance, charity, diligence, kindness, patience, and humility • Seven Joys and Seven Sorrows of the Virgin Mary • Seven Sleepers of Christian myth • Seven Sacraments in the Catholic Church (though some traditions assign a different number) Islam References to the number seven in Islamic knowledge and practice include: • Seven ayat in surat al-Fatiha, the first book of the holy Qur'an • Seven circumambulations of Muslim pilgrims around the Kaaba in Mecca during the Hajj and the Umrah • Seven walks between Al-Safa and Al-Marwah performed Muslim pilgrims during the Hajj and the Umrah • Seven doors to hell (for heaven the number of doors is eight) • Seven Earths and seven Heavens (plural of sky) mentioned in Qur'an (S. 65:12) • Night Journey to the Seventh Heaven, (reported ascension to heaven to meet God) Isra' and Mi'raj of the Qur'an and surah Al-Isra'. • Seventh day naming ceremony held for babies • Seven enunciators of divine revelation (nāṭiqs) according to the celebrated Fatimid Ismaili dignitary Nasir Khusraw[56] • Circle Seven Koran, the holy scripture of the Moorish Science Temple of America Hinduism References to the number seven in Hindu knowledge and practice include: • Seven worlds in the universe and seven seas in the world in Hindu cosmology • Seven sages or Saptarishi and their seven wives or Sapta Matrka in Hindu mythology • Seven Chakras in eastern philosophy • Seven stars in a constellation called "Saptharishi Mandalam" in Indian astronomy • Seven promises, or Saptapadi, and seven circumambulations around a fire at Hindu weddings • Seven virgin goddesses or Saptha Kannimar worshipped in temples in Tamil Nadu, India[57][58] • Seven hills at Tirumala known as Yedu Kondalavadu in Telugu, or ezhu malaiyan in Tamil, meaning "Sevenhills God" • Seven steps taken by the Buddha at birth • Seven divine ancestresses of humankind in Khasi mythology • Seven octets or Saptak Swaras in Indian Music as the basis for Ragas compositions • Seven Social Sins listed by Mahatma Gandhi Eastern tradition Other references to the number seven in Eastern traditions include: • Seven Lucky Gods or gods of good fortune in Japanese mythology • Seven-Branched Sword in Japanese mythology • Seven Sages of the Bamboo Grove in China • Seven minor symbols of yang in Taoist yin-yang Other references Other references to the number seven in traditions from around the world include: • The number seven had mystical and religious significance in Mesopotamian culture by the 22nd century BCE at the latest. This was likely because in the Sumerian sexagesimal number system, dividing by seven was the first division which resulted in infinitely repeating fractions.[59] • Seven palms in an Egyptian Sacred Cubit • Seven ranks in Mithraism • Seven hills of Istanbul • Seven islands of Atlantis • Seven Cherokee clans • Seven lives of cats in Iran and German and Romance language-speaking cultures[60] • Seven fingers on each hand, seven toes on each foot and seven pupils in each eye of the Irish epic hero Cúchulainn • Seventh sons will be werewolves in Galician folklore, or the son of a woman and a werewolf in other European folklores • Seventh sons of a seventh son will be magicians with special powers of healing and clairvoyance in some cultures, or vampires in others • Seven prominent legendary monsters in Guaraní mythology • Seven gateways traversed by Inanna during her descent into the underworld • Seven Wise Masters, a cycle of medieval stories • Seven sister goddesses or fates in Baltic mythology called the Deivės Valdytojos.[61] • Seven legendary Cities of Gold, such as Cibola, that the Spanish thought existed in South America • Seven years spent by Thomas the Rhymer in the faerie kingdom in the eponymous British folk tale • Seven-year cycle in which the Queen of the Fairies pays a tithe to Hell (or possibly Hel) in the tale of Tam Lin • Seven Valleys, a text by the Prophet-Founder Bahá'u'lláh in the Bahá'í faith • Seven superuniverses in the cosmology of Urantia[62] • Seven psychological types called the Seven Rays in the teachings of Alice A. Bailey • Seven, the sacred number of Yemaya[63] • Seven holes representing eyes (سبع عيون) in an Assyrian evil eye bead – though occasionally two, and sometimes nine [64] In culture In literature • Seven Dwarfs • The Seven Brothers, an 1870 novel by Aleksis Kivi • Seven features prominently in A Song of Ice and Fire by George R. R. Martin, namely, the Seven Kingdoms and the Faith of the Seven In visual art • The Group of Seven Canadian landscape painters In sports • Sports with seven players per side • Kabaddi • Rugby sevens • Water Polo • Netball • Handball • Flag Football • Ultimate Frisbee • Seven is the least number of players a soccer team must have on the field in order for a match to start and continue. • A touchdown plus an extra point is worth seven points. See also Wikimedia Commons has media related to 7 (number). Look up seven in Wiktionary, the free dictionary. • Diatonic scale (7 notes) • Seven colors in the rainbow • Seven continents • Seven liberal arts • Seven Wonders of the Ancient World • Seven days of the Week • Septenary (numeral system) • Year Seven (School) • Se7en (disambiguation) • Sevens (disambiguation) • One-seventh area triangle • Z with stroke (Ƶ) • List of highways numbered 7 Notes 1. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.67 2. Eeva Törmänen (September 8, 2011). "Aamulehti: Opetushallitus harkitsee numero 7 viivan palauttamista". Tekniikka & Talous (in Finnish). Archived from the original on September 17, 2011. Retrieved September 9, 2011. 3. "Mengapa orang Indonesia menambahkan garis kecil pada penulisan angka tujuh (7)?" (in Indonesian). Quora. Retrieved June 12, 2023. 4. "Education writing numerals in grade 1." Archived 2008-10-02 at the Wayback Machine(Russian) 5. "Example of teaching materials for pre-schoolers"(French) 6. Elli Harju (August 6, 2015). ""Nenosen seiska" teki paluun: Tiesitkö, mistä poikkiviiva on peräisin?". Iltalehti (in Finnish). 7. "Μαθηματικά Α' Δημοτικού" [Mathematics for the First Grade] (PDF) (in Greek). Ministry of Education, Research, and Religions. p. 33. Retrieved May 7, 2018. 8. Weisstein, Eric W. "Double Mersenne Number". mathworld.wolfram.com. Retrieved 2020-08-06. 9. "Sloane's A088165 : NSW primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 10. "Sloane's A050918 : Woodall primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 11. "Sloane's A088054 : Factorial primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 12. "Sloane's A031157 : Numbers that are both lucky and prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 13. "Sloane's A035497 : Happy primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 14. "Sloane's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 15. Heyden, Anders; Sparr, Gunnar; Nielsen, Mads; Johansen, Peter (2003-08-02). Computer Vision - ECCV 2002: 7th European Conference on Computer Vision, Copenhagen, Denmark, May 28-31, 2002. Proceedings. Part II. Springer. p. 661. ISBN 978-3-540-47967-3. A frieze pattern can be classified into one of the 7 frieze groups... 16. Grünbaum, Branko; Shephard, G. C. (1987). "Section 1.4 Symmetry Groups of Tilings". Tilings and Patterns. New York: W. H. Freeman and Company. pp. 40–45. doi:10.2307/2323457. ISBN 0-7167-1193-1. JSTOR 2323457. OCLC 13092426. S2CID 119730123. 17. Sloane, N. J. A. (ed.). "Sequence A004029 (Number of n-dimensional space groups.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-30. 18. Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-01. 19. Weisstein, Eric W. "Heptagon". mathworld.wolfram.com. Retrieved 2020-08-25. 20. Weisstein, Eric W. "7". mathworld.wolfram.com. Retrieved 2020-08-07. 21. Sloane, N. J. A. (ed.). "Sequence A000566 (Heptagonal numbers (or 7-gonal numbers))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-09. 22. Sloane, N. J. A. (ed.). "Sequence A003215". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. 23. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. Taylor & Francis, Ltd. 50 (5): 231. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006. 24. Jardine, Kevin. "Shield - a 3.7.42 tiling". Imperfect Congruence. Retrieved 2023-01-09. 3.7.42 as a unit facet in an irregular tiling. 25. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. Taylor & Francis, Ltd. 50 (5): 229-230. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006. 26. Dallas, Elmslie William (1855). "Part II. (VII): Of the Circle, with its Inscribed and Circumscribed Figures − Equal Division and the Construction of Polygons". The Elements of Plane Practical Geometry. London: John W. Parker & Son, West Strand. p. 134. "...It will thus be found that, including the employment of the same figures, there are seventeen different combinations of regular polygons by which this may be effected; namely, — When three polygons are employed , there are ten ways; viz., 6,6,6 – 3.7.42 — 3,8,24 – 3,9,18 — 3,10,15 — 3,12,12 — 4,5,20 — 4,6,12 — 4,8,8 — 5,5,10. With four polygons there are four ways, viz., 4,4,4,4 — 3,3,4,12 — 3,3,6,6 — 3,4,4,6. With five polygons there are two ways, viz., 3,3,3,4,4 — 3,3,3,3,6. With six polygons one way — all equilateral triangles [ 3.3.3.3.3.3 ]." Note: the only four other configurations from the same combinations of polygons are: 3.4.3.12, (3.6)2, 3.4.6.4, and 3.3.4.3.4. 27. Coxeter, H. S. M. (1999). "Chapter 3: Wythoff's Construction for Uniform Polytopes". The Beauty of Geometry: Twelve Essays. Mineola, NY: Dover Publications. pp. 326–339. ISBN 9780486409191. OCLC 41565220. S2CID 227201939. Zbl 0941.51001. 28. Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.1: Regular and uniform tilings". Tilings and Patterns. New York: W. H. Freeman and Company. pp. 62–64. doi:10.2307/2323457. ISBN 0-7167-1193-1. JSTOR 2323457. OCLC 13092426. S2CID 119730123. 29. Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.9 Archimedean and uniform colorings". Tilings and Patterns. New York: W. H. Freeman and Company. pp. 102–107. doi:10.2307/2323457. ISBN 0-7167-1193-1. JSTOR 2323457. OCLC 13092426. 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Kubovy, Michael; Psotka, Joseph (May 1976). "The predominance of seven and the apparent spontaneity of numerical choices". Journal of Experimental Psychology: Human Perception and Performance. 2 (2): 291–294. doi:10.1037/0096-1523.2.2.291. Retrieved 20 February 2022. 55. "Number symbolism - 7". 56. "Nāṣir-i Khusraw", An Anthology of Philosophy in Persia, I.B.Tauris, 2001, doi:10.5040/9780755610068.ch-008, ISBN 978-1-84511-542-5, retrieved 2020-11-17 57. Rajarajan, R.K.K. (2020). "Peerless Manifestations of Devī". Carcow Indological Studies (Cracow, Poland). XXII.1: 221–243. doi:10.12797/CIS.22.2020.01.09. S2CID 226326183. 58. Rajarajan, R.K.K. (2020). "Sempiternal "Pattiṉi": Archaic Goddess of the vēṅkai-tree to Avant-garde Acaṉāmpikai". Studia Orientalia Electronica (Helsinki, Finland). 8 (1): 120–144. doi:10.23993/store.84803. S2CID 226373749. 59. The Origin of the Mystical Number Seven in Mesopotamian Culture: Division by Seven in the Sexagesimal Number System 60. "Encyclopædia Britannica "Number Symbolism"". Britannica.com. Retrieved 2012-09-07. 61. Klimka, Libertas (2012-03-01). "Senosios baltų mitologijos ir religijos likimas". Lituanistica. 58 (1). doi:10.6001/lituanistica.v58i1.2293. ISSN 0235-716X. 62. "Chapter I. The Creative Thesis of Perfection by William S. Sadler, Jr. - Urantia Book - Urantia Foundation". urantia.org. 17 August 2011. 63. Yemaya. Santeria Church of the Orishas. Retrieved 25 November 2022 64. Ergil, Leyla Yvonne (2021-06-10). "Turkey's talisman superstitions: Evil eyes, pomegranates and more". Daily Sabah. Retrieved 2023-04-05. References • Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group (1987): 70–71 Integers 0s • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 100s • 100 • 101 • 102 • 103 • 104 • 105 • 106 • 107 • 108 • 109 • 110 • 111 • 112 • 113 • 114 • 115 • 116 • 117 • 118 • 119 • 120 • 121 • 122 • 123 • 124 • 125 • 126 • 127 • 128 • 129 • 130 • 131 • 132 • 133 • 134 • 135 • 136 • 137 • 138 • 139 • 140 • 141 • 142 • 143 • 144 • 145 • 146 • 147 • 148 • 149 • 150 • 151 • 152 • 153 • 154 • 155 • 156 • 157 • 158 • 159 • 160 • 161 • 162 • 163 • 164 • 165 • 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331 • 332 • 333 • 334 • 335 • 336 • 337 • 338 • 339 • 340 • 341 • 342 • 343 • 344 • 345 • 346 • 347 • 348 • 349 • 350 • 351 • 352 • 353 • 354 • 355 • 356 • 357 • 358 • 359 • 360 • 361 • 362 • 363 • 364 • 365 • 366 • 367 • 368 • 369 • 370 • 371 • 372 • 373 • 374 • 375 • 376 • 377 • 378 • 379 • 380 • 381 • 382 • 383 • 384 • 385 • 386 • 387 • 388 • 389 • 390 • 391 • 392 • 393 • 394 • 395 • 396 • 397 • 398 • 399 400s • 400 • 401 • 402 • 403 • 404 • 405 • 406 • 407 • 408 • 409 • 410 • 411 • 412 • 413 • 414 • 415 • 416 • 417 • 418 • 419 • 420 • 421 • 422 • 423 • 424 • 425 • 426 • 427 • 428 • 429 • 430 • 431 • 432 • 433 • 434 • 435 • 436 • 437 • 438 • 439 • 440 • 441 • 442 • 443 • 444 • 445 • 446 • 447 • 448 • 449 • 450 • 451 • 452 • 453 • 454 • 455 • 456 • 457 • 458 • 459 • 460 • 461 • 462 • 463 • 464 • 465 • 466 • 467 • 468 • 469 • 470 • 471 • 472 • 473 • 474 • 475 • 476 • 477 • 478 • 479 • 480 • 481 • 482 • 483 • 484 • 485 • 486 • 487 • 488 • 489 • 490 • 491 • 492 • 493 • 494 • 495 • 496 • 497 • 498 • 499 500s • 500 • 501 • 502 • 503 • 504 • 505 • 506 • 507 • 508 • 509 • 510 • 511 • 512 • 513 • 514 • 515 • 516 • 517 • 518 • 519 • 520 • 521 • 522 • 523 • 524 • 525 • 526 • 527 • 528 • 529 • 530 • 531 • 532 • 533 • 534 • 535 • 536 • 537 • 538 • 539 • 540 • 541 • 542 • 543 • 544 • 545 • 546 • 547 • 548 • 549 • 550 • 551 • 552 • 553 • 554 • 555 • 556 • 557 • 558 • 559 • 560 • 561 • 562 • 563 • 564 • 565 • 566 • 567 • 568 • 569 • 570 • 571 • 572 • 573 • 574 • 575 • 576 • 577 • 578 • 579 • 580 • 581 • 582 • 583 • 584 • 585 • 586 • 587 • 588 • 589 • 590 • 591 • 592 • 593 • 594 • 595 • 596 • 597 • 598 • 599 600s • 600 • 601 • 602 • 603 • 604 • 605 • 606 • 607 • 608 • 609 • 610 • 611 • 612 • 613 • 614 • 615 • 616 • 617 • 618 • 619 • 620 • 621 • 622 • 623 • 624 • 625 • 626 • 627 • 628 • 629 • 630 • 631 • 632 • 633 • 634 • 635 • 636 • 637 • 638 • 639 • 640 • 641 • 642 • 643 • 644 • 645 • 646 • 647 • 648 • 649 • 650 • 651 • 652 • 653 • 654 • 655 • 656 • 657 • 658 • 659 • 660 • 661 • 662 • 663 • 664 • 665 • 666 • 667 • 668 • 669 • 670 • 671 • 672 • 673 • 674 • 675 • 676 • 677 • 678 • 679 • 680 • 681 • 682 • 683 • 684 • 685 • 686 • 687 • 688 • 689 • 690 • 691 • 692 • 693 • 694 • 695 • 696 • 697 • 698 • 699 700s • 700 • 701 • 702 • 703 • 704 • 705 • 706 • 707 • 708 • 709 • 710 • 711 • 712 • 713 • 714 • 715 • 716 • 717 • 718 • 719 • 720 • 721 • 722 • 723 • 724 • 725 • 726 • 727 • 728 • 729 • 730 • 731 • 732 • 733 • 734 • 735 • 736 • 737 • 738 • 739 • 740 • 741 • 742 • 743 • 744 • 745 • 746 • 747 • 748 • 749 • 750 • 751 • 752 • 753 • 754 • 755 • 756 • 757 • 758 • 759 • 760 • 761 • 762 • 763 • 764 • 765 • 766 • 767 • 768 • 769 • 770 • 771 • 772 • 773 • 774 • 775 • 776 • 777 • 778 • 779 • 780 • 781 • 782 • 783 • 784 • 785 • 786 • 787 • 788 • 789 • 790 • 791 • 792 • 793 • 794 • 795 • 796 • 797 • 798 • 799 800s • 800 • 801 • 802 • 803 • 804 • 805 • 806 • 807 • 808 • 809 • 810 • 811 • 812 • 813 • 814 • 815 • 816 • 817 • 818 • 819 • 820 • 821 • 822 • 823 • 824 • 825 • 826 • 827 • 828 • 829 • 830 • 831 • 832 • 833 • 834 • 835 • 836 • 837 • 838 • 839 • 840 • 841 • 842 • 843 • 844 • 845 • 846 • 847 • 848 • 849 • 850 • 851 • 852 • 853 • 854 • 855 • 856 • 857 • 858 • 859 • 860 • 861 • 862 • 863 • 864 • 865 • 866 • 867 • 868 • 869 • 870 • 871 • 872 • 873 • 874 • 875 • 876 • 877 • 878 • 879 • 880 • 881 • 882 • 883 • 884 • 885 • 886 • 887 • 888 • 889 • 890 • 891 • 892 • 893 • 894 • 895 • 896 • 897 • 898 • 899 900s • 900 • 901 • 902 • 903 • 904 • 905 • 906 • 907 • 908 • 909 • 910 • 911 • 912 • 913 • 914 • 915 • 916 • 917 • 918 • 919 • 920 • 921 • 922 • 923 • 924 • 925 • 926 • 927 • 928 • 929 • 930 • 931 • 932 • 933 • 934 • 935 • 936 • 937 • 938 • 939 • 940 • 941 • 942 • 943 • 944 • 945 • 946 • 947 • 948 • 949 • 950 • 951 • 952 • 953 • 954 • 955 • 956 • 957 • 958 • 959 • 960 • 961 • 962 • 963 • 964 • 965 • 966 • 967 • 968 • 969 • 970 • 971 • 972 • 973 • 974 • 975 • 976 • 977 • 978 • 979 • 980 • 981 • 982 • 983 • 984 • 985 • 986 • 987 • 988 • 989 • 990 • 991 • 992 • 993 • 994 • 995 • 996 • 997 • 998 • 999 ≥1000 • 1000 • 2000 • 3000 • 4000 • 5000 • 6000 • 7000 • 8000 • 9000 • 10,000 • 20,000 • 30,000 • 40,000 • 50,000 • 60,000 • 70,000 • 80,000 • 90,000 • 100,000 • 1,000,000 • 10,000,000 • 100,000,000 • 1,000,000,000 Authority control: National • Germany • Israel • United States
9 9 (nine) is the natural number following 8 and preceding 10. ← 8 9 10 → −1 0 1 2 3 4 5 6 7 8 9 → • List of numbers • Integers ← 0 10 20 30 40 50 60 70 80 90 → Cardinalnine Ordinal9th (ninth) Numeral systemnonary Factorization32 Divisors1,3,9 Greek numeralΘ´ Roman numeralIX, ix Greek prefixennea- Latin prefixnona- Binary10012 Ternary1003 Senary136 Octal118 Duodecimal912 Hexadecimal916 Amharic፱ Arabic, Kurdish, Persian, Sindhi, Urdu٩ Armenian numeralԹ Bengali৯ Chinese numeral九, 玖 Devanāgarī९ Greek numeralθ´ Hebrew numeralט Tamil numerals௯ Khmer៩ Telugu numeral౯ Thai numeral๙ Malayalam൯ Evolution of the Hindu–Arabic digit See also: Hindu–Arabic numeral system Circa 300 BCE, as part of the Brahmi numerals, various Indians wrote a digit 9 similar in shape to the modern closing question mark without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the sign @ encircles a lowercase a. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller. Soon, all that was left of the 3-look-alike was a squiggle. The Arabs simply connected that squiggle to the downward stroke at the middle and subsequent European change was purely cosmetic. While the shape of the glyph for the digit 9 has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in . The modern digit resembles an inverted 6. To disambiguate the two on objects and documents that can be inverted, they are often underlined. Another distinction from the 6 is that it is sometimes handwritten with two strokes and a straight stem, resembling a raised lower-case letter q. In seven-segment display, the number 9 can be constructed either with a hook at the end of its stem or without one. Most LCD calculators use the former, but some VFD models use the latter. Mathematics Nine is the fourth composite number, and the first composite number that is odd. Nine is the third square number (32), and the second non-unitary square prime of the form p2, and, the first that is odd, with all subsequent squares of this form odd as well. Nine has the even aliquot sum of 4, and with a composite number sequence of two (9, 4, 3, 1, 0) within the 3-aliquot tree. There are nine Heegner numbers, or square-free positive integers $n$ that yield an imaginary quadratic field $\mathbb {Q} \left[{\sqrt {-n}}\right]$ whose ring of integers has a unique factorization, or class number of 1.[1] 9 is the sum of the cubes of the first two non-zero positive integers $1^{3}+2^{3}$ which makes it the first cube-sum number greater than one.[2] It is also the sum of the first three nonzero factorials $1!+2!+3!$ and equal to the third exponential factorial, since $9=3^{2^{1}}.$[3] By Mihăilescu's theorem, 9 is the only positive perfect power that is one more than another positive perfect power, since the square of 3 is one more than the cube of 2.[4][5] Nine is the number of derangements of 4, or the number of permutations of four elements with no fixed points.[6] 9 is the fourth refactorable number, as it has exactly three positive divisors, and 3 is one of them.[7] A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes.[8] If an odd perfect number exists, it will have at least nine distinct prime factors.[9] 9 is a Motzkin number, for the number of ways of drawing non-intersecting chords between four points on a circle.[10] The first non-trivial magic square is a $3$ x $3$ magic square made of nine cells, with a magic constant of 15.[11] Meanwhile, a $9$ x $9$ magic square has a magic constant of 369.[12] A polygon with nine sides is called a nonagon.[13] Since 9 can be written in the form $2^{m}3^{n}p$, for any nonnegative natural integers $m$ and $n$ with $p$ a product of Pierpont primes, a regular nonagon can be constructed with a regular compass, straightedge, and angle trisector.[14] Also an enneagon, a regular nonagon is able to fill a plane-vertex alongside an equilateral triangle and a regular 18-sided octadecagon (3.9.18), and as such, it is one of only nine polygons that are able to fill a plane-vertex without uniformly tiling the plane.[15] There are nine distinct uniform colorings of the triangular tiling and the square tiling, which are the two simplest regular tilings; the hexagonal tiling, on the other hand, has three distinct uniform colorings. There are a maximum of nine semiregular Archimedean tilings by convex regular polygons, when including chiral forms of the snub hexagonal tiling. There are nine uniform edge-transitive convex polyhedra in three dimensions: • the five regular Platonic solids: the tetrahedron, octahedron, cube, dodecahedron and icosahedron; • the two quasiregular Archimedean solids: the cuboctahedron and the icosidodecahedron; and • two Catalan solids: the rhombic dodecahedron and the rhombic triacontahedron, which are duals to the only two quasiregular polyhedra. Nine distinct stellation's by Miller's rules are produced by the truncated tetrahedron.[16] It is the simplest Archimedean solid, with a total of four equilateral triangular and four hexagonal faces. In four-dimensional space, there are nine paracompact hyperbolic honeycomb Coxeter groups, as well as nine regular compact hyperbolic honeycombs from regular convex and star polychora.[17] There are also nine uniform demitesseractic ($\mathrm {D} _{4}$) Euclidean honeycombs in the fourth dimension. There are only three types of Coxeter groups of uniform figures in dimensions nine and thereafter, aside from the many families of prisms and proprisms: the $\mathrm {A} _{n}$ simplex groups, the $\mathrm {B} _{n}$ hypercube groups, and the $\mathrm {D} _{n}$ demihypercube groups. The ninth dimension is also the final dimension that contains Coxeter-Dynkin diagrams as uniform solutions in hyperbolic space. Inclusive of compact hyperbolic solutions, there are a total of 238 compact and paracompact Coxeter-Dynkin diagrams between dimensions two and nine, or equivalently between ranks three and ten. The most important of the last ${\tilde {E}}_{9}$ paracompact groups is the group ${\tilde {T}}_{9}$ with 1023 total honeycombs, the simplest of which is 621 whose vertex figure is the 521 honeycomb: the vertex arrangement of the densest-possible packing of spheres in 8 dimensions which forms the $\mathbb {E} _{8}$ lattice. The 621 honeycomb is made of 9-simplexes and 9-orthoplexes, with 1023 total polytope elements making up each 9-simplex. It is the final honeycomb figure with infinite facets and vertex figures in the k21 family of semiregular polytopes, first defined by Thorold Gosset in 1900. In decimal 9 is the highest single-digit number in the decimal system. A positive number is divisible by nine if and only if its digital root is nine: • 9 × 2 = 18 (1 + 8 = 9) • 9 × 3 = 27 (2 + 7 = 9) • 9 × 9 = 81 (8 + 1 = 9) • 9 × 121 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9) • 9 × 234 = 2106 (2 + 1 + 0 + 6 = 9) • 9 × 578329 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = 27; 2 + 7 = 9) • 9 × 482729235601 = 4344563120409 (4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45; 4 + 5 = 9) That is, if any natural number is multiplied by 9, and the digits of the answer are repeatedly added until it is just one digit, the sum will be nine.[18] In base-$N$, the divisors of $N-1$ have such a property, which makes 3 the only other number aside from 9 in decimal that shares this property. Another consequence of 9 being 10 − 1 is that it is a Kaprekar number. There are other interesting patterns involving multiples of nine: • 9 × 12345679 = 111111111 • 18 × 12345679 = 222222222 • 81 × 12345679 = 999999999 The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Examples: • The sum of the digits of 41 is 5, and 41 − 5 = 36. The digital root of 36 is 3 + 6 = 9. • The sum of the digits of 35967930 is 3 + 5 + 9 + 6 + 7 + 9 + 3 + 0 = 42, and 35967930 − 42 = 35967888. The digital root of 35967888 is 3 + 5 + 9 + 6 + 7 + 8 + 8 + 8 = 54, 5 + 4 = 9. If dividing a number by the amount of 9s corresponding to its number of digits, the number is turned into a repeating decimal. (e.g. 274/999 = 0.274274274274...) Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers known as long ago as the 12th century.[19] Six recurring nines appear in the decimal places 762 through 767 of π. (See six nines in pi). List of basic calculations Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 20 25 50 100 1000 9 × x 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 180 225 450 900 9000 Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 9 ÷ x 9 4.5 3 2.25 1.8 1.5 1.285714 1.125 1 0.9 0.81 0.75 0.692307 0.6428571 0.6 x ÷ 9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 1.1 1.2 1.3 1.4 1.5 1.6 Exponentiation 1 2 3 4 5 6 7 8 9 10 9x 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 x9 1 512 19683 262144 1953125 10077696 40353607 134217728 387420489 1000000000 Radix 1 5 10 15 20 25 30 40 50 60 70 80 90 100 110 120 130 140 150 200 250 500 1000 10000 100000 1000000 x9 1 5 119 169 229 279 339 449 559 669 779 889 1109 1219 1329 1439 1549 1659 1769 2429 3079 6159 13319 146419 1621519 17836619 Alphabets and codes • In the NATO phonetic alphabet, the digit 9 is called "Niner". • Five-digit produce PLU codes that begin with 9 indicate organic foods. Culture and mythology Indian culture Nine is a number that appears often in Indian culture and mythology. Some instances are enumerated below. • Nine influencers are attested in Indian astrology. • In the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements: Earth, Water, Air, Fire, Ether, Time, Space, Soul, and Mind. • Navaratri is a nine-day festival dedicated to the nine forms of Durga. • Navaratna, meaning "nine jewels" may also refer to Navaratnas – accomplished courtiers, Navratan – a kind of dish, or a form of architecture. • In Indian aesthetics, there are nine kinds of Rasa. Chinese culture • Nine (九; pinyin: jiǔ) is considered a good number in Chinese culture because it sounds the same as the word "long-lasting" (久; pinyin: jiǔ).[20] • Nine is strongly associated with the Chinese dragon, a symbol of magic and power. There are nine forms of the dragon, it is described in terms of nine attributes, and it has nine children. It has 117 scales – 81 yang (masculine, heavenly) and 36 yin (feminine, earthly). All three numbers are multiples of 9 (9 × 13 = 117, 9 × 9 = 81, 9 × 4 = 36)[21] as well as having the same digital root of 9. • The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City. • The circular altar platform (Earthly Mount) of the Temple of Heaven has one circular marble plate in the center, surrounded by a ring of nine plates, then by a ring of 18 plates, and so on, for a total of nine rings, with the outermost having 81 = 9 × 9 plates. • The name of the area called Kowloon in Hong Kong literally means: nine dragons. • The nine-dotted line (Chinese: 南海九段线; pinyin: nánhǎi jiǔduàn xiàn; lit. 'Nine-segment line of the South China Sea') delimits certain island claims by China in the South China Sea. • The nine-rank system was a civil service nomination system used during certain Chinese dynasties. • 9 Points of the Heart (Heal) / Heart Master (Immortality) Channels in Traditional Chinese Medicine. Ancient Egypt • The nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt. • The Ennead is a group of nine Egyptian deities, who, in some versions of the Osiris myth, judged whether Horus or Set should inherit Egypt. European culture • The Nine Worthies are nine historical, or semi-legendary figures who, in the Middle Ages, were believed to personify the ideals of chivalry. • In Norse mythology, the universe is divided into nine worlds which are all connected by the world tree Yggdrasil • In Norse mythology as well, the number nine is associated with Odin, as that is how many days he hung from the world tree Yggdrasil before attaining knowledge of the runes. Greek mythology • The nine Muses in Greek mythology are Calliope (epic poetry), Clio (history), Erato (erotic poetry), Euterpe (lyric poetry), Melpomene (tragedy), Polyhymnia (song), Terpsichore (dance), Thalia (comedy), and Urania (astronomy). • It takes nine days (for an anvil) to fall from heaven to earth, and nine more to fall from earth to Tartarus. • Leto labored for nine days and nine nights for Apollo, according to the Homeric Hymn to Delian Apollo. Mesoamerican mythology • The Lords of the Night, is a group of nine deities who each ruled over every ninth night forming a calendrical cycle Aztec mythology • Mictlan the underworld in Aztec mythology, consists of nine levels. Mayan mythology • The Mayan underworld Xibalba consists of nine levels. • El Castillo, the Mayan step-pyramid in Chichén Itzá, consists of nine steps. It is said that this was done to represent the nine levels of Xibalba. Australian culture The Pintupi Nine, a group of 9 Aboriginal Australian women who remained unaware of European colonisation of Australia and lived a traditional desert-dwelling life in Australia's Gibson Desert until 1984. Anthropology Idioms • "to go the whole nine yards-" • "A cat-o'-nine-tails suggests perfect punishment and atonement." – Robert Ripley. • "A cat has nine lives" • "to be on cloud nine" • "A stitch in time saves nine" • "found true 9 out of 10 times" • "possession is nine tenths of the law" • The word "K-9" pronounces the same as canine and is used in many US police departments to denote the police dog unit. Despite not sounding like the translation of the word canine in other languages, many police and military units around the world use the same designation. • Someone dressed "to the nines" is dressed up as much as they can be. • In North American urban culture, "nine" is a slang word for a 9mm pistol or homicide, the latter from the Illinois Criminal Code for homicide. Society • The 9 on Yahoo!, hosted by Maria Sansone, was a daily video compilation show, or vlog, on Yahoo! featuring the nine top "web finds" of the day. • Nine justices sit on the United States Supreme Court. • Nine justices sit on the Supreme Court of Canada. Technique • Stanines, a method of scaling test scores, range from 1 to 9. • There are 9 square feet in a square yard. Pseudoscience • In Pythagorean numerology the number 9 symbolizes the end of one cycle and the beginning of another. • The modern day's Enneagram model of human psyche defines nine interconnected personality types. Literature • There are nine circles of Hell in Dante's Divine Comedy. • The Nine Bright Shiners, characters in Garth Nix's Old Kingdom trilogy. The Nine Bright Shiners was a 1930s book of poems by Anne Ridler[22] and a 1988 fiction book by Anthea Fraser;[23] the name derives from "a very curious old semi-pagan, semi-Christian" song.[24] • The Nine Tailors is a 1934 mystery novel by British writer Dorothy L. Sayers, her ninth featuring sleuth Lord Peter Wimsey. • Nine Unknown Men are, in occult legend, the custodians of the sciences of the world since ancient times. • In J. R. R. Tolkien's Middle-earth, there are nine rings of power given to men, and consequently, nine ringwraiths. Additionally, The Fellowship of the Ring consists of nine companions. • In Lorien Legacies there are nine Garde sent to Earth. • Number Nine is a character in Lorien Legacies. • In the series A Song of Ice and Fire, there are nine regions of Westeros (the Crownlands, the North, the Riverlands, the Westerlands, the Reach, the Stormlands, the Vale of Arryn, the Iron Islands and Dorne). Additionally, there is a group of nine city-states in western Essos known collectively as the Free Cities (Braavos, Lorath, Lys, Myr, Norvos, Pentos, Qohor, Tyrosh and Volantis). • In The Wheel of Time series, Daughter of the Nine Moons is the title given to the heir to the throne of Seanchan, and the Court of the Nine Moons serves as the throne room of the Seanchan rulers themselves. Additionally, the nation of Illian is partially governed by a body known as the Council of Nine, and the flag of Illian displays nine golden bees on it. Furthermore, in the Age of Legends, the Nine Rods of Dominion were nine regional governors who administered individual areas of the world under the ruling world government. Organizations • Divine Nine – The National Pan-Hellenic Council (NPHC) is a collaborative organization of nine historically African American, international Greek-lettered fraternities and sororities. Places and thoroughfares • List of highways numbered 9 • Ninth Avenue is a major avenue in Manhattan. • South Africa has 9 provinces • Negeri Sembilan, a Malaysian state located in Peninsular Malaysia, is named as such as it was historically a confederation of nine (Malay: sembilan) settlements (nagari) of the Minangkabau migrated from West Sumatra. Religion and philosophy Islam There are three verses that refer to nine in the Quran. We surely gave Moses nine clear signs.1 ˹You, O Prophet, can˺ ask the Children of Israel. When Moses came to them, Pharaoh said to him, “I really think that you, O Moses, are bewitched.” — Surah Al-Isra (The Night Journey/Banī Isrāʾīl):101[25] Note 1: The nine signs of Moses are: the staff, the hand (both mentioned in 20:17-22), famine, shortage of crops, floods, locusts, lice, frogs, and blood (all mentioned in 7:130-133). These signs came as proofs for Pharaoh and the Egyptians. Otherwise, Moses had some other signs such as water gushing out of the rock after he hit it with his staff, and splitting the sea. Now put your hand through ˹the opening of˺ your collar, it will come out ˹shining˺ white, unblemished.2 ˹These are two˺ of nine signs for Pharaoh and his people. They have truly been a rebellious people.” — Surah Al-Naml (The Ant):12[26] Note 2: Moses, who was dark-skinned, was asked to put his hand under his armpit. When he took it out it was shining white, but not out of a skin condition like melanoma. And there were in the city nine ˹elite˺ men who spread corruption in the land, never doing what is right. — Surah Al-Naml (The Ant):48[27] • Ramadan, the month of fasting and prayer, is the ninth month of the Islamic calendar. • Nine, as the highest single-digit number (in base ten), symbolizes completeness in the Baháʼí Faith. In addition, the word Baháʼ in the Abjad notation has a value of 9, and a 9-pointed star is used to symbolize the religion. • The number 9 is revered in Hinduism and considered a complete, perfected and divine number because it represents the end of a cycle in the decimal system, which originated from the Indian subcontinent as early as 3000 BC. • In Buddhism, Gautama Buddha was believed to have nine virtues, which he was (1) Accomplished, (2) Perfectly Enlightened, (3) Endowed with knowledge and Conduct or Practice, (4) Well-gone or Well-spoken, (5) the Knower of worlds, (6) the Guide Unsurpassed of men to be tamed, (7) the Teacher of gods and men, (8) Enlightened, and (9) Blessed. • Important Buddhist rituals usually involve nine monks. • The first nine days of the Hebrew month of Av are collectively known as "The Nine Days" (Tisha HaYamim), and are a period of semi-mourning leading up to Tisha B'Av, the ninth day of Av on which both Temples in Jerusalem were destroyed. • Nine is a significant number in Norse Mythology. Odin hung himself on an ash tree for nine days to learn the runes. • The Fourth Way Enneagram is one system of knowledge which shows the correspondence between the 9 integers and the circle. • In the Christian angelic hierarchy there are 9 choirs of angels. • Tian's Trigram Number, of Feng Shui, in Taoism. • In Christianity there are nine Fruit of the Holy Spirit which followers are expected to have: love, joy, peace, patience, kindness, goodness, faithfulness, gentleness, and self-control. • The Bible recorded that Christ died at the 9th hour of the day (3 pm).[28] Science Astronomy • Before 2006 (when Pluto was officially designated as a non-planet), there were nine planets in the Solar System. • Messier object M9 is a magnitude 9.0 globular cluster in the constellation Ophiuchus. • The New General Catalogue object NGC 9, a spiral galaxy in the constellation Pegasus. Chemistry • The purity of chemicals (see Nine (purity)). • Nine is the atomic number of fluorine. Physiology A human pregnancy normally lasts nine months, the basis of Naegele's rule. Psychology Common terminal digit in psychological pricing. Sports • Nine-ball is the standard professional pocket billiards variant played in the United States. • In association football (soccer), the centre-forward/striker traditionally (since at least the fifties) wears the number 9 shirt. • In baseball: • There are nine players on the field including the pitcher. • There are nine innings in a standard game. • 9 represents the right fielder's position. • NINE: A Journal of Baseball History and Culture, published by the University of Nebraska Press[29] • In rugby league, the jersey number assigned to the hooker in most competitions. (An exception is the Super League, which uses static squad numbering.) • In rugby union, the number worn by the starting scrum-half. Technology • ISO 9 is the ISO's standard for the transliteration of Cyrillic characters into Latin characters • In the Rich Text Format specification, 9 is the language code for the English language. All codes for regional variants of English are congruent to 9 mod 256. • The9 Limited (owner of the9.com) is a company in the video-game industry, including former ties to the extremely popular MMORPG World of Warcraft. Music • "Revolution 9", a sound collage which appears on The Beatles' eponymous 1968 album The Beatles (aka The White Album), prominently features a loop of a man's voice repeating the phrase "Number nine".[30] • There are 9 semitones in a Major 6th interval in music.[31] • There was a superstition among some notable classical music composers that they would die after completing their ninth symphony. Some composers who died after composing their ninth symphony include Ludwig van Beethoven, Anton Bruckner, Antonin Dvorak and Gustav Mahler.[32] • Beethoven's Symphony No. 9 is regarded as a masterpiece, and one of the most frequently performed symphonies in the world. See also Look up nine in Wiktionary, the free dictionary. • 9 (disambiguation) • 0.999... • Cloud Nine • List of highways numbered 9 References 1. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93 2. Sloane, N. J. A. (ed.). "Sequence A000537 (Sum of first n cubes; or n-th triangular number squared.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 19 June 2023. 3. "Sloane's A049384 : a(0)=1, a(n+1) = (n+1)^a(n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 1 June 2016. 4. Mihăilescu, Preda (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. Berlin: De Gruyter. 572: 167–195. doi:10.1515/crll.2004.048. MR 2076124. S2CID 121389998. 5. Metsänkylä, Tauno (2004). "Catalan's conjecture: another old Diophantine problem solved" (PDF). Bulletin of the American Mathematical Society. Providence, R.I.: American Mathematical Society. 41 (1): 43–57. doi:10.1090/S0273-0979-03-00993-5. MR 2015449. S2CID 17998831. Zbl 1081.11021. 6. Sloane, N. J. A. (ed.). "Sequence A000166 (Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 10 December 2022. 7. Sloane, N. J. A. (ed.). "Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 19 June 2023. 8. Avagyan, Armen; Dallakyan, Gurgen (2018). "A new method in the problem of three cubes". Universal Journal of Computational Mathematics. 5 (3): 45–56. arXiv:1802.06776. doi:10.13189/ujcmj.2017.050301. S2CID 36818799. 9. Pace P., Nielsen (2007). "Odd perfect numbers have at least nine distinct prime factors". Mathematics of Computation. Providence, R.I.: American Mathematical Society. 76 (260): 2109–2126. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. MR 2336286. S2CID 2767519. Zbl 1142.11086. 10. "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 1 June 2016. 11. William H. Richardson. "Magic Squares of Order 3". Wichita State University Dept. of Mathematics. Retrieved 6 November 2022. 12. Sloane, N. J. A. (ed.). "Sequence A006003 (Also the sequence M(n) of magic constants for n X n magic squares)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 December 2022. 13. Robert Dixon, Mathographics. New York: Courier Dover Publications: 24 14. Gleason, Andrew M. (1988). "Angle trisection, the heptagon, and the triskaidecagon". American Mathematical Monthly. Taylor & Francis, Ltd. 95 (3): 191–194. doi:10.2307/2323624. JSTOR 2323624. MR 0935432. S2CID 119831032. 15. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. Taylor & Francis, Ltd. 50 (5): 228-234. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006. 16. Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 25 November 2022. Retrieved 15 December 2022. 17. Coxeter, H. S. M. (1956), "Regular honeycombs in hyperbolic space", Proceedings of the International Congress of Mathematicians, vol. III, Amsterdam: North-Holland Publishing Co., pp. 167–169, MR 0087114 18. Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155 19. Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91 20. "Lucky Number Nine, Meaning of Number 9 in Chinese Culture". www.travelchinaguide.com. Retrieved 15 January 2021. 21. Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN 1-4179-6429-4. 22. Jane Dowson (1996). Women's Poetry of the 1930s: A Critical Anthology. Routledge. ISBN 0-415-13095-6. 23. Anthea Fraser (1988). The Nine Bright Shiners. Doubleday. ISBN 0-385-24323-5. 24. Charles Herbert Malden (1905). Recollections of an Eton Colleger, 1898–1902. Spottiswoode. p. 182. nine-bright-shiners. 25. "Surah Al-Isra - 101". Quran.com. Retrieved 17 August 2023. 26. "Surah An-Naml - 12". Quran.com. Retrieved 17 August 2023. 27. "Surah An-Naml - 48". Quran.com. Retrieved 17 August 2023. 28. "Meaning of Numbers in the Bible The Number 9". Bible Study. Archived from the original on 17 November 2007. 29. "Web site for NINE: A Journal of Baseball History & Culture". Archived from the original on 4 November 2009. Retrieved 20 February 2013. 30. Glover, Diane (9 October 2019). "#9 Dream: John Lennon and numerology". www.beatlesstory.com. Beatles Story. Retrieved 6 November 2022. Perhaps the most significant use of the number 9 in John's music was the White Album's 'Revolution 9', an experimental sound collage influenced by the avant-garde style of Yoko Ono and composers such as Edgard Varèse and Karlheinz Stockhausen. It featured a series of tape loops including one with a recurring 'Number Nine' announcement. John said of 'Revolution 9': 'It's an unconscious picture of what I actually think will happen when it happens; just like a drawing of a revolution. One thing was an engineer's testing voice saying, 'This is EMI test series number nine.' I just cut up whatever he said and I'd number nine it. Nine turned out to be my birthday and my lucky number and everything. I didn't realise it: it was just so funny the voice saying, 'number nine'; it was like a joke, bringing number nine into it all the time, that's all it was.' 31. Truax, Barry (2001). Handbook for Acoustic Ecology (Interval). Burnaby: Simon Fraser University. ISBN 1-56750-537-6.. 32. "The Curse of the Ninth Haunted These Composers \| WQXR Editorial". WQXR. 17 October 2016. Retrieved 16 January 2022. Further reading • Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ISBN 3-7913-1933-7, ISBN 978-3-7913-1933-9 Integers 0s • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 100s • 100 • 101 • 102 • 103 • 104 • 105 • 106 • 107 • 108 • 109 • 110 • 111 • 112 • 113 • 114 • 115 • 116 • 117 • 118 • 119 • 120 • 121 • 122 • 123 • 124 • 125 • 126 • 127 • 128 • 129 • 130 • 131 • 132 • 133 • 134 • 135 • 136 • 137 • 138 • 139 • 140 • 141 • 142 • 143 • 144 • 145 • 146 • 147 • 148 • 149 • 150 • 151 • 152 • 153 • 154 • 155 • 156 • 157 • 158 • 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986 • 987 • 988 • 989 • 990 • 991 • 992 • 993 • 994 • 995 • 996 • 997 • 998 • 999 ≥1000 • 1000 • 2000 • 3000 • 4000 • 5000 • 6000 • 7000 • 8000 • 9000 • 10,000 • 20,000 • 30,000 • 40,000 • 50,000 • 60,000 • 70,000 • 80,000 • 90,000 • 100,000 • 1,000,000 • 10,000,000 • 100,000,000 • 1,000,000,000 Authority control: National • Germany • Israel • United States
Celsius The degree Celsius is the unit of temperature on the Celsius scale[1] (originally known as the centigrade scale outside Sweden),[2] one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The degree Celsius (symbol: °C) can refer to a specific temperature on the Celsius scale or a unit to indicate a difference or range between two temperatures. It is named after the Swedish astronomer Anders Celsius (1701–1744), who developed a variant of it in 1742. The unit was called centigrade in several languages (from the Latin centum, which means 100, and gradus, which means steps) for many years. In 1948, the International Committee for Weights and Measures[3] renamed it to honor Celsius and also to remove confusion with the term for one hundredth of a gradian in some languages. Most countries use this scale; the other major scale, Fahrenheit, is still used in the United States, some island territories, and Liberia. The Kelvin scale is of use in the sciences, with 0 K (−273.15 °C) representing absolute zero. degree Celsius A thermometer calibrated in degrees Celsius, showing a temperature of −17 °C General information Unit systemSI Unit oftemperature Symbol°C Named afterAnders Celsius Conversions x °C in ...... corresponds to ... SI base units (x + 273.15) K Imperial/US units (9/5x + 32) °F Since 1743, the Celsius scale has been based on 0 °C for the freezing point of water and 100 °C for the boiling point of water at 1 atm pressure. Prior to 1743 the values were reversed (i.e. the boiling point was 0 degrees and the freezing point was 100 degrees). The 1743 scale reversal was proposed by Jean-Pierre Christin. By international agreement, between 1954 and 2019 the unit degree Celsius and the Celsius scale were defined by absolute zero and the triple point of water. After 2007, it was clarified that this definition referred to Vienna Standard Mean Ocean Water (VSMOW), a precisely defined water standard.[4] This definition also precisely related the Celsius scale to the scale of the kelvin, the SI base unit of thermodynamic temperature with symbol K. Absolute zero, the lowest temperature possible, is defined as being exactly 0 K and −273.15 °C. Until 19 May 2019, the temperature of the triple point of water was defined as exactly 273.16 K (0.01 °C).[5] On 20 May 2019, the kelvin was redefined so that its value is now determined by the definition of the Boltzmann constant rather than being defined by the triple point of VSMOW. This means that the triple point is now a measured value, not a defined value. The newly-defined exact value of the Boltzmann constant was selected so that the measured value of the VSMOW triple point is exactly the same as the older defined value to within the limits of accuracy of contemporary metrology. The temperature in degree Celsius is now defined as the temperature in kelvins subtracted by 273.15,[6][7] meaning that a temperature difference of one degree Celsius and that of one kelvin are exactly the same,[8] and that the degree Celsius remains exactly equal to the kelvin (i.e., 0 °C remains exactly 273.15 K). History In 1742, Swedish astronomer Anders Celsius (1701–1744) created a temperature scale that was the reverse of the scale now known as "Celsius": 0 represented the boiling point of water, while 100 represented the freezing point of water.[9] In his paper Observations of two persistent degrees on a thermometer, he recounted his experiments showing that the melting point of ice is essentially unaffected by pressure. He also determined with remarkable precision how the boiling point of water varied as a function of atmospheric pressure. He proposed that the zero point of his temperature scale, being the boiling point, would be calibrated at the mean barometric pressure at mean sea level. This pressure is known as one standard atmosphere. The BIPM's 10th General Conference on Weights and Measures (CGPM) in 1954 defined one standard atmosphere to equal precisely 1,013,250 dynes per square centimeter (101.325 kPa).[10] In 1743, the Lyonnais physicist Jean-Pierre Christin, permanent secretary of the Academy of Lyon, inverted the Celsius scale so that 0 represented the freezing point of water and 100 represented the boiling point of water. Some credit Christin for independently inventing the reverse of Celsius's original scale, while others believe Christin merely reversed Celsius's scale.[11][12] On 19 May 1743 he published the design of a mercury thermometer, the "Thermometer of Lyon" built by the craftsman Pierre Casati that used this scale.[13][14][15] In 1744, coincident with the death of Anders Celsius, the Swedish botanist Carl Linnaeus (1707–1778) reversed Celsius's scale.[16] His custom-made "linnaeus-thermometer", for use in his greenhouses, was made by Daniel Ekström, Sweden's leading maker of scientific instruments at the time, whose workshop was located in the basement of the Stockholm observatory. As often happened in this age before modern communications, numerous physicists, scientists, and instrument makers are credited with having independently developed this same scale;[17] among them were Pehr Elvius, the secretary of the Royal Swedish Academy of Sciences (which had an instrument workshop) and with whom Linnaeus had been corresponding; Daniel Ekström, the instrument maker; and Mårten Strömer (1707–1770) who had studied astronomy under Anders Celsius. The first known Swedish document[18] reporting temperatures in this modern "forward" Celsius scale is the paper Hortus Upsaliensis dated 16 December 1745 that Linnaeus wrote to a student of his, Samuel Nauclér. In it, Linnaeus recounted the temperatures inside the orangery at the University of Uppsala Botanical Garden: ... since the caldarium (the hot part of the greenhouse) by the angle of the windows, merely from the rays of the sun, obtains such heat that the thermometer often reaches 30 degrees, although the keen gardener usually takes care not to let it rise to more than 20 to 25 degrees, and in winter not under 15 degrees ... Centigrade vis-à-vis Celsius Since the 19th century, the scientific and thermometry communities worldwide have used the phrase "centigrade scale" and temperatures were often reported simply as "degrees" or, when greater specificity was desired, as "degrees centigrade", with the symbol °C. In the French language, the term centigrade also means one hundredth of a gradian, when used for angular measurement. The term centesimal degree was later introduced for temperatures[19] but was also problematic, as it means gradian (one hundredth of a right angle) in the French and Spanish languages. The risk of confusion between temperature and angular measurement was eliminated in 1948 when the 9th meeting of the General Conference on Weights and Measures and the Comité International des Poids et Mesures (CIPM) formally adopted "degree Celsius" for temperature.[20][lower-alpha 1] While "Celsius" is the term commonly used in scientific work, "centigrade" remains in common use in English-speaking countries, especially in informal contexts.[21] While in Australia from 1 September 1972, only Celsius measurements were given for temperature in weather reports/forecasts,[22] it was not until February 1985 that the weather forecasts issued by the BBC switched from "centigrade" to "Celsius".[23] Common temperatures All phase transitions are at standard atmosphere. Figures are either by definition, or approximated from empirical measurements. Key scale relations Kelvin (K)Celsius (°C)Fahrenheit (°F)Rankine (°R) Absolute zero[upper-alpha 1] 0 −273.15 −459.67 0 Intersection of Celsius and Fahrenheit scales[upper-alpha 1] 233.15 −40 −40 419.67 Boiling point of water[lower-alpha 2] 373.1339 99.9839 211.971 671.6410 Boiling point of liquid nitrogen 77.4 −195.8[24] −320.4 139.3 Melting point of ice[25] 273.1499 −0.0001 31.9998 491.6698 Sublimation point of dry ice 195.1 −78 −108.4 351.2 Room temperature[upper-alpha 2][26] 293.15 20.0 68.0 527.69 Average normal human body temperature[27] 310.15 37.0 98.6 558.27 1. Exact value, by SI definition of the kelvin 2. Exact value, by NIST standard definition Name and symbol typesetting The "degree Celsius" has been the only SI unit whose full unit name contains an uppercase letter since 1967, when the SI base unit for temperature became the kelvin, replacing the capitalized term degrees Kelvin. The plural form is "degrees Celsius".[28] The general rule of the International Bureau of Weights and Measures (BIPM) is that the numerical value always precedes the unit, and a space is always used to separate the unit from the number, e.g. "30.2 °C" (not "30.2°C" or "30.2° C").[29] The only exceptions to this rule are for the unit symbols for degree, minute, and second for plane angle (°, ′, and ″, respectively), for which no space is left between the numerical value and the unit symbol.[30] Other languages, and various publishing houses, may follow different typographical rules. Unicode character Unicode provides the Celsius symbol at code point U+2103 ℃ DEGREE CELSIUS. However, this is a compatibility character provided for roundtrip compatibility with legacy encodings. It easily allows correct rendering for vertically written East Asian scripts, such as Chinese. The Unicode standard explicitly discourages the use of this character: "In normal use, it is better to represent degrees Celsius '°C' with a sequence of U+00B0 ° DEGREE SIGN + U+0043 C LATIN CAPITAL LETTER C, rather than U+2103 ℃ DEGREE CELSIUS. For searching, treat these two sequences as identical."[31] Temperatures and intervals The degree Celsius is subject to the same rules as the kelvin with regard to the use of its unit name and symbol. Thus, besides expressing specific temperatures along its scale (e.g. "Gallium melts at 29.7646 °C" and "The temperature outside is 23 degrees Celsius"), the degree Celsius is also suitable for expressing temperature intervals: differences between temperatures or their uncertainties (e.g. "The output of the heat exchanger is hotter by 40 degrees Celsius", and "Our standard uncertainty is ±3 °C").[32] Because of this dual usage, one must not rely upon the unit name or its symbol to denote that a quantity is a temperature interval; it must be unambiguous through context or explicit statement that the quantity is an interval.[lower-alpha 3] This is sometimes solved by using the symbol °C (pronounced "degrees Celsius") for a temperature, and C° (pronounced "Celsius degrees") for a temperature interval, although this usage is non-standard.[33] Another way to express the same is "40 °C ± 3 K", which can be commonly found in literature. Celsius measurement follows an interval system but not a ratio system; and it follows a relative scale not an absolute scale. For example, an object at 20 °C does not have twice the energy of when it is 10 °C; and 0 °C is not the lowest Celsius value. Thus, degrees Celsius is a useful interval measurement but does not possess the characteristics of ratio measures like weight or distance.[34] Coexistence of Kelvin and Celsius scales In science and in engineering, the Celsius scale and the Kelvin scale are often used in combination in close contexts, e.g. "a measured value was 0.01023 °C with an uncertainty of 70 μK". This practice is permissible because the magnitude of the degree Celsius is equal to that of the kelvin. Notwithstanding the official endorsement provided by decision no. 3 of Resolution 3 of the 13th CGPM,[35] which stated "a temperature interval may also be expressed in degrees Celsius", the practice of simultaneously using both °C and K remains widespread throughout the scientific world as the use of SI-prefixed forms of the degree Celsius (such as "μ°C" or "microdegrees Celsius") to express a temperature interval has not been well adopted. Melting and boiling points of water Celsius temperature conversion formulae from Celsius to Celsius Fahrenheit x °C ≘ (x × 9/5 + 32) °F x °F ≘ (x − 32) × 5/9 °C Kelvin x °C ≘ (x + 273.15) K x K ≘ (x − 273.15) °C Rankine x °C ≘ (x + 273.15) × 9/5 °R x °R ≘ (x − 491.67) × 5/9 °C For temperature intervals rather than specific temperatures, 1 °C = 1 K = 9/5 °F = 9/5 °R Conversion between temperature scales The melting and boiling points of water are no longer part of the definition of the Celsius scale. In 1948, the definition was changed to use the triple point of water.[36] In 2005 the definition was further refined to use water with precisely defined isotopic composition (VSMOW) for the triple point. In 2019, the definition was changed to use the Boltzmann constant, completely decoupling the definition of the kelvin from the properties of water. Each of these formal definitions left the numerical values of the Celsius scale identical to the prior definition to within the limits of accuracy of the metrology of the time. When the melting and boiling points of water ceased being part of the definition, they became measured quantities instead. This is also true of the triple point. In 1948 when the 9th General Conference on Weights and Measures (CGPM) in Resolution 3 first considered using the triple point of water as a defining point, the triple point was so close to being 0.01 °C greater than water's known melting point, it was simply defined as precisely 0.01 °C. However, later measurements showed that the difference between the triple and melting points of VSMOW is actually very slightly (< 0.001 °C) greater than 0.01 °C. Thus, the actual melting point of ice is very slightly (less than a thousandth of a degree) below 0 °C. Also, defining water's triple point at 273.16 K precisely defined the magnitude of each 1 °C increment in terms of the absolute thermodynamic temperature scale (referencing absolute zero). Now decoupled from the actual boiling point of water, the value "100 °C" is hotter than 0 °C – in absolute terms – by a factor of exactly 373.15/273.15 (approximately 36.61% thermodynamically hotter). When adhering strictly to the two-point definition for calibration, the boiling point of VSMOW under one standard atmosphere of pressure was actually 373.1339 K (99.9839 °C). When calibrated to ITS-90 (a calibration standard comprising many definition points and commonly used for high-precision instrumentation), the boiling point of VSMOW was slightly less, about 99.974 °C.[37] This boiling-point difference of 16.1 millikelvins between the Celsius scale's original definition and the previous one (based on absolute zero and the triple point) has little practical meaning in common daily applications because water's boiling point is very sensitive to variations in barometric pressure. For example, an altitude change of only 28 cm (11 in) causes the boiling point to change by one millikelvin. See also • Comparison of temperature scales • Degree of frost • Thermodynamic temperature Notes 1. According to The Oxford English Dictionary (OED), the term "Celsius thermometer" had been used at least as early as 1797. Further, the term "The Celsius or Centigrade thermometer" was again used in reference to a particular type of thermometer at least as early as 1850. The OED also cites this 1928 reporting of a temperature: "My altitude was about 5,800 metres, the temperature was 28° Celsius." However, dictionaries seek to find the earliest use of a word or term and are not a useful resource as regards to the terminology used throughout the history of science. According to several writings of Dr. Terry Quinn CBE FRS, Director of the BIPM (1988–2004), including "Temperature Scales from the early days of thermometry to the 21st century" (PDF). Archived from the original (PDF) on 26 December 2010. Retrieved 31 May 2016. (146 KiB) as well as Temperature (2nd Edition/1990/Academic Press/0125696817), the term Celsius in connection with the centigrade scale was not used whatsoever by the scientific or thermometry communities until after the CIPM and CGPM adopted the term in 1948. The BIPM was not even aware that "degree Celsius" was in sporadic, non-scientific use before that time. It is also noteworthy that the twelve-volume, 1933 edition of OED didn't even have a listing for the word Celsius (but did have listings for both centigrade and centesimal in the context of temperature measurement). The 1948 adoption of Celsius accomplished three objectives: 1. All common temperature scales would have their units named after someone closely associated with them; namely, Kelvin, Celsius, Fahrenheit, Réaumur and Rankine. 2. Notwithstanding the important contribution of Linnaeus who gave the Celsius scale its modern form, Celsius's name was the obvious choice because it began with the letter C. Thus, the symbol °C that for centuries had been used in association with the name centigrade could remain in use and would simultaneously inherit an intuitive association with the new name. 3. The new name eliminated the ambiguity of the term "centigrade", freeing it to refer exclusively to the French-language name for the unit of angular measurement. 2. For Vienna Standard Mean Ocean Water at one standard atmosphere (101.325 kPa) when calibrated solely per the two-point definition of thermodynamic temperature. Older definitions of the Celsius scale once defined the boiling point of water under one standard atmosphere as being precisely 100 °C. However, the current definition results in a boiling point that is actually 16.1 mK less. For more about the actual boiling point of water, see VSMOW in temperature measurement. A different approximation uses ITS-90, which approximates the temperature to 99.974 °C 3. In 1948, Resolution 7 of the 9th CGPM stated, "To indicate a temperature interval or difference, rather than a temperature, the word 'degree' in full, or the abbreviation 'deg' must be used." This resolution was abrogated in 1967/1968 by Resolution 3 of the 13th CGPM, which stated that ["The names "degree Kelvin" and "degree", the symbols "°K" and "deg" and the rules for their use given in Resolution 7 of the 9th CGPM (1948),] ...and the designation of the unit to express an interval or a difference of temperatures are abrogated, but the usages which derive from these decisions remain permissible for the time being." Consequently, there is now wide freedom in usage regarding how to indicate a temperature interval. The most important thing is that one's intention must be clear and the basic rule of the SI must be followed; namely that the unit name or its symbol must not be relied upon to indicate the nature of the quantity. Thus, if a temperature interval is, say, 10 K or 10 °C (which may be written 10 kelvins or 10 degrees Celsius), it must be unambiguous through obvious context or explicit statement that the quantity is an interval. Rules governing the expressing of temperatures and intervals are covered in the BIPM's "SI Brochure, 8th edition" (PDF). (1.39 MiB). References 1. "Celsius temperature scale". Encyclopædia Britannica. Retrieved 19 February 2012. Celsius temperature scale, also called centigrade temperature scale, scale based on 0 ° for the melting point of water and 100 ° for the boiling point of water at 1 atm pressure. 2. Helmenstine, Anne Marie (15 December 2014). "What Is the Difference Between Celsius and Centigrade?". Chemistry.about.com. About.com. Retrieved 25 April 2020. 3. "Proceedings of the 42nd CIPM (1948), 1948, p. 88". Bureau International des Poids et Mesures. 1948. Retrieved 19 August 2023. 4. "Resolution 10 of the 23rd CGPM (2007)". Retrieved 27 December 2021. 5. "SI brochure, section 2.1.1.5". International Bureau of Weights and Measures. Archived from the original on 26 September 2007. Retrieved 9 May 2008. 6. "SI Brochure: The International System of Units (SI) – 9th edition". BIPM. Retrieved 21 February 2022. 7. "SI base unit: kelvin (K)". bipm.org. BIPM. Retrieved 5 March 2022. 8. "Essentials of the SI: Base & derived units". Retrieved 9 May 2008. 9. Celsius, Anders (1742) "Observationer om twänne beständiga grader på en thermometer" (Observations about two stable degrees on a thermometer), Kungliga Svenska Vetenskapsakademiens Handlingar (Proceedings of the Royal Swedish Academy of Sciences), 3 : 171–180 and Fig. 1. 10. "Resolution 4 of the 10th meeting of the CGPM (1954)". 11. Don Rittner; Ronald A. Bailey (2005): Encyclopedia of Chemistry. Facts On File, Manhattan, New York City. p. 43. 12. Smith, Jacqueline (2009). "Appendix I: Chronology". The Facts on File Dictionary of Weather and Climate. Infobase Publishing. p. 246. ISBN 978-1-4381-0951-0. 1743 Jean-Pierre Christin inverts the fixed points on Celsius' scale, to produce the scale used today. 13. Mercure de France (1743): MEMOIRE sur la dilatation du Mercure dans le Thermométre. Chaubert; Jean de Nully, Pissot, Duchesne, Paris. pp. 1609–1610. 14. Journal helvétique (1743): LION. Imprimerie des Journalistes, Neuchâtel. pp. 308–310. 15. Memoires pour L'Histoire des Sciences et des Beaux Arts (1743): DE LYON. Chaubert, París. pp. 2125–2128. 16. Citation: Uppsala University (Sweden), Linnaeus' thermometer 17. Citation for Christin of Lyons: Le Moyne College, Glossary, (Celsius scale); citation for Linnaeus's connection with Pehr Elvius and Daniel Ekström: Uppsala University (Sweden), Linnaeus' thermometer; general citation: The Uppsala Astronomical Observatory, History of the Celsius temperature scale 18. Citations: University of Wisconsin–Madison, Linnæus & his Garden and; Uppsala University, Linnaeus' thermometer 19. Comptes rendus des séances de la cinquième conférence générale des poids et mesures, réunie à Paris en 1913. Bureau international des poids et mesures. 1913. pp. 55, 57, 59. Retrieved 10 June 2021. p. 60: ...à la température de 20° centésimaux 20. "CIPM, 1948 and 9th CGPM, 1948". International Bureau of Weights and Measures. Archived from the original on 5 April 2021. Retrieved 9 May 2008. 21. "centigrade, adj. and n." Oxford English Dictionary. Oxford University Press. Retrieved 20 November 2011. 22. "Temperature and Pressure go Metric" (PDF). Commonwealth Bureau of Meteorology. 1 September 1972. Retrieved 16 February 2022. 23. 1985 BBC Special: A Change In The Weather on YouTube 24. Lide, D.R., ed. (1990–1991). Handbook of Chemistry and Physics. 71st ed. CRC Press. p. 4–22. 25. The ice point of purified water has been measured at 0.000089(10) degrees Celsius – see Magnum, B.W. (June 1995). "Reproducibility of the Temperature of the Ice Point in Routine Measurements" (PDF). Nist Technical Note. 1411. Archived from the original (PDF) on 10 July 2007. Retrieved 11 February 2007. 26. "SI Units – Temperature". NIST Office of Weights and Measures. 2010. Retrieved 21 July 2022. 27. Elert, Glenn (2005). "Temperature of a Healthy Human (Body Temperature)". The Physics Factbook. Retrieved 22 August 2007. 28. "Unit of thermodynamic temperature (kelvin)". The NIST Reference on Constants, Units, and Uncertainty: Historical context of the SI. National Institute of Standards and Technology (NIST). 2000. Archived from the original on 11 November 2004. Retrieved 16 November 2011. 29. BIPM, SI Brochure, Section 5.3.3. 30. For more information on conventions used in technical writing, see the informative SI Unit rules and style conventions by the NIST as well as the BIPM's SI brochure: Subsection 5.3.3, Formatting the value of a quantity. Archived 5 July 2014 at the Wayback Machine 31. "22.2". The Unicode Standard, Version 9.0 (PDF). Mountain View, CA, USA: The Unicode Consortium. July 2016. ISBN 978-1-936213-13-9. Retrieved 20 April 2017. 32. Decision No. 3 of Resolution 3 of the 13th CGPM. 33. H.D. Young, R. A. Freedman (2008). University Physics with Modern Physics (12th ed.). Addison Wesley. p. 573. 34. This fact is demonstrated in the book Biostatistics: A Guide to Design, Analysis, and Discovery By Ronald N. Forthofer, Eun Sul Lee and Mike Hernandez 35. "Resolution 3 of the 13th CGPM (1967)". 36. "Resolution 3 of the 9th CGPM (1948)". International Bureau of Weights and Measures. Retrieved 9 May 2008. 37. Citation: London South Bank University, Water Structure and Behavior, notes c1 and c2 External links The dictionary definition of Celsius at Wiktionary • NIST, Basic unit definitions: Kelvin • The Uppsala Astronomical Observatory, History of the Celsius temperature scale • London South Bank University, Water, scientific data • BIPM, SI brochure, section 2.1.1.5, Unit of thermodynamic temperature Scales of temperature • Celsius • Delisle • Fahrenheit • Gas mark • Kelvin • Leiden • Newton • Rankine • Réaumur • Rømer • Wedgwood Conversion formulas and comparison SI units Base units • ampere • candela • kelvin • kilogram • metre • mole • second Derived units with special names • becquerel • coulomb • degree Celsius • farad • gray • henry • hertz • joule • katal • lumen • lux • newton • ohm • pascal • radian • siemens • sievert • steradian • tesla • volt • watt • weber Other accepted units • astronomical unit • dalton • day • decibel • degree of arc • electronvolt • hectare • hour • litre • minute • minute and second of arc • neper • tonne See also • Conversion of units • Metric prefixes • 2005–2019 definition • 2019 redefinition • Systems of measurement • Category
ℓ-adic sheaf In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme X is an inverse system consisting of $\mathbb {Z} /\ell ^{n}$-modules $F_{n}$ in the étale topology and $F_{n+1}\to F_{n}$ inducing $F_{n+1}\otimes _{\mathbb {Z} /\ell ^{n+1}}\mathbb {Z} /\ell ^{n}{\overset {\simeq }{\to }}F_{n}$.[1][2] Bhatt–Scholze's pro-étale topology gives an alternative approach.[3] Motivation The development of étale cohomology as a whole was fueled by the desire to produce a 'topological' theory of cohomology for algebraic varieties, i.e. a Weil cohomology theory that works in any characteristic. An essential feature of such a theory is that it admits coefficients in a field of characteristic 0. However, constant étale sheaves with no torsion have no interesting cohomology. For example, if $X$ is a smooth variety over a field $k$, then $H^{i}(X_{\text{ét}},\mathbb {Q} )=0$ for all positive $i$. On the other hand, the constant sheaves $\mathbb {Z} /m$ do produce the 'correct' cohomology, as long as $m$ is invertible in the ground field $k$. So one takes a prime $\ell $ for which this is true and defines $\ell $-adic cohomology as $H^{i}(X_{\text{ét}},\mathbb {Z} _{\ell }):=\varprojlim _{n}H^{i}(X_{\text{ét}},\mathbb {Z} /\ell ^{n}){\text{, and }}H^{i}(X_{\text{ét}},\mathbb {Q} _{\ell }):=\varprojlim _{n}H^{i}(X_{\text{ét}},\mathbb {Z} /\ell ^{n})\otimes \mathbb {Q} $. This definition, however, is not completely satisfactory: As in the classical case of topological spaces, one might want to consider cohomology with coefficients in a local system of $\mathbb {Q} _{\ell }$-vector spaces, and there should be a category equivalence between such local systems and continuous $\mathbb {Q} _{\ell }$-representations of the étale fundamental group. Another problem with the definition above is that it behaves well only when $k$ is a separably closed. In this case, all the groups occurring in the inverse limit are finitely generated and taking the limit is exact. But if $k$ is for example a number field, the cohomology groups $H^{i}(X_{\text{ét}},\mathbb {Z} /\ell ^{n})$ will often be infinite and the limit not exact, which causes issues with functoriality. For instance, there is in general no Hochschild-Serre spectral sequence relating $H^{i}(X_{\text{ét}},\mathbb {Z} _{\ell })$ to the Galois cohomology of $H^{i}((X_{k^{\text{sep}}})_{\text{ét}},\mathbb {Z} _{\ell })$.[4] These considerations lead one to consider the category of inverse systems of sheaves as described above. One has then the desired equivalence of categories with representations of the fundamental group (for $\mathbb {Z} _{\ell }$-local systems, and when $X$ is normal for $\mathbb {Q} _{\ell }$-systems as well), and the issue in the last paragraph is resolved by so-called continuous étale cohomology, where one takes the derived functor of the composite functor of taking the limit over global sections of the system. Constructible and lisse ℓ-adic sheaves An ℓ-adic sheaf $\{F_{n}\}_{\geq 0}$ is said to be • constructible if each $F_{n}$ is constructible. • lisse if each $F_{n}$ is constructible and locally constant. Some authors (e.g., those of SGA 41⁄2)[5] assume an ℓ-adic sheaf to be constructible. Given a connected scheme X with a geometric point x, SGA 1 defines the étale fundamental group $\pi _{1}^{\text{ét}}(X,x)$ of X at x to be the group classifying finite Galois coverings of X. Then the category of lisse ℓ-adic sheaves on X is equivalent to the category of continuous representations of $\pi _{1}^{\text{ét}}(X,x)$ on finite free $\mathbb {Z} _{l}$-modules. This is an analog of the correspondence between local systems and continuous representations of the fundament group in algebraic topology (because of this, a lisse ℓ-adic sheaf is sometimes also called a local system). ℓ-adic cohomology An ℓ-adic cohomology groups is an inverse limit of étale cohomology groups with certain torsion coefficients. The "derived category" of constructible ℓ-adic sheaves In a way similar to that for ℓ-adic cohomology, the derived category of constructible ${\overline {\mathbb {Q} }}_{\ell }$-sheaves is defined essentially as $D_{c}^{b}(X,{\overline {\mathbb {Q} }}_{\ell }):=(\varprojlim _{n}D_{c}^{b}(X,\mathbb {Z} /\ell ^{n}))\otimes _{\mathbb {Z} _{\ell }}{\overline {\mathbb {Q} }}_{\ell }.$ (Scholze & Bhatt 2013) writes "in daily life, one pretends (without getting into much trouble) that $D_{c}^{b}(X,{\overline {\mathbb {Q} }}_{\ell })$ is simply the full subcategory of some hypothetical derived category $D(X,{\overline {\mathbb {Q} }}_{\ell })$ ..." See also • Fourier–Deligne transform References 1. Milne, James S. (1980-04-21). Etale Cohomology (PMS-33). Princeton University Press. p. 163. ISBN 978-0-691-08238-7.{{cite book}}: CS1 maint: url-status (link) 2. Stacks Project, Tag 03UL. 3. Scholze, Peter; Bhatt, Bhargav (2013-09-04). "The pro-étale topology for schemes". arXiv:1309.1198v2 [math.AG]. 4. Jannsen, Uwe (1988). "Continuous Étale Cohomology". Mathematische Annalen. 280 (2): 207–246. ISSN 0025-5831. 5. Deligne, Pierre (1977). Séminaire de Géométrie Algébrique du Bois Marie – Cohomologie étale – (SGA 4½). Lecture Notes in Mathematics (in French). Vol. 569. Berlin; New York: Springer-Verlag. pp. iv+312. doi:10.1007/BFb0091516. ISBN 978-3-540-08066-4. MR 0463174. • Exposé V, VI of Illusie, Luc, ed. (1977). Séminaire de Géométrie Algébrique du Bois Marie – 1965–66 – Cohomologie ℓ-adique et Fonctions L – (SGA 5). Lecture notes in mathematics (in French). Vol. 589. Berlin; New York: Springer-Verlag. xii+484. doi:10.1007/BFb0096802. ISBN 3-540-08248-4. MR 0491704. • J. S. Milne (1980), Étale cohomology, Princeton, N.J: Princeton University Press, ISBN 0-691-08238-3 External links • Mathoverflow: A nice explanation of what is a smooth (ℓ-adic) sheaf? • Number theory learning seminar 2016–2017 at Stanford
Module (mathematics) In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Algebraic structure → Ring theory Ring theory Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring $\mathbb {Z} $ • Terminal ring $0=\mathbb {Z} _{1}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree p-adic number theory and decimals • Direct limit/Inverse limit • Zero ring $\mathbb {Z} _{1}$ • Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$ • Base-p circle ring $\mathbb {T} $ • Base-p integers $\mathbb {Z} $ • p-adic rationals $\mathbb {Z} [1/p]$ • Base-p real numbers $\mathbb {R} $ • p-adic integers $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • p-adic solenoid $\mathbb {T} _{p}$ Algebraic geometry • Affine variety Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra Algebraic structures Group-like • Group • Semigroup / Monoid • Rack and quandle • Quasigroup and loop • Abelian group • Magma • Lie group Group theory Ring-like • Ring • Rng • Semiring • Near-ring • Commutative ring • Domain • Integral domain • Field • Division ring • Lie ring Ring theory Lattice-like • Lattice • Semilattice • Complemented lattice • Total order • Heyting algebra • Boolean algebra • Map of lattices • Lattice theory Module-like • Module • Group with operators • Vector space • Linear algebra Algebra-like • Algebra • Associative • Non-associative • Composition algebra • Lie algebra • Graded • Bialgebra • Hopf algebra Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules. Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do, free modules, need not have a unique rank if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the axiom of choice in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as Lp spaces.) Formal definition Suppose that R is a ring, and 1 is its multiplicative identity. A left R-module M consists of an abelian group (M, +) and an operation · : R × M → M such that for all r, s in R and x, y in M, we have 1. $r\cdot (x+y)=r\cdot x+r\cdot y$ 2. $(r+s)\cdot x=r\cdot x+s\cdot x$ 3. $(rs)\cdot x=r\cdot (s\cdot x)$ 4. $1\cdot x=x.$ The operation · is called scalar multiplication. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in R. One may write RM to emphasize that M is a left R-module. A right R-module MR is defined similarly in terms of an operation · : M × R → M. Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left R-modules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be unital.[1] An (R,S)-bimodule is an abelian group together with both a left scalar multiplication · by elements of R and a right scalar multiplication ∗ by elements of S, making it simultaneously a left R-module and a right S-module, satisfying the additional condition (r · x) ∗ s = r ⋅ (x ∗ s) for all r in R, x in M, and s in S. If R is commutative, then left R-modules are the same as right R-modules and are simply called R-modules. Examples • If K is a field, then K-vector spaces (vector spaces over K) and K-modules are identical. • If K is a field, and K[x] a univariate polynomial ring, then a K[x]-module M is a K-module with an additional action of x on M that commutes with the action of K on M. In other words, a K[x]-module is a K-vector space M combined with a linear map from M to M. Applying the structure theorem for finitely generated modules over a principal ideal domain to this example shows the existence of the rational and Jordan canonical forms. • The concept of a Z-module agrees with the notion of an abelian group. That is, every abelian group is a module over the ring of integers Z in a unique way. For n > 0, let n ⋅ x = x + x + ... + x (n summands), 0 ⋅ x = 0, and (−n) ⋅ x = −(n ⋅ x). Such a module need not have a basis—groups containing torsion elements do not. (For example, in the group of integers modulo 3, one cannot find even one element which satisfies the definition of a linearly independent set since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a finite field is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.) • The decimal fractions (including negative ones) form a module over the integers. Only singletons are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no rank. • If R is any ring and n a natural number, then the cartesian product Rn is both a left and right R-module over R if we use the component-wise operations. Hence when n = 1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial R-module {0} consisting only of its identity element. Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module. • If Mn(R) is the ring of n × n matrices over a ring R, M is an Mn(R)-module, and ei is the n × n matrix with 1 in the (i, i)-entry (and zeros elsewhere), then eiM is an R-module, since reim = eirm ∈ eiM. So M breaks up as the direct sum of R-modules, M = e1M ⊕ ... ⊕ enM. Conversely, given an R-module M0, then M0⊕n is an Mn(R)-module. In fact, the category of R-modules and the category of Mn(R)-modules are equivalent. The special case is that the module M is just R as a module over itself, then Rn is an Mn(R)-module. • If S is a nonempty set, M is a left R-module, and MS is the collection of all functions f : S → M, then with addition and scalar multiplication in MS defined pointwise by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(s), MS is a left R-module. The right R-module case is analogous. In particular, if R is commutative then the collection of R-module homomorphisms h : M → N (see below) is an R-module (and in fact a submodule of NM). • If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C∞(X). The set of all smooth vector fields defined on X form a module over C∞(X), and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over C∞(X), and by Swan's theorem, every projective module is isomorphic to the module of sections of some bundle; the category of C∞(X)-modules and the category of vector bundles over X are equivalent. • If R is any ring and I is any left ideal in R, then I is a left R-module, and analogously right ideals in R are right R-modules. • If R is a ring, we can define the opposite ring Rop which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in Rop. Any left R-module M can then be seen to be a right module over Rop, and any right module over R can be considered a left module over Rop. • Modules over a Lie algebra are (associative algebra) modules over its universal enveloping algebra. • If R and S are rings with a ring homomorphism φ : R → S, then every S-module M is an R-module by defining rm = φ(r)m. In particular, S itself is such an R-module. Submodules and homomorphisms Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or more explicitly an R-submodule) if for any n in N and any r in R, the product r ⋅ n (or n ⋅ r for a right R-module) is in N. If X is any subset of an R-module M, then the submodule spanned by X is defined to be $ \langle X\rangle =\,\bigcap _{N\supseteq X}N$ where N runs over the submodules of M which contain X, or explicitly $ \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}$, which is important in the definition of tensor products.[2] The set of submodules of a given module M, together with the two binary operations + and ∩, forms a lattice which satisfies the modular law: Given submodules U, N1, N2 of M such that N1 ⊂ N2, then the following two submodules are equal: (N1 + U) ∩ N2 = N1 + (U ∩ N2). If M and N are left R-modules, then a map f : M → N is a homomorphism of R-modules if for any m, n in M and r, s in R, $f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)$. This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects. Another name for a homomorphism of R-modules is an R-linear map. A bijective module homomorphism f : M → N is called a module isomorphism, and the two modules M and N are called isomorphic. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements. The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f, and the image of f is the submodule of N consisting of values f(m) for all elements m of M.[3] The isomorphism theorems familiar from groups and vector spaces are also valid for R-modules. Given a ring R, the set of all left R-modules together with their module homomorphisms forms an abelian category, denoted by R-Mod (see category of modules). Types of modules Finitely generated An R-module M is finitely generated if there exist finitely many elements x1, ..., xn in M such that every element of M is a linear combination of those elements with coefficients from the ring R. Cyclic A module is called a cyclic module if it is generated by one element. Free A free R-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces. Projective Projective modules are direct summands of free modules and share many of their desirable properties. Injective Injective modules are defined dually to projective modules. Flat A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible.[4] Semisimple A semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called completely reducible. Indecomposable An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. uniform modules). Faithful A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. r ⋅ x ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal. Torsion-free A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies r = 0 or m = 0. Noetherian A Noetherian module is a module which satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated. Artinian An Artinian module is a module which satisfies the descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps. Graded A graded module is a module with a decomposition as a direct sum M = ⨁x Mx over a graded ring R = ⨁x Rx such that RxMy ⊂ Mx+y for all x and y. Uniform A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection. Further notions Relation to representation theory A representation of a group G over a field k is a module over the group ring k[G]. If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case of a right module), and is necessarily a group endomorphism of the abelian group (M, +). The set of all group endomorphisms of M is denoted EndZ(M) and forms a ring under addition and composition, and sending a ring element r of R to its action actually defines a ring homomorphism from R to EndZ(M). Such a ring homomorphism R → EndZ(M) is called a representation of R over the abelian group M; an alternative and equivalent way of defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R over it. Such a representation R → EndZ(M) may also be called a ring action of R on M. A representation is called faithful if and only if the map R → EndZ(M) is injective. In terms of modules, this means that if r is an element of R such that rx = 0 for all x in M, then r = 0. Every abelian group is a faithful module over the integers or over some ring of integers modulo n, Z/nZ. Generalizations A ring R corresponds to a preadditive category R with a single object. With this understanding, a left R-module is just a covariant additive functor from R to the category Ab of abelian groups, and right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C. These functors form a functor category C-Mod which is the natural generalization of the module category R-Mod. Modules over commutative rings can be generalized in a different direction: take a ringed space (X, OX) and consider the sheaves of OX-modules (see sheaf of modules). These form a category OX-Mod, and play an important role in modern algebraic geometry. If X has only a single point, then this is a module category in the old sense over the commutative ring OX(X). One can also consider modules over a semiring. Modules over rings are abelian groups, but modules over semirings are only commutative monoids. Most applications of modules are still possible. In particular, for any semiring S, the matrices over S form a semiring over which the tuples of elements from S are a module (in this generalized sense only). This allows a further generalization of the concept of vector space incorporating the semirings from theoretical computer science. Over near-rings, one can consider near-ring modules, a nonabelian generalization of modules. See also • Group ring • Algebra (ring theory) • Module (model theory) • Module spectrum • Annihilator Notes 1. Dummit, David S. & Foote, Richard M. (2004). Abstract Algebra. Hoboken, NJ: John Wiley & Sons, Inc. ISBN 978-0-471-43334-7. 2. Mcgerty, Kevin (2016). "ALGEBRA II: RINGS AND MODULES" (PDF). 3. Ash, Robert. "Module Fundamentals" (PDF). Abstract Algebra: The Basic Graduate Year. 4. Jacobson (1964), p. 4, Def. 1 References • F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, ISBN 0-387-97845-3, ISBN 3-540-97845-3 • Nathan Jacobson. Structure of rings. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964, ISBN 978-0-8218-1037-8 External links • "Module", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • module at the nLab Authority control International • FAST National • Spain • France • BnF data • Israel • United States • Japan • Czech Republic Other • IdRef
Maplet A maplet or maplet arrow (symbol: ↦, commonly pronounced "maps to") is a symbol consisting of a vertical line with a rightward-facing arrow. It is used in mathematics and in computer science to denote functions (the expression x ↦ y is also called a maplet). One example of use of the maplet is in Z notation, a formal specification language used in software development.[1] ↦ Maplet In the Unicode character set, the maplet is at the point U+21A6.[2] See also • Arrow notation – e.g., $x\mapsto x+1$, also known as map References 1. Mikušiak, Luboš; Miroslav Adámy; Thomas Seidmann (1997). "Publishing formal specifications in Z notation on world wide web". TAPSOFT '97: Theory and Practice of Software Development. Lecture Notes in Computer Science. Vol. 1214. pp. 871–874. doi:10.1007/BFb0030650. ISBN 978-3-540-62781-4. 2. Unicode Character 'RIGHTWARDS ARROW FROM BAR' (U+21A6) Common punctuation marks and other typographical marks or symbols • space • , comma • : colon • ; semicolon • ‐ hyphen • ’ ' apostrophe • ′ ″ ‴ prime • . full stop • & ampersand • @ at sign • ^ caret • / slash • \ backslash • … ellipsis • * asterisk • ⁂ asterism • * * * dinkus • - hyphen-minus • ‒ – — dash • = ⸗ double hyphen • ? question mark • ! exclamation mark • ‽ interrobang • ¡ ¿ inverted ! and ? • ⸮ irony punctuation • # number sign • № numero sign • º ª ordinal indicator • % percent sign • ‰ per mille • ‱ basis point • ° degree symbol • ⌀ diameter sign • + − plus and minus signs • × multiplication sign • ÷ division sign • ~ tilde • ± plus–minus sign • ∓ minus-plus sign • _ underscore • ⁀ tie • \| ¦ ‖ vertical bar • • bullet • · interpunct • © copyright symbol • © copyleft • ℗ sound recording copyright • ® registered trademark • SM service mark symbol • TM trademark symbol • ‘ ’ “ ” ' ' " " quotation mark • ‹ › « » guillemet • ( ) [ ] { } ⟨ ⟩ bracket • ” 〃 ditto mark • † ‡ dagger • ❧ hedera/floral heart • ☞ manicule • ◊ lozenge • ¶ ⸿ pilcrow (paragraph mark) • ※ reference mark • § section mark • Version of this table as a sortable list • Currency symbols • Diacritics (accents) • Logic symbols • Math symbols • Whitespace • Chinese punctuation • Hebrew punctuation • Japanese punctuation • Korean punctuation
If and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence),[1] and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, P if and only if Q means that P is true whenever Q is true, and the only case in which P is true is if Q is also true, whereas in the case of P if Q, there could be other scenarios where P is true and Q is false. ↔⇔≡⟺ Logical symbols representing iff In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q.[2] Some authors regard "iff" as unsuitable in formal writing;[3] others consider it a "borderline case" and tolerate its use.[4] In logical formulae, logical symbols, such as $\leftrightarrow $ and $\Leftrightarrow $,[5] are used instead of these phrases; see § Notation below. Definition The truth table of P $\Leftrightarrow $ Q is as follows:[6][7] Truth table P Q P $\Rightarrow $ Q P $\Leftarrow $ Q P $\Leftrightarrow $ Q TTTTT TFFTF FTTFF FFTTT It is equivalent to that produced by the XNOR gate, and opposite to that produced by the XOR gate.[8] Usage Notation The corresponding logical symbols are "$\leftrightarrow $", "$\Leftrightarrow $",[5] and $\equiv $,[9] and sometimes "iff". These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas (e.g., in metalogic). In Łukasiewicz's Polish notation, it is the prefix symbol $E$.[10] Another term for the logical connective, i.e., the symbol in logic formulas, is exclusive nor. In TeX, "if and only if" is shown as a long double arrow: $\iff $ via command \iff or \Longleftrightarrow.[11] Proofs In most logical systems, one proves a statement of the form "P iff Q" by proving either "if P, then Q" and "if Q, then P", or "if P, then Q" and "if not-P, then not-Q". Proving these pairs of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. An alternative is to prove the disjunction "(P and Q) or (not-P and not-Q)", which itself can be inferred directly from either of its disjuncts—that is, because "iff" is truth-functional, "P iff Q" follows if P and Q have been shown to be both true, or both false. Origin of iff and pronunciation Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology.[12] Its invention is often credited to Paul Halmos, who wrote "I invented 'iff,' for 'if and only if'—but I could never believe I was really its first inventor."[13] It is somewhat unclear how "iff" was meant to be pronounced. In current practice, the single 'word' "iff" is almost always read as the four words "if and only if". However, in the preface of General Topology, Kelley suggests that it should be read differently: "In some cases where mathematical content requires 'if and only if' and euphony demands something less I use Halmos' 'iff'". The authors of one discrete mathematics textbook suggest:[14] "Should you need to pronounce iff, really hang on to the 'ff' so that people hear the difference from 'if'", implying that "iff" could be pronounced as [ɪfː]. Usage in definitions Technically, definitions are "if and only if" statements; some texts — such as Kelley's General Topology — follow the strict demands of logic, and use "if and only if" or iff in definitions of new terms.[15] However, this logically correct usage of "if and only if" is relatively uncommon and overlooks the linguistic fact that the "if" of a definition is interpreted as meaning "if and only if". The majority of textbooks, research papers and articles (including English Wikipedia articles) follow the linguistic convention to interpret "if" as "if and only if" whenever a mathematical definition is involved (as in "a topological space is compact if every open cover has a finite subcover").[16] Distinction from "if" and "only if" • "Madison will eat the fruit if it is an apple." (equivalent to "Only if Madison will eat the fruit, can it be an apple" or "Madison will eat the fruit ← the fruit is an apple") This states that Madison will eat fruits that are apples. It does not, however, exclude the possibility that Madison might also eat bananas or other types of fruit. All that is known for certain is that she will eat any and all apples that she happens upon. That the fruit is an apple is a sufficient condition for Madison to eat the fruit. • "Madison will eat the fruit only if it is an apple." (equivalent to "If Madison will eat the fruit, then it is an apple" or "Madison will eat the fruit → the fruit is an apple") This states that the only fruit Madison will eat is an apple. It does not, however, exclude the possibility that Madison will refuse an apple if it is made available, in contrast with (1), which requires Madison to eat any available apple. In this case, that a given fruit is an apple is a necessary condition for Madison to be eating it. It is not a sufficient condition since Madison might not eat all the apples she is given. • "Madison will eat the fruit if and only if it is an apple." (equivalent to "Madison will eat the fruit ↔ the fruit is an apple") This statement makes it clear that Madison will eat all and only those fruits that are apples. She will not leave any apple uneaten, and she will not eat any other type of fruit. That a given fruit is an apple is both a necessary and a sufficient condition for Madison to eat the fruit. Sufficiency is the converse of necessity. That is to say, given P→Q (i.e. if P then Q), P would be a sufficient condition for Q, and Q would be a necessary condition for P. Also, given P→Q, it is true that ¬Q→¬P (where ¬ is the negation operator, i.e. "not"). This means that the relationship between P and Q, established by P→Q, can be expressed in the following, all equivalent, ways: P is sufficient for Q Q is necessary for P ¬Q is sufficient for ¬P ¬P is necessary for ¬Q As an example, take the first example above, which states P→Q, where P is "the fruit in question is an apple" and Q is "Madison will eat the fruit in question". The following are four equivalent ways of expressing this very relationship: If the fruit in question is an apple, then Madison will eat it. Only if Madison will eat the fruit in question, is it an apple. If Madison will not eat the fruit in question, then it is not an apple. Only if the fruit in question is not an apple, will Madison not eat it. Here, the second example can be restated in the form of if...then as "If Madison will eat the fruit in question, then it is an apple"; taking this in conjunction with the first example, we find that the third example can be stated as "If the fruit in question is an apple, then Madison will eat it; and if Madison will eat the fruit, then it is an apple". In terms of Euler diagrams • A is a proper subset of B. A number is in A only if it is in B; a number is in B if it is in A. • C is a subset but not a proper subset of B. A number is in B if and only if it is in C, and a number is in C if and only if it is in B. Euler diagrams show logical relationships among events, properties, and so forth. "P only if Q", "if P then Q", and "P→Q" all mean that P is a subset, either proper or improper, of Q. "P if Q", "if Q then P", and Q→P all mean that Q is a proper or improper subset of P. "P if and only if Q" and "Q if and only if P" both mean that the sets P and Q are identical to each other. More general usage Iff is used outside the field of logic as well. Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition). The elements of X are all and only the elements of Y means: "For any z in the domain of discourse, z is in X if and only if z is in Y." See also • Equivalence relation • Logical biconditional • Logical equality • Logical equivalence • Polysyllogism References 1. Copi, I. M.; Cohen, C.; Flage, D. E. (2006). Essentials of Logic (Second ed.). Upper Saddle River, NJ: Pearson Education. p. 197. ISBN 978-0-13-238034-8. 2. Weisstein, Eric W. "Iff." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Iff.html Archived 13 November 2018 at the Wayback Machine 3. E.g. Daepp, Ulrich; Gorkin, Pamela (2011), Reading, Writing, and Proving: A Closer Look at Mathematics, Undergraduate Texts in Mathematics, Springer, p. 52, ISBN 9781441994790, While it can be a real time-saver, we don't recommend it in formal writing. 4. Rothwell, Edward J.; Cloud, Michael J. (2014), Engineering Writing by Design: Creating Formal Documents of Lasting Value, CRC Press, p. 98, ISBN 9781482234312, It is common in mathematical writing 5. Peil, Timothy. "Conditionals and Biconditionals". web.mnstate.edu. Archived from the original on 24 October 2020. Retrieved 4 September 2020. 6. p <=> q Archived 18 October 2016 at the Wayback Machine. Wolfram\|Alpha 7. If and only if, UHM Department of Mathematics, archived from the original on 5 May 2000, retrieved 16 October 2016, Theorems which have the form "P if and only Q" are much prized in mathematics. They give what are called "necessary and sufficient" conditions, and give completely equivalent and hopefully interesting new ways to say exactly the same thing. 8. "XOR/XNOR/Odd Parity/Even Parity Gate". www.cburch.com. Archived from the original on 7 April 2022. Retrieved 22 October 2019. 9. Weisstein, Eric W. "Equivalent". mathworld.wolfram.com. Archived from the original on 3 October 2020. Retrieved 4 September 2020. 10. "Jan Łukasiewicz > Łukasiewicz's Parenthesis-Free or Polish Notation (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Archived from the original on 9 August 2019. Retrieved 22 October 2019. 11. "LaTeX:Symbol". Art of Problem Solving. Archived from the original on 22 October 2019. Retrieved 22 October 2019. 12. General Topology, reissue ISBN 978-0-387-90125-1 13. Nicholas J. Higham (1998). Handbook of writing for the mathematical sciences (2nd ed.). SIAM. p. 24. ISBN 978-0-89871-420-3. 14. Maurer, Stephen B.; Ralston, Anthony (2005). Discrete Algorithmic Mathematics (3rd ed.). Boca Raton, Fla.: CRC Press. p. 60. ISBN 1568811667. 15. For instance, from General Topology, p. 25: "A set is countable iff it is finite or countably infinite." [boldface in original] 16. Krantz, Steven G. (1996), A Primer of Mathematical Writing, American Mathematical Society, p. 71, ISBN 978-0-8218-0635-7 External links Wikimedia Commons has media related to If and only if. • "Tables of truth for if and only if". Archived from the original on 5 May 2000. • Language Log: "Just in Case" • Southern California Philosophy for philosophy graduate students: "Just in Case" Common logical symbols ∧ or & and ∨ or ¬ or ~ not → implies ⊃ implies, superset ↔ or ≡ iff \| nand ∀ universal quantification ∃ existential quantification ⊤ true, tautology ⊥ false, contradiction ⊢ entails, proves ⊨ entails, therefore ∴ therefore ∵ because Philosophy portal Mathematics portal
Partial function In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then f is said to be a total function. More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to exactly one element of the second set. A partial function is often used when its exact domain of definition is not known or difficult to specify. This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a function. In computability theory, a general recursive function is a partial function from the integers to the integers; no algorithm can exist for deciding whether an arbitrary such function is in fact total. When arrow notation is used for functions, a partial function $f$ from $X$ to $Y$ is sometimes written as $f:X\rightharpoonup Y,$ $f:X\nrightarrow Y,$ or $f:X\hookrightarrow Y.$ However, there is no general convention, and the latter notation is more commonly used for inclusion maps or embeddings. Specifically, for a partial function $f:X\rightharpoonup Y,$ and any $x\in X,$ one has either: • $f(x)=y\in Y$ (it is a single element in Y), or • $f(x)$ is undefined. For example, if $f$ is the square root function restricted to the integers $f:\mathbb {Z} \to \mathbb {N} ,$ defined by: $f(n)=m$ if, and only if, $m^{2}=n,$ $m\in \mathbb {N} ,n\in \mathbb {Z} ,$ then $f(n)$ is only defined if $n$ is a perfect square (that is, $0,1,4,9,16,\ldots $). So $f(25)=5$ but $f(26)$ is undefined. Basic concepts A partial function arises from the consideration of maps between two sets X and Y that may not be defined on the entire set X. A common example is the square root operation on the real numbers $\mathbb {R} $: because negative real numbers do not have real square roots, the operation can be viewed as a partial function from $\mathbb {R} $ to $\mathbb {R} .$ The domain of definition of a partial function is the subset S of X on which the partial function is defined; in this case, the partial function may also be viewed as a function from S to Y. In the example of the square root operation, the set S consists of the nonnegative real numbers $[0,+\infty ).$ The notion of partial function is particularly convenient when the exact domain of definition is unknown or even unknowable. For a computer-science example of the latter, see Halting problem. In case the domain of definition S is equal to the whole set X, the partial function is said to be total. Thus, total partial functions from X to Y coincide with functions from X to Y. Many properties of functions can be extended in an appropriate sense of partial functions. A partial function is said to be injective, surjective, or bijective when the function given by the restriction of the partial function to its domain of definition is injective, surjective, bijective respectively. Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective.[1] An injective partial function may be inverted to an injective partial function, and a partial function which is both injective and surjective has an injective function as inverse. Furthermore, a function which is injective may be inverted to a bijective partial function. The notion of transformation can be generalized to partial functions as well. A partial transformation is a function $f:A\rightharpoonup B,$ where both $A$ and $B$ are subsets of some set $X.$[1] Function spaces For convenience, denote the set of all partial functions $f:X\rightharpoonup Y$ from a set $X$ to a set $Y$ by $[X\rightharpoonup Y].$ This set is the union of the sets of functions defined on subsets of $X$ with same codomain $Y$: $[X\rightharpoonup Y]=\bigcup _{D\subseteq X}[D\to Y],$ the latter also written as $ \bigcup _{D\subseteq {X}}Y^{D}.$ In finite case, its cardinality is $\|[X\rightharpoonup Y]\|=(\|Y\|+1)^{\|X\|},$ because any partial function can be extended to a function by any fixed value $c$ not contained in $Y,$ so that the codomain is $Y\cup \{c\},$ an operation which is injective (unique and invertible by restriction). Discussion and examples The first diagram at the top of the article represents a partial function that is not a function since the element 1 in the left-hand set is not associated with anything in the right-hand set. Whereas, the second diagram represents a function since every element on the left-hand set is associated with exactly one element in the right hand set. Natural logarithm Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore, the natural logarithm function is not a function when viewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only include the positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals), then the natural logarithm is a function. Subtraction of natural numbers Subtraction of natural numbers (non-negative integers) can be viewed as a partial function: $f:\mathbb {N} \times \mathbb {N} \rightharpoonup \mathbb {N} $ $f(x,y)=x-y.$ It is defined only when $x\geq y.$ Bottom element In denotational semantics a partial function is considered as returning the bottom element when it is undefined. In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. The IEEE floating point standard defines a not-a-number value which is returned when a floating point operation is undefined and exceptions are suppressed, e.g. when the square root of a negative number is requested. In a programming language where function parameters are statically typed, a function may be defined as a partial function because the language's type system cannot express the exact domain of the function, so the programmer instead gives it the smallest domain which is expressible as a type and contains the domain of definition of the function. In category theory In category theory, when considering the operation of morphism composition in concrete categories, the composition operation $\circ \;:\;\hom(C)\times \hom(C)\to \hom(C)$ is a function if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): \operatorname {ob} (C) has one element. The reason for this is that two morphisms $f:X\to Y$ and $g:U\to V$ can only be composed as $g\circ f$ if $Y=U,$ that is, the codomain of $f$ must equal the domain of $g.$ The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps.[2] One textbook notes that "This formal completion of sets and partial maps by adding “improper,” “infinite” elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science."[3] The category of sets and partial bijections is equivalent to its dual.[4] It is the prototypical inverse category.[5] In abstract algebra Partial algebra generalizes the notion of universal algebra to partial operations. An example would be a field, in which the multiplicative inversion is the only proper partial operation (because division by zero is not defined).[6] The set of all partial functions (partial transformations) on a given base set, $X,$ forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on $X$), typically denoted by ${\mathcal {PT}}_{X}.$[7][8][9] The set of all partial bijections on $X$ forms the symmetric inverse semigroup.[7][8] Charts and atlases for manifolds and fiber bundles Charts in the atlases which specify the structure of manifolds and fiber bundles are partial functions. In the case of manifolds, the domain is the point set of the manifold. In the case of fiber bundles, the domain is the space of the fiber bundle. In these applications, the most important construction is the transition map, which is the composite of one chart with the inverse of another. The initial classification of manifolds and fiber bundles is largely expressed in terms of constraints on these transition maps. The reason for the use of partial functions instead of functions is to permit general global topologies to be represented by stitching together local patches to describe the global structure. The "patches" are the domains where the charts are defined. See also Function x ↦ f (x) Examples of domains and codomains • $X$ → $\mathbb {B} $, $\mathbb {B} $ → $X$, $\mathbb {B} ^{n}$ → $X$ • $X$ → $\mathbb {Z} $, $\mathbb {Z} $ → $X$ • $X$ → $\mathbb {R} $, $\mathbb {R} $ → $X$, $\mathbb {R} ^{n}$ → $X$ • $X$ → $\mathbb {C} $, $\mathbb {C} $ → $X$, $\mathbb {C} ^{n}$ → $X$ Classes/properties • Constant • Identity • Linear • Polynomial • Rational • Algebraic • Analytic • Smooth • Continuous • Measurable • Injective • Surjective • Bijective Constructions • Restriction • Composition • λ • Inverse Generalizations • Partial • Multivalued • Implicit • space • Analytic continuation – Extension of the domain of an analytic function (mathematics) • Multivalued function – Generalized mathematical function • Densely defined operator – Function that is defined almost everywhere (mathematics) References 1. Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1. 2. Lutz Schröder (2001). "Categories: a free tour". In Jürgen Koslowski and Austin Melton (ed.). Categorical Perspectives. Springer Science & Business Media. p. 10. ISBN 978-0-8176-4186-3. 3. Neal Koblitz; B. Zilber; Yu. I. Manin (2009). A Course in Mathematical Logic for Mathematicians. Springer Science & Business Media. p. 290. ISBN 978-1-4419-0615-1. 4. Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge University Press. p. 289. ISBN 978-0-521-44179-7. 5. Marco Grandis (2012). Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups. World Scientific. p. 55. ISBN 978-981-4407-06-9. 6. Peter Burmeister (1993). "Partial algebras – an introductory survey". In Ivo G. Rosenberg; Gert Sabidussi (eds.). Algebras and Orders. Springer Science & Business Media. ISBN 978-0-7923-2143-9. 7. Alfred Hoblitzelle Clifford; G. B. Preston (1967). The Algebraic Theory of Semigroups. Volume II. American Mathematical Soc. p. xii. ISBN 978-0-8218-0272-4. 8. Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press, Incorporated. p. 4. ISBN 978-0-19-853577-5. 9. Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. pp. 16 and 24. ISBN 978-1-84800-281-4. • Martin Davis (1958), Computability and Unsolvability, McGraw–Hill Book Company, Inc, New York. Republished by Dover in 1982. ISBN 0-486-61471-9. • Stephen Kleene (1952), Introduction to Meta-Mathematics, North-Holland Publishing Company, Amsterdam, Netherlands, 10th printing with corrections added on 7th printing (1974). ISBN 0-7204-2103-9. • Harold S. Stone (1972), Introduction to Computer Organization and Data Structures, McGraw–Hill Book Company, New York.
Uniqueness quantification In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition.[1] This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!"[2] or "∃=1". For example, the formal statement $\exists !n\in \mathbb {N} \,(n-2=4)$ may be read as "there is exactly one natural number $n$ such that $n-2=4$". Proving uniqueness The most common technique to prove the unique existence of a certain object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, $a$ and $b$) must be equal to each other (i.e. $a=b$). For example, to show that the equation $x+2=5$ has exactly one solution, one would first start by establishing that at least one solution exists, namely 3; the proof of this part is simply the verification that the equation below holds: $3+2=5.$ To establish the uniqueness of the solution, one would then proceed by assuming that there are two solutions, namely $a$ and $b$, satisfying $x+2=5$. That is, $a+2=5{\text{ and }}b+2=5.$ By transitivity of equality, $a+2=b+2.$ Subtracting 2 from both sides then yields $a=b.$ which completes the proof that 3 is the unique solution of $x+2=5$. In general, both existence (there exists at least one object) and uniqueness (there exists at most one object) must be proven, in order to conclude that there exists exactly one object satisfying a said condition. An alternative way to prove uniqueness is to prove that there exists an object $a$ satisfying the condition, and then to prove that every object satisfying the condition must be equal to $a$. Reduction to ordinary existential and universal quantification Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic, by defining the formula $\exists !xP(x)$ to mean $\exists x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x)),$ which is logically equivalent to $\exists x\,(P(x)\wedge \forall y\,(P(y)\to y=x)).$ An equivalent definition that separates the notions of existence and uniqueness into two clauses, at the expense of brevity, is $\exists x\,P(x)\wedge \forall y\,\forall z\,[(P(y)\wedge P(z))\to y=z].$ Another equivalent definition, which has the advantage of brevity, is $\exists x\,\forall y\,(P(y)\leftrightarrow y=x).$ Generalizations The uniqueness quantification can be generalized into counting quantification (or numerical quantification[3]). This includes both quantification of the form "exactly k objects exist such that …" as well as "infinitely many objects exist such that …" and "only finitely many objects exist such that…". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary first-order logic.[4] Uniqueness depends on a notion of equality. Loosening this to some coarser equivalence relation yields quantification of uniqueness up to that equivalence (under this framework, regular uniqueness is "uniqueness up to equality"). For example, many concepts in category theory are defined to be unique up to isomorphism. The exclamation mark $!$ !} can be also used as a separate quantification symbol, so $(\exists !x.P(x))\leftrightarrow ((\exists x.P(x))\land (!x.P(x)))$, where $(!x.P(x)):=(\forall a\forall b.P(a)\land P(b)\rightarrow a=b)$. E.g. it can be safely used in the replacement axiom, instead of $\exists !$ !} . See also • Essentially unique • One-hot • Singleton (mathematics) • Uniqueness theorem References 1. Weisstein, Eric W. "Uniqueness Theorem". mathworld.wolfram.com. Retrieved 2019-12-15. 2. "2.5 Uniqueness Arguments". www.whitman.edu. Retrieved 2019-12-15. 3. Helman, Glen (August 1, 2013). "Numerical quantification" (PDF). persweb.wabash.edu. Retrieved 2019-12-14. 4. This is a consequence of the compactness theorem. Bibliography • Kleene, Stephen (1952). Introduction to Metamathematics. Ishi Press International. p. 199. • Andrews, Peter B. (2002). An introduction to mathematical logic and type theory to truth through proof (2. ed.). Dordrecht: Kluwer Acad. Publ. p. 233. ISBN 1-4020-0763-9. Mathematical logic General • Axiom • list • Cardinality • First-order logic • Formal proof • Formal semantics • Foundations of mathematics • Information theory • Lemma • Logical consequence • Model • Theorem • Theory • Type theory Theorems (list) & Paradoxes • Gödel's completeness and incompleteness theorems • Tarski's undefinability • Banach–Tarski paradox • Cantor's theorem, paradox and diagonal argument • Compactness • Halting problem • Lindström's • Löwenheim–Skolem • Russell's paradox Logics Traditional • Classical logic • Logical truth • Tautology • Proposition • Inference • Logical equivalence • Consistency • Equiconsistency • Argument • Soundness • Validity • Syllogism • Square of opposition • Venn diagram Propositional • Boolean algebra • Boolean functions • Logical connectives • Propositional calculus • Propositional formula • Truth tables • Many-valued logic • 3 • Finite • ∞ Predicate • First-order • list • Second-order • Monadic • Higher-order • Free • Quantifiers • Predicate • Monadic predicate calculus Set theory • Set • Hereditary • Class • (Ur-)Element • Ordinal number • Extensionality • Forcing • Relation • Equivalence • Partition • Set operations: • Intersection • Union • Complement • Cartesian product • Power set • Identities Types of Sets • Countable • Uncountable • Empty • Inhabited • Singleton • Finite • Infinite • Transitive • Ultrafilter • Recursive • Fuzzy • Universal • Universe • Constructible • Grothendieck • Von Neumann Maps & Cardinality • Function/Map • Domain • Codomain • Image • In/Sur/Bi-jection • Schröder–Bernstein theorem • Isomorphism • Gödel numbering • Enumeration • Large cardinal • Inaccessible • Aleph number • Operation • Binary Set theories • Zermelo–Fraenkel • Axiom of choice • Continuum hypothesis • General • Kripke–Platek • Morse–Kelley • Naive • New Foundations • Tarski–Grothendieck • Von Neumann–Bernays–Gödel • Ackermann • Constructive Formal systems (list), Language & Syntax • Alphabet • Arity • Automata • Axiom schema • Expression • Ground • Extension • by definition • Conservative • Relation • Formation rule • Grammar • Formula • Atomic • Closed • Ground • Open • Free/bound variable • Language • Metalanguage • Logical connective • ¬ • ∨ • ∧ • → • ↔ • = • Predicate • Functional • Variable • Propositional variable • Proof • Quantifier • ∃ • ! • ∀ • rank • Sentence • Atomic • Spectrum • Signature • String • Substitution • Symbol • Function • Logical/Constant • Non-logical • Variable • Term • Theory • list Example axiomatic systems (list) • of arithmetic: • Peano • second-order • elementary function • primitive recursive • Robinson • Skolem • of the real numbers • Tarski's axiomatization • of Boolean algebras • canonical • minimal axioms • of geometry: • Euclidean: • Elements • Hilbert's • Tarski's • non-Euclidean • Principia Mathematica Proof theory • Formal proof • Natural deduction • Logical consequence • Rule of inference • Sequent calculus • Theorem • Systems • Axiomatic • Deductive • Hilbert • list • Complete theory • Independence (from ZFC) • Proof of impossibility • Ordinal analysis • Reverse mathematics • Self-verifying theories Model theory • Interpretation • Function • of models • Model • Equivalence • Finite • Saturated • Spectrum • Submodel • Non-standard model • of arithmetic • Diagram • Elementary • Categorical theory • Model complete theory • Satisfiability • Semantics of logic • Strength • Theories of truth • Semantic • Tarski's • Kripke's • T-schema • Transfer principle • Truth predicate • Truth value • Type • Ultraproduct • Validity Computability theory • Church encoding • Church–Turing thesis • Computably enumerable • Computable function • Computable set • Decision problem • Decidable • Undecidable • P • NP • P versus NP problem • Kolmogorov complexity • Lambda calculus • Primitive recursive function • Recursion • Recursive set • Turing machine • Type theory Related • Abstract logic • Category theory • Concrete/Abstract Category • Category of sets • History of logic • History of mathematical logic • timeline • Logicism • Mathematical object • Philosophy of mathematics • Supertask Mathematics portal
Function composition In mathematics, function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in domain X to g(f(x)) in codomain Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.[nb 1] "Ring operator" redirects here. Not to be confused with operator ring or operator assistance. Function x ↦ f (x) Examples of domains and codomains • $X$ → $\mathbb {B} $, $\mathbb {B} $ → $X$, $\mathbb {B} ^{n}$ → $X$ • $X$ → $\mathbb {Z} $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): \mathbb {Z} → $X$ • $X$ → $\mathbb {R} $, $\mathbb {R} $ → $X$, $\mathbb {R} ^{n}$ → $X$ • $X$ → $\mathbb {C} $, $\mathbb {C} $ → $X$, $\mathbb {C} ^{n}$ → $X$ Classes/properties • Constant • Identity • Linear • Polynomial • Rational • Algebraic • Analytic • Smooth • Continuous • Measurable • Injective • Surjective • Bijective Constructions • Restriction • Composition • λ • Inverse Generalizations • Partial • Multivalued • Implicit • space The notation g ∘ f is read as "g of f ", "g after f ", "g circle f ", "g round f ", "g about f ", "g composed with f ", "g following f ", "f then g", or "g on f ", or "the composition of g and f ". Intuitively, composing functions is a chaining process in which the output of function f feeds the input of function g. The composition of functions is a special case of the composition of relations, sometimes also denoted by $\circ $. As a result, all properties of composition of relations are true of composition of functions,[1] such as the property of associativity. Composition of functions is different from multiplication of functions (if defined at all), and has some quite different properties; in particular, composition of functions is not commutative.[2] Examples • Composition of functions on a finite set: If f = {(1, 1), (2, 3), (3, 1), (4, 2)}, and g = {(1, 2), (2, 3), (3, 1), (4, 2)}, then g ∘ f = {(1, 2), (2, 1), (3, 2), (4, 3)}, as shown in the figure. • Composition of functions on an infinite set: If f: R → R (where R is the set of all real numbers) is given by f(x) = 2x + 4 and g: R → R is given by g(x) = x3, then: (f ∘ g)(x) = f(g(x)) = f(x3) = 2x3 + 4, and (g ∘ f)(x) = g(f(x)) = g(2x + 4) = (2x + 4)3. • If an airplane's altitude at time t is a(t), and the air pressure at altitude x is p(x), then (p ∘ a)(t) is the pressure around the plane at time t. Properties The composition of functions is always associative—a property inherited from the composition of relations.[1] That is, if f, g, and h are composable, then f ∘ (g ∘ h) = (f ∘ g) ∘ h.[3] Since the parentheses do not change the result, they are generally omitted. In a strict sense, the composition g ∘ f is only meaningful if the codomain of f equals the domain of g; in a wider sense, it is sufficient that the former be an improper subset of the latter.[nb 2] Moreover, it is often convenient to tacitly restrict the domain of f, such that f produces only values in the domain of g. For example, the composition g ∘ f of the functions f : R → (−∞,+9] defined by f(x) = 9 − x2 and g : [0,+∞) → R defined by $g(x)={\sqrt {x}}$ can be defined on the interval [−3,+3]. The functions g and f are said to commute with each other if g ∘ f = f ∘ g. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, \|x\| + 3 = \|x + 3\| only when x ≥ 0. The picture shows another example. The composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that (f ∘ g)−1 = g−1∘ f−1.[4] Derivatives of compositions involving differentiable functions can be found using the chain rule. Higher derivatives of such functions are given by Faà di Bruno's formula.[3] Composition monoids Main article: Transformation monoid Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and codomain; these are often called transformations. Then one can form chains of transformations composed together, such as f ∘ f ∘ g ∘ f. Such chains have the algebraic structure of a monoid, called a transformation monoid or (much more seldom) a composition monoid. In general, transformation monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions f: X → X is called the full transformation semigroup[5] or symmetric semigroup[6] on X. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.[7]) If the transformations are bijective (and thus invertible), then the set of all possible combinations of these functions forms a transformation group; and one says that the group is generated by these functions. A fundamental result in group theory, Cayley's theorem, essentially says that any group is in fact just a subgroup of a permutation group (up to isomorphism).[8] The set of all bijective functions f: X → X (called permutations) forms a group with respect to function composition. This is the symmetric group, also sometimes called the composition group. In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the symmetric semigroup is a regular semigroup.[9] Functional powers Main article: Iterated function If Y ⊆ X, then f: X→Y may compose with itself; this is sometimes denoted as f 2. That is: (f ∘ f)(x) = f(f(x)) = f 2(x) (f ∘ f ∘ f)(x) = f(f(f(x))) = f 3(x) (f ∘ f ∘ f ∘ f)(x) = f(f(f(f(x)))) = f 4(x) More generally, for any natural number n ≥ 2, the nth functional power can be defined inductively by f n = f ∘ f n−1 = f n−1 ∘ f, a notation introduced by Hans Heinrich Bürmann[10][11] and John Frederick William Herschel.[12][10][13][11] Repeated composition of such a function with itself is called iterated function. • By convention, f 0 is defined as the identity map on f 's domain, idX. • If even Y = X and f: X → X admits an inverse function f −1, negative functional powers f −n are defined for n > 0 as the negated power of the inverse function: f −n = (f −1)n.[12][10][11] Note: If f takes its values in a ring (in particular for real or complex-valued f ), there is a risk of confusion, as f n could also stand for the n-fold product of f, e.g. f 2(x) = f(x) · f(x).[11] For trigonometric functions, usually the latter is meant, at least for positive exponents.[11] For example, in trigonometry, this superscript notation represents standard exponentiation when used with trigonometric functions: sin2(x) = sin(x) · sin(x). However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., tan−1 = arctan ≠ 1/tan. In some cases, when, for a given function f, the equation g ∘ g = f has a unique solution g, that function can be defined as the functional square root of f, then written as g = f 1/2. More generally, when gn = f has a unique solution for some natural number n > 0, then f m/n can be defined as gm. Under additional restrictions, this idea can be generalized so that the iteration count becomes a continuous parameter; in this case, such a system is called a flow, specified through solutions of Schröder's equation. Iterated functions and flows occur naturally in the study of fractals and dynamical systems. To avoid ambiguity, some mathematicians choose to use ∘ to denote the compositional meaning, writing f∘n(x) for the n-th iterate of the function f(x), as in, for example, f∘3(x) meaning f(f(f(x))). For the same purpose, f[n](x) was used by Benjamin Peirce[14][11] whereas Alfred Pringsheim and Jules Molk suggested nf(x) instead.[15][11][nb 3] Alternative notations Many mathematicians, particularly in group theory, omit the composition symbol, writing gf for g ∘ f.[16] In the mid-20th century, some mathematicians decided that writing "g ∘ f " to mean "first apply f, then apply g" was too confusing and decided to change notations. They write "xf " for "f(x)" and "(xf)g" for "g(f(x))".[17] This can be more natural and seem simpler than writing functions on the left in some areas – in linear algebra, for instance, when x is a row vector and f and g denote matrices and the composition is by matrix multiplication. This alternative notation is called postfix notation. The order is important because function composition is not necessarily commutative (e.g. matrix multiplication). Successive transformations applying and composing to the right agrees with the left-to-right reading sequence. Mathematicians who use postfix notation may write "fg", meaning first apply f and then apply g, in keeping with the order the symbols occur in postfix notation, thus making the notation "fg" ambiguous. Computer scientists may write "f ; g" for this,[18] thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in the Z notation the ⨾ character is used for left relation composition.[19] Since all functions are binary relations, it is correct to use the [fat] semicolon for function composition as well (see the article on composition of relations for further details on this notation). Composition operator Main article: Composition operator Given a function g, the composition operator Cg is defined as that operator which maps functions to functions as $C_{g}f=f\circ g.$ Composition operators are studied in the field of operator theory. In programming languages Function composition appears in one form or another in numerous programming languages. Multivariate functions Partial composition is possible for multivariate functions. The function resulting when some argument xi of the function f is replaced by the function g is called a composition of f and g in some computer engineering contexts, and is denoted f \|xi = g $f\|_{x_{i}=g}=f(x_{1},\ldots ,x_{i-1},g(x_{1},x_{2},\ldots ,x_{n}),x_{i+1},\ldots ,x_{n}).$ When g is a simple constant b, composition degenerates into a (partial) valuation, whose result is also known as restriction or co-factor.[20] $f\|_{x_{i}=b}=f(x_{1},\ldots ,x_{i-1},b,x_{i+1},\ldots ,x_{n}).$ In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of primitive recursive function. Given f, a n-ary function, and n m-ary functions g1, ..., gn, the composition of f with g1, ..., gn, is the m-ary function $h(x_{1},\ldots ,x_{m})=f(g_{1}(x_{1},\ldots ,x_{m}),\ldots ,g_{n}(x_{1},\ldots ,x_{m})).$ This is sometimes called the generalized composite or superposition of f with g1, ..., gn.[21] The partial composition in only one argument mentioned previously can be instantiated from this more general scheme by setting all argument functions except one to be suitably chosen projection functions. Here g1, ..., gn can be seen as a single vector/tuple-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition.[22] A set of finitary operations on some base set X is called a clone if it contains all projections and is closed under generalized composition. Note that a clone generally contains operations of various arities.[21] The notion of commutation also finds an interesting generalization in the multivariate case; a function f of arity n is said to commute with a function g of arity m if f is a homomorphism preserving g, and vice versa i.e.:[21] $f(g(a_{11},\ldots ,a_{1m}),\ldots ,g(a_{n1},\ldots ,a_{nm}))=g(f(a_{11},\ldots ,a_{n1}),\ldots ,f(a_{1m},\ldots ,a_{nm})).$ A unary operation always commutes with itself, but this is not necessarily the case for a binary (or higher arity) operation. A binary (or higher arity) operation that commutes with itself is called medial or entropic.[21] Generalizations Composition can be generalized to arbitrary binary relations. If R ⊆ X × Y and S ⊆ Y × Z are two binary relations, then their composition R∘S is the relation defined as {(x, z) ∈ X × Z : ∃y ∈ Y. (x, y) ∈ R ∧ (y, z) ∈ S}. Considering a function as a special case of a binary relation (namely functional relations), function composition satisfies the definition for relation composition. A small circle R∘S has been used for the infix notation of composition of relations, as well as functions. When used to represent composition of functions $(g\circ f)(x)\ =\ g(f(x))$ however, the text sequence is reversed to illustrate the different operation sequences accordingly. The composition is defined in the same way for partial functions and Cayley's theorem has its analogue called the Wagner–Preston theorem.[23] The category of sets with functions as morphisms is the prototypical category. The axioms of a category are in fact inspired from the properties (and also the definition) of function composition.[24] The structures given by composition are axiomatized and generalized in category theory with the concept of morphism as the category-theoretical replacement of functions. The reversed order of composition in the formula (f ∘ g)−1 = (g−1 ∘ f −1) applies for composition of relations using converse relations, and thus in group theory. These structures form dagger categories. Typography The composition symbol ∘ is encoded as U+2218 ∘ RING OPERATOR (&compfn;, &SmallCircle;); see the Degree symbol article for similar-appearing Unicode characters. In TeX, it is written \circ. See also • Cobweb plot – a graphical technique for functional composition • Combinatory logic • Composition ring, a formal axiomatization of the composition operation • Flow (mathematics) • Function composition (computer science) • Function of random variable, distribution of a function of a random variable • Functional decomposition • Functional square root • Higher-order function • Infinite compositions of analytic functions • Iterated function • Lambda calculus Notes 1. Some authors use f ∘ g : X → Z, defined by (f ∘ g )(x) = g(f(x)) instead. This is common when a postfix notation is used, especially if functions are represented by exponents, as, for instance, in the study of group actions. See Dixon, John D.; Mortimer, Brian (1996). Permutation groups. Springer. p. 5. ISBN 0-387-94599-7. 2. The strict sense is used, e.g., in category theory, where a subset relation is modelled explicitly by an inclusion function. 3. Alfred Pringsheim's and Jules Molk's (1907) notation nf(x) to denote function compositions must not be confused with Rudolf von Bitter Rucker's (1982) notation nx, introduced by Hans Maurer (1901) and Reuben Louis Goodstein (1947) for tetration, or with David Patterson Ellerman's (1995) nx pre-superscript notation for roots. References 1. Velleman, Daniel J. (2006). How to Prove It: A Structured Approach. Cambridge University Press. p. 232. ISBN 978-1-139-45097-3. 2. "3.4: Composition of Functions". Mathematics LibreTexts. 2020-01-16. Retrieved 2020-08-28. 3. Weisstein, Eric W. "Composition". mathworld.wolfram.com. Retrieved 2020-08-28. 4. Rodgers, Nancy (2000). Learning to Reason: An Introduction to Logic, Sets, and Relations. John Wiley & Sons. pp. 359–362. ISBN 978-0-471-37122-9. 5. Hollings, Christopher (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 334. ISBN 978-1-4704-1493-1. 6. Grillet, Pierre A. (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 2. ISBN 978-0-8247-9662-4. 7. Dömösi, Pál; Nehaniv, Chrystopher L. (2005). Algebraic Theory of Automata Networks: An introduction. SIAM. p. 8. ISBN 978-0-89871-569-9. 8. Carter, Nathan (2009-04-09). Visual Group Theory. MAA. p. 95. ISBN 978-0-88385-757-1. 9. Ganyushkin, Olexandr; Mazorchuk, Volodymyr (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 24. ISBN 978-1-84800-281-4. 10. Herschel, John Frederick William (1820). "Part III. Section I. Examples of the Direct Method of Differences". A Collection of Examples of the Applications of the Calculus of Finite Differences. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 [5–6]. Archived from the original on 2020-08-04. Retrieved 2020-08-04. (NB. Inhere, Herschel refers to his 1813 work and mentions Hans Heinrich Bürmann's older work.) 11. Cajori, Florian (1952) [March 1929]. "§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions". A History of Mathematical Notations. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, USA: Open court publishing company. pp. 108, 176–179, 336, 346. ISBN 978-1-60206-714-1. Retrieved 2016-01-18. […] §473. Iterated logarithms […] We note here the symbolism used by Pringsheim and Molk in their joint Encyclopédie article: "2logb a = logb (logb a), …, k+1logb a = logb (klogb a)."[a] […] §533. John Herschel's notation for inverse functions, sin−1 x, tan−1 x, etc., was published by him in the Philosophical Transactions of London, for the year 1813. He says (p. 10): "This notation cos.−1 e must not be understood to signify 1/cos. e, but what is usually written thus, arc (cos.=e)." He admits that some authors use cos.m A for (cos. A)m, but he justifies his own notation by pointing out that since d2 x, Δ3 x, Σ2 x mean dd x, ΔΔΔ x, ΣΣ x, we ought to write sin.2 x for sin. sin. x, log.3 x for log. log. log. x. Just as we write d−n V=∫n V, we may write similarly sin.−1 x=arc (sin.=x), log.−1 x.=cx. Some years later Herschel explained that in 1813 he used fn(x), f−n(x), sin.−1 x, etc., "as he then supposed for the first time. The work of a German Analyst, Burmann, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He[Burmann], however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan−1, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption."[b] […] §535. Persistence of rival notations for inverse function.— […] The use of Herschel's notation underwent a slight change in Benjamin Peirce's books, to remove the chief objection to them; Peirce wrote: "cos[−1] x," "log[−1] x."[c] […] §537. Powers of trigonometric functions.—Three principal notations have been used to denote, say, the square of sin x, namely, (sin x)2, sin x2, sin2 x. The prevailing notation at present is sin2 x, though the first is least likely to be misinterpreted. In the case of sin2 x two interpretations suggest themselves; first, sin x · sin x; second,[d] sin (sin x). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log2 x, where log x · log x and log (log x) are of frequent occurrence in analysis. […] The notation sinn x for (sin x)n has been widely used and is now the prevailing one. […] (xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.) 12. Herschel, John Frederick William (1813) [1812-11-12]. "On a Remarkable Application of Cotes's Theorem". Philosophical Transactions of the Royal Society of London. London: Royal Society of London, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall. 103 (Part 1): 8–26 [10]. doi:10.1098/rstl.1813.0005. JSTOR 107384. S2CID 118124706. 13. Peano, Giuseppe (1903). Formulaire mathématique (in French). Vol. IV. p. 229. 14. Peirce, Benjamin (1852). Curves, Functions and Forces. Vol. I (new ed.). Boston, USA. p. 203.{{cite book}}: CS1 maint: location missing publisher (link) 15. Pringsheim, Alfred; Molk, Jules (1907). Encyclopédie des sciences mathématiques pures et appliquées (in French). Vol. I. p. 195. Part I. 16. Ivanov, Oleg A. (2009-01-01). Making Mathematics Come to Life: A Guide for Teachers and Students. American Mathematical Society. pp. 217–. ISBN 978-0-8218-4808-1. 17. Gallier, Jean (2011). Discrete Mathematics. Springer. p. 118. ISBN 978-1-4419-8047-2. 18. Barr, Michael; Wells, Charles (1998). Category Theory for Computing Science (PDF). p. 6. Archived from the original (PDF) on 2016-03-04. Retrieved 2014-08-23. (NB. This is the updated and free version of book originally published by Prentice Hall in 1990 as ISBN 978-0-13-120486-7.) 19. ISO/IEC 13568:2002(E), p. 23 20. Bryant, R. E. (August 1986). "Logic Minimization Algorithms for VLSI Synthesis" (PDF). IEEE Transactions on Computers. C-35 (8): 677–691. doi:10.1109/tc.1986.1676819. S2CID 10385726. 21. Bergman, Clifford (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 79–80, 90–91. ISBN 978-1-4398-5129-6. 22. Tourlakis, George (2012). Theory of Computation. John Wiley & Sons. p. 100. ISBN 978-1-118-31533-0. 23. Lipscomb, S. (1997). Symmetric Inverse Semigroups. AMS Mathematical Surveys and Monographs. p. xv. ISBN 0-8218-0627-0. 24. Hilton, Peter; Wu, Yel-Chiang (1989). A Course in Modern Algebra. John Wiley & Sons. p. 65. ISBN 978-0-471-50405-4. External links • "Composite function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • "Composition of Functions" by Bruce Atwood, the Wolfram Demonstrations Project, 2007.
Radical symbol In mathematics, the radical symbol, radical sign, root symbol, radix, or surd is a symbol for the square root or higher-order root of a number. The square root of a number x is written as ${\sqrt {11}},$ while the nth root of x is written as ${\sqrt[{n}]{x}}.$ It is also used for other meanings in more advanced mathematics, such as the radical of an ideal. In linguistics, the symbol is used to denote a root word. Principal square root Each positive real number has two square roots, one positive and the other negative. The square root symbol refers to the principal square root, which is the positive one. The two square roots of a negative number are both imaginary numbers, and the square root symbol refers to the principal square root, the one with a positive imaginary part. For the definition of the principal square root of other complex numbers, see Square root#Principal square root of a complex number. Origin The origin of the root symbol √ is largely speculative. Some sources imply that the symbol was first used by Arab mathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī (1421–1486). Legend has it that it was taken from the Arabic letter "ج" (ǧīm), which is the first letter in the Arabic word "جذر" (jadhir, meaning "root").[1] However, Leonhard Euler[2] believed it originated from the letter "r", the first letter of the Latin word "radix" (meaning "root"), referring to the same mathematical operation. The symbol was first seen in print without the vinculum (the horizontal "bar" over the numbers inside the radical symbol) in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician. In 1637 Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.[3] Encoding The Unicode and HTML character codes for the radical symbols are: ReadCharacterUnicode[4]XMLURLHTML[5] Square root√U+221A√ or √%E2%88%9A√ or &Sqrt; Cube root∛U+221B∛ or ∛%E2%88%9B Fourth root∜U+221C∜ or ∜%E2%88%9C However, these characters differ in appearance from most mathematical typesetting by omitting the overline connected to the radical symbol, which surrounds the argument of the square root function. The OpenType math table allows adding this overline following the radical symbol. Legacy encodings of the square root character U+221A include: • 0xC3 in Mac OS Roman and Mac OS Cyrillic • 0xFB (Alt+251) in Code page 437 and Code page 866 (but not Code page 850) on DOS and the Windows console • 0xD6 in the Symbol font encoding[6] • 02-69 (7-bit 0x2265, SJIS 0x81E3, EUC 0xA2E5) in Japanese JIS X 0208[7] • 01-78 (EUC/UHC 0xA1EE) in Korean Wansung code[8] • 01-44 (EUC 0xA1CC) in Mainland Chinese GB 2312 or GBK[9] • Traditional Chinese: 0xA1D4 in Big5[10][11] or 1-2235 (kuten 01-02-21, EUC 0xA2B5 or 0x8EA1A2B5) in CNS 11643[11][12] The Symbol font displays the character without any vinculum whatsoever; the overline may be a separate character at 0x60.[13] The JIS,[14] Wansung[15] and CNS 11643[11][16] code charts include a short overline attached to the radical symbol, whereas the GB 2312[17] and GB 18030 charts do not.[18] Additionally a "Radical Symbol Bottom" (U+23B7, ⎷) is available in the Miscellaneous Technical block.[19] This was used in contexts where box-drawing characters are used, such as in the technical character set of DEC terminals, to join up with box drawing characters on the line above to create the vinculum.[20] In LaTeX the square root symbol may be generated by the \sqrt macro,[21] and the square root symbol without the overline may be generated by the \surd macro. References 1. "Language Log: Ab surd". Retrieved 22 June 2012. 2. Leonhard Euler (1755). Institutiones calculi differentialis (in Latin). 3. Cajori, Florian (2012) [1928], A History of Mathematical Notations, vol. I, Dover, p. 208, ISBN 978-0-486-67766-8 4. Unicode Consortium (2022-09-16). "Mathematical Operators" (PDF). The Unicode Standard (15.0 ed.). Retrieved 2023-07-16. 5. Web Hypertext Application Technology Working Group (2023-07-14). "Named Character References". HTML Living Standard. Retrieved 2023-07-16. 6. Apple Computer (2005-04-05) [1995-04-15]. Map (external version) from Mac OS Symbol character set to Unicode 4.0 and later. Unicode Consortium. SYMBOL.TXT. 7. Unicode Consortium (2015-12-02) [1994-03-08]. JIS X 0208 (1990) to Unicode. JIS0208.TXT. 8. Unicode Consortium (2011-10-14) [1995-07-24]. Unified Hangeul(KSC5601-1992) to Unicode table. KSC5601.TXT. 9. IBM (2002). "windows-936-2000". International Components for Unicode. 10. Unicode Consortium (2015-12-02) [1994-02-11]. BIG5 to Unicode table (complete). BIG5.TXT. 11. "[√] 1-2235". Word Information. National Development Council. 12. IBM (2014). "euc-tw-2014". International Components for Unicode. 13. IBM. Code Page 01038 (PDF). Archived from the original (PDF) on 2015-07-08. 14. ISO/IEC JTC 1/SC 2 (1992-07-13). Japanese Graphic Character Set for Information Interchange (PDF). ITSCJ/IPSJ. ISO-IR-168. 15. Korea Bureau of Standards (1988-10-01). Korean Graphic Character Set for Information Interchange (PDF). ITSCJ/IPSJ. ISO-IR-149. 16. ECMA (1994). Chinese Standard Interchange Code (CSIC) - Set 1 (PDF). ITSCJ/IPSJ. ISO-IR-171. 17. China Association for Standardization (1980). Coded Chinese Graphic Character Set for Information Interchange (PDF). ITSCJ/IPSJ. ISO-IR-58. 18. Standardization Administration of China (2005). Information Technology—Chinese coded character set. p. 8. GB 18030-2005. 19. Unicode Consortium (2022-09-16). "Miscellaneous Technical" (PDF). The Unicode Standard (15.0 ed.). Retrieved 2023-07-16. 20. Williams, Paul Flo (2002). "DEC Technical Character Set (TCS)". VT100.net. Retrieved 2023-07-16. 21. Braams, Johannes; et al. (2023-06-01). "The LATEX 2ε Sources" (PDF) (2023-06-01 Patch Level 1 ed.). § ltmath.dtx: Math Environments. Retrieved 2023-07-16.
Right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or $\pi $/2 radians[1] corresponding to a quarter turn.[2] If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.[3] The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line. Types of angles 2D angles Right Interior Exterior 2D angle pairs Adjacent Vertical Complementary Supplementary Transversal 3D angles Dihedral Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles,[4] making the right angle basic to trigonometry. Etymology The meaning of right in right angle possibly refers to the Latin adjective rectus 'erect, straight, upright, perpendicular'. A Greek equivalent is orthos 'straight; perpendicular' (see orthogonality). In elementary geometry A rectangle is a quadrilateral with four right angles. A square has four right angles, in addition to equal-length sides. The Pythagorean theorem states how to determine when a triangle is a right triangle. Symbols In Unicode, the symbol for a right angle is U+221F ∟ RIGHT ANGLE (&angrt;). It should not be confused with the similarly shaped symbol U+231E ⌞ BOTTOM LEFT CORNER (&dlcorn;, &llcorner;). Related symbols are U+22BE ⊾ RIGHT ANGLE WITH ARC (&angrtvb;), U+299C ⦜ RIGHT ANGLE VARIANT WITH SQUARE (&vangrt;), and U+299D ⦝ MEASURED RIGHT ANGLE WITH DOT (&angrtvbd;).[5] In diagrams, the fact that an angle is a right angle is usually expressed by adding a small right angle that forms a square with the angle in the diagram, as seen in the diagram of a right triangle (in British English, a right-angled triangle) to the right. The symbol for a measured angle, an arc, with a dot, is used in some European countries, including German-speaking countries and Poland, as an alternative symbol for a right angle.[6] Euclid Right angles are fundamental in Euclid's Elements. They are defined in Book 1, definition 10, which also defines perpendicular lines. Definition 10 does not use numerical degree measurements but rather touches at the very heart of what a right angle is, namely two straight lines intersecting to form two equal and adjacent angles.[7] The straight lines which form right angles are called perpendicular.[8] Euclid uses right angles in definitions 11 and 12 to define acute angles (those smaller than a right angle) and obtuse angles (those greater than a right angle).[9] Two angles are called complementary if their sum is a right angle.[10] Book 1 Postulate 4 states that all right angles are equal, which allows Euclid to use a right angle as a unit to measure other angles with. Euclid's commentator Proclus gave a proof of this postulate using the previous postulates, but it may be argued that this proof makes use of some hidden assumptions. Saccheri gave a proof as well but using a more explicit assumption. In Hilbert's axiomatization of geometry this statement is given as a theorem, but only after much groundwork. One may argue that, even if postulate 4 can be proven from the preceding ones, in the order that Euclid presents his material it is necessary to include it since without it postulate 5, which uses the right angle as a unit of measure, makes no sense.[11] Conversion to other units A right angle may be expressed in different units: • 1/4 turn • 90° (degrees) • π/2 radians • 100 grad (also called grade, gradian, or gon) • 8 points (of a 32-point compass rose) • 6 hours (astronomical hour angle) Rule of 3-4-5 Throughout history, carpenters and masons have known a quick way to confirm if an angle is a true "right angle". It is based on the most widely known Pythagorean triple (3, 4, 5) and so called the "rule of 3-4-5". From the angle in question, running a straight line along one side exactly 3 units in length, and along the second side exactly 4 units in length, will create a hypotenuse (the longer line opposite the right angle that connects the two measured endpoints) of exactly 5 units in length. This measurement can be made quickly and without technical instruments. The geometric law behind the measurement is the Pythagorean theorem ("The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides"). Thales' theorem Construction of the perpendicular to the half-line h from the point P (applicable not only at the end point A, M is freely selectable), animation at the end with pause 10 s Alternative construction if P outside of the half-line h and the distance A to P' is small (B is freely selectable), animation at the end with pause 10 s Main article: Thales' theorem Thales' theorem states that an angle inscribed in a semicircle (with a vertex on the semicircle and its defining rays going through the endpoints of the semicircle) is a right angle. Two application examples in which the right angle and the Thales' theorem are included (see animations). See also Wikimedia Commons has media related to Right angles. • Cartesian coordinate system • Types of angles References 1. "Right Angle". Math Open Reference. Retrieved 26 April 2017. 2. Wentworth p. 11 3. Wentworth p. 8 4. Wentworth p. 40 5. Unicode 5.2 Character Code Charts Mathematical Operators, Miscellaneous Mathematical Symbols-B 6. Müller-Philipp, Susanne; Gorski, Hans-Joachim (2011). Leitfaden Geometrie [Handbook Geometry] (in German). Springer. ISBN 9783834886163. 7. Heath p. 181 8. Heath p. 181 9. Heath p. 181 10. Wentworth p. 9 11. Heath pp. 200–201 for the paragraph • Wentworth, G.A. (1895). A Text-Book of Geometry. Ginn & Co. • Euclid, commentary and trans. by T. L. Heath Elements Vol. 1 (1908 Cambridge) Google Books Authority control: National • Germany
Internal and external angles In geometry, an angle of a polygon is formed by two adjacent sides. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. "Interior angle" redirects here. For interior angles on the same side of the transversal, see Transversal line. Types of angles 2D angles Right Interior Exterior 2D angle pairs Adjacent Vertical Complementary Supplementary Transversal 3D angles Dihedral If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the polygon is called convex. In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.[1]: pp. 261-264 Properties • The sum of the internal angle and the external angle on the same vertex is π radians (180°). • The sum of all the internal angles of a simple polygon is π(n−2) radians or 180(n–2) degrees, where n is the number of sides. The formula can be proved by using mathematical induction: starting with a triangle, for which the angle sum is 180°, then replacing one side with two sides connected at another vertex, and so on. • The sum of the external angles of any simple convex or non-convex polygon, if only one of the two external angles is assumed at each vertex, is 2π radians (360°). • The measure of the exterior angle at a vertex is unaffected by which side is extended: the two exterior angles that can be formed at a vertex by extending alternately one side or the other are vertical angles and thus are equal. Extension to crossed polygons The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles. In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360°) revolutions one undergoes by walking around the perimeter of the polygon. In other words, the sum of all the exterior angles is 2πk radians or 360k degrees. Example: for ordinary convex polygons and concave polygons, k = 1, since the exterior angle sum is 360°, and one undergoes only one full revolution by walking around the perimeter. References 1. Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012. External links • Internal angles of a triangle • Interior angle sum of polygons: a general formula - Provides an interactive Java activity that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons.
Monus In mathematics, monus is an operator on certain commutative monoids that are not groups. A commutative monoid on which a monus operator is defined is called a commutative monoid with monus, or CMM. The monus operator may be denoted with the − symbol because the natural numbers are a CMM under subtraction; it is also denoted with the $\mathop {\dot {-}} $ symbol to distinguish it from the standard subtraction operator. Notation glyph Unicode name Unicode code point[1] HTML character entity reference HTML/XML numeric character references TeX ∸ DOT MINUS U+2238 ∸ \dot - − MINUS SIGN U+2212 − − - Definition Let $(M,+,0)$ be a commutative monoid. Define a binary relation $\leq $ on this monoid as follows: for any two elements $a$ and $b$, define $a\leq b$ if there exists an element $c$ such that $a+c=b$. It is easy to check that $\leq $ is reflexive[2] and that it is transitive.[3] $M$ is called naturally ordered if the $\leq $ relation is additionally antisymmetric and hence a partial order. Further, if for each pair of elements $a$ and $b$, a unique smallest element $c$ exists such that $a\leq b+c$, then M is called a commutative monoid with monus[4]: 129 and the monus $a\mathop {\dot {-}} b$ of any two elements $a$ and $b$ can be defined as this unique smallest element $c$ such that $a\leq b+c$. An example of a commutative monoid that is not naturally ordered is $(\mathbb {Z} ,+,0)$, the commutative monoid of the integers with usual addition, as for any $a,b\in \mathbb {Z} $ there exists $c$ such that $a+c=b$, so $a\leq b$ holds for any $a,b\in \mathbb {Z} $, so $\leq $ is not a partial order. There are also examples of monoids which are naturally ordered but are not semirings with monus.[5] Other structures Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid[6]) is a semiring where the commutative monoid induced by the addition operator is naturally ordered. When this monoid is a commutative monoid with monus, the semiring is called a semiring with monus, or m-semiring. Examples If M is an ideal in a Boolean algebra, then M is a commutative monoid with monus under $a+b=a\vee b$ and $a\mathop {\dot {-}} b=a\wedge \neg b$.[4]: 129 Natural numbers The natural numbers including 0 form a commutative monoid with monus, with their ordering being the usual order of natural numbers and the monus operator being a saturating variant of standard subtraction, variously referred to as truncated subtraction,[7] limited subtraction, proper subtraction, doz (difference or zero),[8] and monus.[9] Truncated subtraction is usually defined as[7] $a\mathop {\dot {-}} b={\begin{cases}0&{\mbox{if }}a<b\\a-b&{\mbox{if }}a\geq b,\end{cases}}$ where − denotes standard subtraction. For example, 5 − 3 = 2 and 3 − 5 = −2 in regular subtraction, whereas in truncated subtraction 3 ∸ 5 = 0. Truncated subtraction may also be defined as[9] $a\mathop {\dot {-}} b=\max(a-b,0).$ In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):[7] ${\begin{aligned}P(0)&=0\\P(S(a))&=a\\a\mathop {\dot {-}} 0&=a\\a\mathop {\dot {-}} S(b)&=P(a\mathop {\dot {-}} b).\end{aligned}}$ A definition that does not need the predecessor function is: ${\begin{aligned}a\mathop {\dot {-}} 0&=a\\0\mathop {\dot {-}} b&=0\\S(a)\mathop {\dot {-}} S(b)&=a\mathop {\dot {-}} b.\end{aligned}}$ Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers.[7] Truncated subtraction is also used in the definition of the multiset difference operator. Properties The class of all commutative monoids with monus form a variety.[4]: 129 The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms: ${\begin{aligned}a+(b\mathop {\dot {-}} a)&=b+(a\mathop {\dot {-}} b),\\(a\mathop {\dot {-}} b)\mathop {\dot {-}} c&=a\mathop {\dot {-}} (b+c),\\(a\mathop {\dot {-}} a)&=0,\\(0\mathop {\dot {-}} a)&=0.\\\end{aligned}}$ Notes 1. Characters in Unicode are referenced in prose via the "U+" notation. The hexadecimal number after the "U+" is the character's Unicode code point. 2. taking $c$ to be the neutral element of the monoid 3. if $a\leq b$ with witness $d$ and $b\leq c$ with witness $d'$ then $d+d'$ witnesses that $a\leq c$ 4. Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis, 18: 129–131, doi:10.1007/BF01182254 5. M.Monet (2016-10-14). "Example of a naturally ordered semiring which is not an m-semiring". Mathematics Stack Exchange. Retrieved 2016-10-14. 6. Semirings for breakfast, slide 17 7. Vereschchagin, Nikolai K.; Shen, Alexander (2003). Computable Functions. Translated by V. N. Dubrovskii. American Mathematical Society. p. 141. ISBN 0-8218-2732-4. 8. Warren Jr., Henry S. (2013). Hacker's Delight (2 ed.). Addison Wesley - Pearson Education, Inc. ISBN 978-0-321-84268-8. 9. Jacobs, Bart (1996). "Coalgebraic Specifications and Models of Deterministic Hybrid Systems". In Wirsing, Martin; Nivat, Maurice (eds.). Algebraic Methodology and Software Technology. Lecture Notes in Computer Science. Vol. 1101. Springer. p. 522. ISBN 3-540-61463-X.
Wreath product In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used in the classification of permutation groups and also provide a way of constructing interesting examples of groups. Algebraic structure → Group theory Group theory Basic notions • Subgroup • Normal subgroup • Quotient group • (Semi-)direct product Group homomorphisms • kernel • image • direct sum • wreath product • simple • finite • infinite • continuous • multiplicative • additive • cyclic • abelian • dihedral • nilpotent • solvable • action • Glossary of group theory • List of group theory topics Finite groups • Cyclic group Zn • Symmetric group Sn • Alternating group An • Dihedral group Dn • Quaternion group Q • Cauchy's theorem • Lagrange's theorem • Sylow theorems • Hall's theorem • p-group • Elementary abelian group • Frobenius group • Schur multiplier Classification of finite simple groups • cyclic • alternating • Lie type • sporadic • Discrete groups • Lattices • Integers ($\mathbb {Z} $) • Free group Modular groups • PSL(2, $\mathbb {Z} $) • SL(2, $\mathbb {Z} $) • Arithmetic group • Lattice • Hyperbolic group Topological and Lie groups • Solenoid • Circle • General linear GL(n) • Special linear SL(n) • Orthogonal O(n) • Euclidean E(n) • Special orthogonal SO(n) • Unitary U(n) • Special unitary SU(n) • Symplectic Sp(n) • G2 • F4 • E6 • E7 • E8 • Lorentz • Poincaré • Conformal • Diffeomorphism • Loop Infinite dimensional Lie group • O(∞) • SU(∞) • Sp(∞) Algebraic groups • Linear algebraic group • Reductive group • Abelian variety • Elliptic curve Given two groups $A$ and $H$ (sometimes known as the bottom and top[1]), there exist two variations of the wreath product: the unrestricted wreath product $A{\text{ Wr }}H$ and the restricted wreath product $A{\text{ wr }}H$. The general form, denoted by $A{\text{ Wr}}_{\Omega }H$ or $A{\text{ wr}}_{\Omega }H$ respectively, requires that $H$ acts on some set $\Omega $; when unspecified, usually $\Omega =H$ (a regular wreath product), though a different $\Omega $ is sometimes implied. The two variations coincide when $A$, $H$, and $\Omega $ are all finite. Either variation is also denoted as $A\wr H$ (with \wr for the LaTeX symbol) or A ≀ H (Unicode U+2240). The notion generalizes to semigroups and is a central construction in the Krohn–Rhodes structure theory of finite semigroups. Definition Let $A$ be a group and let $H$ be a group acting on a set $\Omega $ (on the left). The direct product $A^{\Omega }$ of $A$ with itself indexed by $\Omega $ is the set of sequences ${\overline {a}}=(a_{\omega })_{\omega \in \Omega }$ in $A$ indexed by $\Omega $, with a group operation given by pointwise multiplication. The action of $H$ on $\Omega $ can be extended to an action on $A^{\Omega }$ by reindexing, namely by defining $h\cdot (a_{\omega })_{\omega \in \Omega }:=(a_{h^{-1}\cdot \omega })_{\omega \in \Omega }$ for all $h\in H$ and all $(a_{\omega })_{\omega \in \Omega }\in A^{\Omega }$. Then the unrestricted wreath product $A{\text{ Wr}}_{\Omega }H$ of $A$ by $H$ is the semidirect product $A^{\Omega }\rtimes H$ with the action of $H$ on $A^{\Omega }$ given above. The subgroup $A^{\Omega }$ of $A^{\Omega }\rtimes H$ is called the base of the wreath product. The restricted wreath product $A{\text{ wr}}_{\Omega }H$ is constructed in the same way as the unrestricted wreath product except that one uses the direct sum as the base of the wreath product. In this case, the base consists of all sequences in $A$ with finitely-many non-identity entries. In the most common case, $\Omega =H$, and $H$ acts on itself by left multiplication. In this case, the unrestricted and restricted wreath product may be denoted by $A{\text{ Wr }}H$ and $A{\text{ wr }}H$ respectively. This is called the regular wreath product. Notation and conventions The structure of the wreath product of A by H depends on the H-set Ω and in case Ω is infinite it also depends on whether one uses the restricted or unrestricted wreath product. However, in literature the notation used may be deficient and one needs to pay attention to the circumstances. • In literature A≀ΩH may stand for the unrestricted wreath product A WrΩ H or the restricted wreath product A wrΩ H. • Similarly, A≀H may stand for the unrestricted regular wreath product A Wr H or the restricted regular wreath product A wr H. • In literature the H-set Ω may be omitted from the notation even if Ω ≠ H. • In the special case that H = Sn is the symmetric group of degree n it is common in the literature to assume that Ω = {1,...,n} (with the natural action of Sn) and then omit Ω from the notation. That is, A≀Sn commonly denotes A≀{1,...,n}Sn instead of the regular wreath product A≀SnSn. In the first case the base group is the product of n copies of A, in the latter it is the product of n! copies of A. Properties Agreement of unrestricted and restricted wreath product on finite Ω Since the finite direct product is the same as the finite direct sum of groups, it follows that the unrestricted A WrΩ H and the restricted wreath product A wrΩ H agree if the H-set Ω is finite. In particular this is true when Ω = H is finite. Subgroup A wrΩ H is always a subgroup of A WrΩ H. Cardinality If A, H and Ω are finite, then \|A≀ΩH\| = \|A\|\|Ω\|\|H\|.[2] Universal embedding theorem Main article: Universal embedding theorem Universal embedding theorem: If G is an extension of A by H, then there exists a subgroup of the unrestricted wreath product A≀H which is isomorphic to G.[3] This is also known as the Krasner–Kaloujnine embedding theorem. The Krohn–Rhodes theorem involves what is basically the semigroup equivalent of this.[4] Canonical actions of wreath products If the group A acts on a set Λ then there are two canonical ways to construct sets from Ω and Λ on which A WrΩ H (and therefore also A wrΩ H) can act. • The imprimitive wreath product action on Λ × Ω. If ((aω),h) ∈ A WrΩ H and (λ,ω′) ∈ Λ × Ω, then $((a_{\omega }),h)\cdot (\lambda ,\omega '):=(a_{h(\omega ')}\lambda ,h\omega ').$ • The primitive wreath product action on ΛΩ. An element in ΛΩ is a sequence (λω) indexed by the H-set Ω. Given an element ((aω), h) ∈ A WrΩ H its operation on (λω) ∈ ΛΩ is given by $((a_{\omega }),h)\cdot (\lambda _{\omega }):=(a_{h^{-1}\omega }\lambda _{h^{-1}\omega }).$ Examples • The Lamplighter group is the restricted wreath product ℤ2≀ℤ. • ℤm≀Sn (Generalized symmetric group). The base of this wreath product is the n-fold direct product ℤmn = ℤm × ... × ℤm of copies of ℤm where the action φ : Sn → Aut(ℤmn) of the symmetric group Sn of degree n is given by φ(σ)(α1,..., αn) := (ασ(1),..., ασ(n)).[5] • S2≀Sn (Hyperoctahedral group). The action of Sn on {1,...,n} is as above. Since the symmetric group S2 of degree 2 is isomorphic to ℤ2 the hyperoctahedral group is a special case of a generalized symmetric group.[6] • The smallest non-trivial wreath product is ℤ2≀ℤ2, which is the two-dimensional case of the above hyperoctahedral group. It is the symmetry group of the square, also called Dih4, the dihedral group of order 8. • Let p be a prime and let n≥1. Let P be a Sylow p-subgroup of the symmetric group Spn. Then P is isomorphic to the iterated regular wreath product Wn = ℤp ≀ ℤp≀...≀ℤp of n copies of ℤp. Here W1 := ℤp and Wk := Wk−1≀ℤp for all k ≥ 2.[7][8] For instance, the Sylow 2-subgroup of S4 is the above ℤ2≀ℤ2 group. • The Rubik's Cube group is a normal subgroup of index 12 in the product of wreath products, (ℤ3≀S8) × (ℤ2≀S12), the factors corresponding to the symmetries of the 8 corners and 12 edges. • The Sudoku validity preserving transformations (VPT) group contains the double wreath product (S3 ≀ S3) ≀ S2, where the factors are the permutation of rows/columns within a 3-row or 3-column band or stack (S3), the permutation of the bands/stacks themselves (S3) and the transposition, which interchanges the bands and stacks (S2). Here, the index sets Ω are the set of bands (resp. stacks) (\|Ω\| = 3) and the set {bands, stacks} (\|Ω\| = 2). Accordingly, \|S3 ≀ S3\| = \|S3\|3\|S3\| = (3!)4 and \|(S3 ≀ S3) ≀ S2\| = \|S3 ≀ S3\|2\|S2\| = (3!)8 × 2. • Wreath products arise naturally in the symmetry group of complete rooted trees and their graphs. For example, the repeated (iterated) wreath product S2 ≀ S2 ≀ ... ≀ S2 is the automorphism group of a complete binary tree. References 1. Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1998), "Wreath products", Notes on Infinite Permutation Groups, Lecture Notes in Mathematics, Berlin, Heidelberg: Springer, pp. 67–76, doi:10.1007/bfb0092558, ISBN 978-3-540-49813-1, retrieved 2021-05-12 2. Joseph J. Rotman, An Introduction to the Theory of Groups, p. 172 (1995) 3. M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de groupes III", Acta Sci. Math. 14, pp. 69–82 (1951) 4. J D P Meldrum (1995). Wreath Products of Groups and Semigroups. Longman [UK] / Wiley [US]. p. ix. ISBN 978-0-582-02693-3. 5. J. W. Davies and A. O. Morris, "The Schur Multiplier of the Generalized Symmetric Group", J. London Math. Soc. (2), 8, (1974), pp. 615–620 6. P. Graczyk, G. Letac and H. Massam, "The Hyperoctahedral Group, Symmetric Group Representations and the Moments of the Real Wishart Distribution", J. Theoret. Probab. 18 (2005), no. 1, 1–42. 7. Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995) 8. L. Kaloujnine, "La structure des p-groupes de Sylow des groupes symétriques finis", Annales Scientifiques de l'École Normale Supérieure. Troisième Série 65, pp. 239–276 (1948) External links • Wreath product in Encyclopedia of Mathematics. • Some Applications of the Wreath Product Construction. Archived 21 February 2014 at the Wayback Machine
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Double turnstile In logic, the symbol ⊨, ⊧ or $\models $ is called the double turnstile. It is often read as "entails", "models", "is a semantic consequence of" or "is stronger than".[1] It is closely related to the turnstile symbol $\vdash $, which has a single bar across the middle, and which denotes syntactic consequence (in contrast to semantic). Meaning The double turnstile is a binary relation. It has several different meanings in different contexts: • To show semantic consequence, with a set of sentences on the left and a single sentence on the right, to denote that if every sentence on the left is true, the sentence on the right must be true, e.g. $\Gamma \vDash \varphi $. This usage is closely related to the single-barred turnstile symbol which denotes syntactic consequence. • To show satisfaction, with a model (or truth-structure) on the left and a set of sentences on the right, to denote that the structure is a model for (or satisfies) the set of sentences, e.g. ${\mathcal {A}}\models \Gamma $. This is typically done inductively along with restricting the range of a variable assignment, a function mapping each variable symbol to a value in ${\mathcal {A}}$ it might hold. [2] • In this context, the semantic consequence in the previous list can be stated as "For a given model ${\mathcal {A}}$, if ${\mathcal {A}}\models \Gamma $ then ${\mathcal {A}}\vDash \varphi $". • To denote a tautology, $\vDash \varphi $. which is to say that the expression $\varphi $ is a semantic consequence of the empty set. • You can also use this symbol as follows: ⊭ to denote the statement 'does not entail'. Typography In TeX, the turnstile symbols $\vDash $ and $\models $ are obtained from the commands \vDash and \models respectively. In Unicode it is encoded at U+22A8 ⊨ TRUE (&DoubleRightTee;, &vDash;) , and the opposite of it is U+22AD ⊭ NOT TRUE (&nvDash;) . In LaTeX there is the turnstile package, which issues this sign in many ways, including the double turnstile, and is capable of putting labels below or above it, in the correct places. The article A Tool for Logicians is a tutorial on using this package. See also • List of logic symbols • List of mathematical symbols • Turnstile ⊢ References 1. Nederpelt, Rob (2004). "Chapter 7: Strengthening and weakening". Logical Reasoning: A First Course (3rd revised ed.). King's College Publications. p. 62. ISBN 0-9543006-7-X. 2. Open Logic Project, First-order logic (p.7). Accessed 4 January 2022. Common logical symbols ∧ or & and ∨ or ¬ or ~ not → implies ⊃ implies, superset ↔ or ≡ iff \| nand ∀ universal quantification ∃ existential quantification ⊤ true, tautology ⊥ false, contradiction ⊢ entails, proves ⊨ entails, therefore ∴ therefore ∵ because Philosophy portal Mathematics portal
Ordered set operators In mathematical notation, ordered set operators indicate whether an object precedes or succeeds another. These relationship operators are denoted by the unicode symbols U+227A-F, along with symbols located unicode blocks U+228x through U+22Ex. Mathematical Operators[1] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+227x ≰ ≱ ≲ ≳ ≴ ≵ ≶ ≷ ≸ ≹ ≺ ≻ ≼ ≽ ≾ ≿ U+228x ⊀ ⊁ ⊂ ⊃ ⊄ ⊅ ⊆ ⊇ ⊈ ⊉ ⊊ ⊋ ⊌ ⊍ ⊎ ⊏ U+22Bx ⊰ ⊱ ⊲ ⊳ ⊴ ⊵ ⊶ ⊷ ⊸ ⊹ ⊺ ⊻ ⊼ ⊽ ⊾ ⊿ U+22Dx ⋐ ⋑ ⋒ ⋓ ⋔ ⋕ ⋖ ⋗ ⋘ ⋙ ⋚ ⋛ ⋜ ⋝ ⋞ ⋟ U+22Ex ⋠ ⋡ ⋢ ⋣ ⋤ ⋥ ⋦ ⋧ ⋨ ⋩ ⋪ ⋫ ⋬ ⋭ ⋮ ⋯ Notes 1.^ As of Unicode version 7.0 Examples • The relationship x precedes y is written x ≺ y. The relation x precedes or is equal to y is written x ≼ y. • The relationship x succeeds (or follows) y is written x ≻ y. The relation x succeeds or is equal to y is written x ≽ y. Use in political science Political scientists use order relations typically in the context of an agent's choice, for example the preferences of a voter over several political candidates. • x ≺ y means that the voter prefers candidate y over candidate x. • x ∼ y means the voter is indifferent between candidates x and y. • x ≲ y means the voter is indifferent or prefers candidate y.[1] References 1. Cooley, Brandon. "Ordered Sets" (PDF) (Lecture note for: Introduction to Mathematics for Political Science (2019) at Princeton University). pp. 2–3. Retrieved 2021-05-11.{{cite web}}: CS1 maint: url-status (link) See also • Order theory • Partially ordered set • Directional symbols • Polynomial-time reduction • Wolfram Mathworld: precedes and succeeds
Normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup)[1] is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup $N$ of the group $G$ is normal in $G$ if and only if $gng^{-1}\in N$ for all $g\in G$ and $n\in N.$ The usual notation for this relation is $N\triangleleft G.$ "Invariant subgroup" redirects here. Not to be confused with Fully invariant subgroup. Algebraic structure → Group theory Group theory Basic notions • Subgroup • Normal subgroup • Quotient group • (Semi-)direct product Group homomorphisms • kernel • image • direct sum • wreath product • simple • finite • infinite • continuous • multiplicative • additive • cyclic • abelian • dihedral • nilpotent • solvable • action • Glossary of group theory • List of group theory topics Finite groups • Cyclic group Zn • Symmetric group Sn • Alternating group An • Dihedral group Dn • Quaternion group Q • Cauchy's theorem • Lagrange's theorem • Sylow theorems • Hall's theorem • p-group • Elementary abelian group • Frobenius group • Schur multiplier Classification of finite simple groups • cyclic • alternating • Lie type • sporadic • Discrete groups • Lattices • Integers ($\mathbb {Z} $) • Free group Modular groups • PSL(2, $\mathbb {Z} $) • SL(2, $\mathbb {Z} $) • Arithmetic group • Lattice • Hyperbolic group Topological and Lie groups • Solenoid • Circle • General linear GL(n) • Special linear SL(n) • Orthogonal O(n) • Euclidean E(n) • Special orthogonal SO(n) • Unitary U(n) • Special unitary SU(n) • Symplectic Sp(n) • G2 • F4 • E6 • E7 • E8 • Lorentz • Poincaré • Conformal • Diffeomorphism • Loop Infinite dimensional Lie group • O(∞) • SU(∞) • Sp(∞) Algebraic groups • Linear algebraic group • Reductive group • Abelian variety • Elliptic curve Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of $G$ are precisely the kernels of group homomorphisms with domain $G,$ which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups.[2] Definitions A subgroup $N$ of a group $G$ is called a normal subgroup of $G$ if it is invariant under conjugation; that is, the conjugation of an element of $N$ by an element of $G$ is always in $N.$[3] The usual notation for this relation is $N\triangleleft G.$ Equivalent conditions For any subgroup $N$ of $G,$ the following conditions are equivalent to $N$ being a normal subgroup of $G.$ Therefore, any one of them may be taken as the definition. • The image of conjugation of $N$ by any element of $G$ is a subset of $N,$[4] i.e., $gNg^{-1}\subseteq N$ for all $g\in G$. • The image of conjugation of $N$ by any element of $G$ is equal to $N,$[4] i.e., $gNg^{-1}=N$ for all $g\in G$. • For all $g\in G,$ the left and right cosets $gN$ and $Ng$ are equal.[4] • The sets of left and right cosets of $N$ in $G$ coincide.[4] • Multiplication in $G$ preserves the equivalence relation "is in the same left coset as". That is, for every $g,g',h,h'\in G$ satisfying $gN=g'N$ and $hN=h'N$, we have $(gh)N=(g'h')N.$ • There exists a group on the set of left cosets of $N$ where multiplication of any two left cosets $gN$ and $hN$ yields the left coset $(gh)N$. (This group is called the quotient group of $G$ modulo $N$, denoted $G/N$.) • $N$ is a union of conjugacy classes of $G.$[2] • $N$ is preserved by the inner automorphisms of $G.$[5] • There is some group homomorphism $G\to H$ whose kernel is $N.$[2] • There exists a group homomorphism $\phi :G\to H$ whose fibers form a group where the identity element is $N$ and multiplication of any two fibers $\phi ^{-1}(h_{1})$ and $\phi ^{-1}(h_{2})$ yields the fiber $\phi ^{-1}(h_{1}h_{2})$. (This group is the same group $G/N$ mentioned above.) • There is some congruence relation on $G$ for which the equivalence class of the identity element is $N$. • For all $n\in N$ and $g\in G,$ the commutator $[n,g]=n^{-1}g^{-1}ng$ is in $N.$ • Any two elements commute modulo the normal subgroup membership relation. That is, for all $g,h\in G,$ $gh\in N$ if and only if $hg\in N.$ Examples For any group $G,$ the trivial subgroup $\{e\}$ consisting of just the identity element of $G$ is always a normal subgroup of $G.$ Likewise, $G$ itself is always a normal subgroup of $G.$ (If these are the only normal subgroups, then $G$ is said to be simple.)[6] Other named normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup $[G,G].$[7][8] More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup.[9] If $G$ is an abelian group then every subgroup $N$ of $G$ is normal, because $gN=\{gn\}_{n\in N}=\{ng\}_{n\in N}=Ng.$ More generally, for any group $G$, every subgroup of the center $Z(G)$ of $G$ is normal in $G$. (In the special case that $G$ is abelian, the center is all of $G$, hence the fact that all subgroups of an abelian group are normal.) A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.[10] A concrete example of a normal subgroup is the subgroup $N=\{(1),(123),(132)\}$ of the symmetric group $S_{3},$ consisting of the identity and both three-cycles. In particular, one can check that every coset of $N$ is either equal to $N$ itself or is equal to $(12)N=\{(12),(23),(13)\}.$ On the other hand, the subgroup $H=\{(1),(12)\}$ is not normal in $S_{3}$ since $(123)H=\{(123),(13)\}\neq \{(123),(23)\}=H(123).$[11] This illustrates the general fact that any subgroup $H\leq G$ of index two is normal. As an example of a normal subgroup within a matrix group, consider the general linear group $\mathrm {GL} _{n}(\mathbf {R} )$ of all invertible $n\times n$ matrices with real entries under the operation of matrix multiplication and its subgroup $\mathrm {SL} _{n}(\mathbf {R} )$ of all $n\times n$ matrices of determinant 1 (the special linear group). To see why the subgroup $\mathrm {SL} _{n}(\mathbf {R} )$ is normal in $\mathrm {GL} _{n}(\mathbf {R} )$, consider any matrix $X$ in $\mathrm {SL} _{n}(\mathbf {R} )$ and any invertible matrix $A$. Then using the two important identities $\det(AB)=\det(A)\det(B)$ and $\det(A^{-1})=\det(A)^{-1}$, one has that $\det(AXA^{-1})=\det(A)\det(X)\det(A)^{-1}=\det(X)=1$, and so $AXA^{-1}\in \mathrm {SL} _{n}(\mathbf {R} )$ as well. This means $\mathrm {SL} _{n}(\mathbf {R} )$ is closed under conjugation in $\mathrm {GL} _{n}(\mathbf {R} )$, so it is a normal subgroup.[lower-alpha 1] In the Rubik's Cube group, the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.[12] The translation group is a normal subgroup of the Euclidean group in any dimension.[13] This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation. By contrast, the subgroup of all rotations about the origin is not a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin. Properties • If $H$ is a normal subgroup of $G,$ and $K$ is a subgroup of $G$ containing $H,$ then $H$ is a normal subgroup of $K.$[14] • A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8.[15] However, a characteristic subgroup of a normal subgroup is normal.[16] A group in which normality is transitive is called a T-group.[17] • The two groups $G$ and $H$ are normal subgroups of their direct product $G\times H.$ • If the group $G$ is a semidirect product $G=N\rtimes H,$ then $N$ is normal in $G,$ though $H$ need not be normal in $G.$ • If $M$ and $N$ are normal subgroups of an additive group $G$ such that $G=M+N$ and $M\cap N=\{0\}$, then $G=M\oplus N.$[18] • Normality is preserved under surjective homomorphisms;[19] that is, if $G\to H$ is a surjective group homomorphism and $N$ is normal in $G,$ then the image $f(N)$ is normal in $H.$ • Normality is preserved by taking inverse images;[19] that is, if $G\to H$ is a group homomorphism and $N$ is normal in $H,$ then the inverse image $f^{-1}(N)$ is normal in $G.$ • Normality is preserved on taking direct products;[20] that is, if $N_{1}\triangleleft G_{1}$ and $N_{2}\triangleleft G_{2},$ then $N_{1}\times N_{2}\;\triangleleft \;G_{1}\times G_{2}.$ • Every subgroup of index 2 is normal. More generally, a subgroup, $H,$ of finite index, $n,$ in $G$ contains a subgroup, $K,$ normal in $G$ and of index dividing $n!$ called the normal core. In particular, if $p$ is the smallest prime dividing the order of $G,$ then every subgroup of index $p$ is normal.[21] • The fact that normal subgroups of $G$ are precisely the kernels of group homomorphisms defined on $G$ accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images,[22] a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup. Lattice of normal subgroups Given two normal subgroups, $N$ and $M,$ of $G,$ their intersection $N\cap M$and their product $NM=\{nm:n\in N\;{\text{ and }}\;m\in M\}$ are also normal subgroups of $G.$ The normal subgroups of $G$ form a lattice under subset inclusion with least element, $\{e\},$ and greatest element, $G.$ The meet of two normal subgroups, $N$ and $M,$ in this lattice is their intersection and the join is their product. The lattice is complete and modular.[20] Normal subgroups, quotient groups and homomorphisms If $N$ is a normal subgroup, we can define a multiplication on cosets as follows: $\left(a_{1}N\right)\left(a_{2}N\right):=\left(a_{1}a_{2}\right)N.$ This relation defines a mapping $G/N\times G/N\to G/N.$ To show that this mapping is well-defined, one needs to prove that the choice of representative elements $a_{1},a_{2}$ does not affect the result. To this end, consider some other representative elements $a_{1}'\in a_{1}N,a_{2}'\in a_{2}N.$ Then there are $n_{1},n_{2}\in N$ such that $a_{1}'=a_{1}n_{1},a_{2}'=a_{2}n_{2}.$ It follows that $a_{1}'a_{2}'N=a_{1}n_{1}a_{2}n_{2}N=a_{1}a_{2}n_{1}'n_{2}N=a_{1}a_{2}N,$ where we also used the fact that $N$ is a normal subgroup, and therefore there is $n_{1}'\in N$ such that $n_{1}a_{2}=a_{2}n_{1}'.$ This proves that this product is a well-defined mapping between cosets. With this operation, the set of cosets is itself a group, called the quotient group and denoted with $G/N.$ There is a natural homomorphism, $f:G\to G/N,$ given by $f(a)=aN.$ This homomorphism maps $N$ into the identity element of $G/N,$ which is the coset $eN=N,$[23] that is, $\ker(f)=N.$ In general, a group homomorphism, $f:G\to H$ sends subgroups of $G$ to subgroups of $H.$ Also, the preimage of any subgroup of $H$ is a subgroup of $G.$ We call the preimage of the trivial group $\{e\}$ in $H$ the kernel of the homomorphism and denote it by $\ker f.$ As it turns out, the kernel is always normal and the image of $G,f(G),$ is always isomorphic to $G/\ker f$ (the first isomorphism theorem).[24] In fact, this correspondence is a bijection between the set of all quotient groups of $G,G/N,$ and the set of all homomorphic images of $G$ (up to isomorphism).[25] It is also easy to see that the kernel of the quotient map, $f:G\to G/N,$ is $N$ itself, so the normal subgroups are precisely the kernels of homomorphisms with domain $G.$[26] See also Operations taking subgroups to subgroups • Normalizer • Conjugate closure • Normal core Subgroup properties complementary (or opposite) to normality • Malnormal subgroup • Contranormal subgroup • Abnormal subgroup • Self-normalizing subgroup Subgroup properties stronger than normality • Characteristic subgroup • Fully characteristic subgroup Subgroup properties weaker than normality • Subnormal subgroup • Ascendant subgroup • Descendant subgroup • Quasinormal subgroup • Seminormal subgroup • Conjugate permutable subgroup • Modular subgroup • Pronormal subgroup • Paranormal subgroup • Polynormal subgroup • C-normal subgroup Related notions in algebra • Ideal (ring theory) Notes 1. In other language: $\det $ is a homomorphism from $\mathrm {GL} _{n}(\mathbf {R} )$ to the multiplicative subgroup $\mathbf {R} ^{\times }$, and $\mathrm {SL} _{n}(\mathbf {R} )$ is the kernel. Both arguments also work over the complex numbers, or indeed over an arbitrary field. References 1. Bradley 2010, p. 12. 2. Cantrell 2000, p. 160. 3. Dummit & Foote 2004. 4. Hungerford 2003, p. 41. 5. Fraleigh 2003, p. 141. 6. Robinson 1996, p. 16. 7. Hungerford 2003, p. 45. 8. Hall 1999, p. 138. 9. Hall 1999, p. 32. 10. Hall 1999, p. 190. 11. Judson 2020, Section 10.1. 12. Bergvall et al. 2010, p. 96. 13. Thurston 1997, p. 218. 14. Hungerford 2003, p. 42. 15. Robinson 1996, p. 17. 16. Robinson 1996, p. 28. 17. Robinson 1996, p. 402. 18. Hungerford 2013, p. 290. 19. Hall 1999, p. 29. 20. Hungerford 2003, p. 46. 21. Robinson 1996, p. 36. 22. Dõmõsi & Nehaniv 2004, p. 7. 23. Hungerford 2003, pp. 42–43. 24. Hungerford 2003, p. 44. 25. Robinson 1996, p. 20. 26. Hall 1999, p. 27. Bibliography • Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010). "On Rubik's Cube" (PDF). KTH. {{cite journal}}: Cite journal requires \|journal= (help) • Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 978-0-521-59180-5. • Dõmõsi, Pál; Nehaniv, Chrystopher L. (2004). Algebraic Theory of Automata Networks. SIAM Monographs on Discrete Mathematics and Applications. SIAM. • Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. • Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley. ISBN 978-0-321-15608-2. • Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8. • Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. • Hungerford, Thomas (2013). Abstract Algebra: An Introduction. Brooks/Cole Cengage Learning. • Judson, Thomas W. (2020). Abstract Algebra: Theory and Applications. • Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001. • Thurston, William (1997). Levy, Silvio (ed.). Three-dimensional geometry and topology, Vol. 1. Princeton Mathematical Series. Princeton University Press. ISBN 978-0-691-08304-9. • Bradley, C. J. (2010). The mathematical theory of symmetry in solids : representation theory for point groups and space groups. Oxford New York: Clarendon Press. ISBN 978-0-19-958258-7. OCLC 859155300. Further reading • I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp. External links • Weisstein, Eric W. "normal subgroup". MathWorld. • Normal subgroup in Springer's Encyclopedia of Mathematics • Robert Ash: Group Fundamentals in Abstract Algebra. The Basic Graduate Year • Timothy Gowers, Normal subgroups and quotient groups • John Baez, What's a Normal Subgroup?
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: $A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {{a}_{ji}}}$ For matrices with symmetry over the real number field, see Symmetric matrix. or in matrix form: $A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}.$ Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix $A$ is denoted by $A^{\mathsf {H}},$ then the Hermitian property can be written concisely as $A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}$ Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are $A^{\mathsf {H}}=A^{\dagger }=A^{\ast },$ although in quantum mechanics, $A^{\ast }$ typically means the complex conjugate only, and not the conjugate transpose. Alternative characterizations Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below: Equality with the adjoint A square matrix $A$ is Hermitian if and only if it is equal to its adjoint, that is, it satisfies $\langle \mathbf {w} ,A\mathbf {v} \rangle =\langle A\mathbf {w} ,\mathbf {v} \rangle ,$ for any pair of vectors $\mathbf {v} ,\mathbf {w} ,$ where $\langle \cdot ,\cdot \rangle $ denotes the inner product operation. This is also the way that the more general concept of self-adjoint operator is defined. Reality of quadratic forms An $n\times {}n$ matrix $A$ is Hermitian if and only if $\langle \mathbf {v} ,A\mathbf {v} \rangle \in \mathbb {R} ,\quad \mathbf {v} \in \mathbb {C} ^{n}.$ Spectral properties A square matrix $A$ is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues. Applications Hermitian matrices are fundamental to quantum mechanics because they describe operators with necessarily real eigenvalues. An eigenvalue $a$ of an operator ${\hat {A}}$ on some quantum state $\|\psi \rangle $ is one of the possible measurement outcomes of the operator, which necessitates the need for operators with real eigenvalues. Examples and solutions In this section, the conjugate transpose of matrix $A$ is denoted as $A^{\mathsf {H}},$ the transpose of matrix $A$ is denoted as $A^{\mathsf {T}}$ and conjugate of matrix $A$ is denoted as ${\overline {A}}.$ See the following example: ${\begin{bmatrix}0&a-ib&c-id\\a+ib&1&m-in\\c+id&m+in&2\end{bmatrix}}$ The diagonal elements must be real, as they must be their own complex conjugate. Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients,[1][2] which results in skew-Hermitian matrices. Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix $A$ equals the product of a matrix with its conjugate transpose, that is, $A=BB^{\mathsf {H}},$ then $A$ is a Hermitian positive semi-definite matrix. Furthermore, if $B$ is row full-rank, then $A$ is positive definite. Properties Main diagonal values are real The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are real. Proof By definition of the Hermitian matrix $H_{ij}={\overline {H}}_{ji}$ so for i = j the above follows. Only the main diagonal entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their off-diagonal elements, as long as diagonally-opposite entries are complex conjugates. Symmetric A matrix that has only real entries is symmetric if and only if it is Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix. Proof $H_{ij}={\overline {H}}_{ji}$ by definition. Thus $H_{ij}=H_{ji}$ (matrix symmetry) if and only if $H_{ij}={\overline {H}}_{ij}$ ($H_{ij}$ is real). So, if a real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit $i,$ then it becomes Hermitian. Normal Every Hermitian matrix is a normal matrix. That is to say, $AA^{\mathsf {H}}=A^{\mathsf {H}}A.$ Proof $A=A^{\mathsf {H}},$ so $AA^{\mathsf {H}}=AA=A^{\mathsf {H}}A.$ Diagonalizable The finite-dimensional spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix A with dimension n are real, and that A has n linearly independent eigenvectors. Moreover, a Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an orthogonal basis of Cn consisting of n eigenvectors of A. Sum of Hermitian matrices The sum of any two Hermitian matrices is Hermitian. Proof $(A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},$ as claimed. Inverse is Hermitian The inverse of an invertible Hermitian matrix is Hermitian as well. Proof If $A^{-1}A=I,$ then $I=I^{\mathsf {H}}=\left(A^{-1}A\right)^{\mathsf {H}}=A^{\mathsf {H}}\left(A^{-1}\right)^{\mathsf {H}}=A\left(A^{-1}\right)^{\mathsf {H}},$ so $A^{-1}=\left(A^{-1}\right)^{\mathsf {H}}$ as claimed. Associative product of Hermitian matrices The product of two Hermitian matrices A and B is Hermitian if and only if AB = BA. Proof $(AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}\ {\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.$ Thus $(AB)^{\mathsf {H}}=AB$ if and only if $AB=BA.$ Thus An is Hermitian if A is Hermitian and n is an integer. ABA Hermitian If A and B are Hermitian, then ABA is also Hermitian. Proof $(ABA)^{\mathsf {H}}=(A(BA))^{\mathsf {H}}=(BA)^{\mathsf {H}}A^{\mathsf {H}}=A^{\mathsf {H}}B^{\mathsf {H}}A^{\mathsf {H}}=ABA$ vHAv is real for complex v For an arbitrary complex valued vector v the product $\mathbf {v} ^{\mathsf {H}}A\mathbf {v} $ is real because of $\mathbf {v} ^{\mathsf {H}}A\mathbf {v} =\left(\mathbf {v} ^{\mathsf {H}}A\mathbf {v} \right)^{\mathsf {H}}.$ This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system, e.g. total spin, which have to be real. Complex Hermitian forms vector space over R The Hermitian complex n-by-n matrices do not form a vector space over the complex numbers, C, since the identity matrix In is Hermitian, but i In is not. However the complex Hermitian matrices do form a vector space over the real numbers R. In the 2n2-dimensional vector space of complex n × n matrices over R, the complex Hermitian matrices form a subspace of dimension n2. If Ejk denotes the n-by-n matrix with a 1 in the j,k position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows: $E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})$ together with the set of matrices of the form ${\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)$ and the matrices ${\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)$ where $i$ denotes the imaginary unit, $i={\sqrt {-1}}~.$ An example is that the four Pauli matrices form a complete basis for the vector space of all complex 2-by-2 Hermitian matrices over R. Eigendecomposition If n orthonormal eigenvectors $\mathbf {u} _{1},\dots ,\mathbf {u} _{n}$ of a Hermitian matrix are chosen and written as the columns of the matrix U, then one eigendecomposition of A is $A=U\Lambda U^{\mathsf {H}}$ where $UU^{\mathsf {H}}=I=U^{\mathsf {H}}U$ and therefore $A=\sum _{j}\lambda _{j}\mathbf {u} _{j}\mathbf {u} _{j}^{\mathsf {H}},$ where $\lambda _{j}$ are the eigenvalues on the diagonal of the diagonal matrix $\Lambda .$ Singular values[3] The singular values of $A$ are the absolute values of its eigenvalues: Since $A$ has an eigendecomposition $A=U\Lambda U^{H}$, where $U$ is a unitary matrix (its columns are orthonormal vectors; see above), a singular value decomposition of $A$ is $A=U\|\Lambda \|{\text{sgn}}(\Lambda )U^{H}$, where $\|\Lambda \|$ and ${\text{sgn}}(\Lambda )$ are diagonal matrices containing the absolute values $\|\lambda \|$ and signs ${\text{sgn}}(\lambda )$ of $A$'s eigenvalues, respectively. $\operatorname {sgn}(\Lambda )U^{H}$ is unitary, since the columns of $U^{H}$ are only getting multiplied by $\pm 1$. $\|\Lambda \|$ contains the singular values of $A$, namely, the absolute values of its eigenvalues. Real determinant The determinant of a Hermitian matrix is real: Proof $\det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}$ Therefore if $A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}.$ (Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.) Decomposition into Hermitian and skew-Hermitian matrices Additional facts related to Hermitian matrices include: • The sum of a square matrix and its conjugate transpose $\left(A+A^{\mathsf {H}}\right)$ is Hermitian. • The difference of a square matrix and its conjugate transpose $\left(A-A^{\mathsf {H}}\right)$ is skew-Hermitian (also called antihermitian). This implies that the commutator of two Hermitian matrices is skew-Hermitian. • An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B. This is known as the Toeplitz decomposition of C.[4]: 227 $C=A+B\quad {\text{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\text{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)$ Rayleigh quotient Main article: Rayleigh quotient In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient[5] $R(M,\mathbf {x} ),$ is defined as:[4]: p. 234 [6] $R(M,\mathbf {x} ):={\frac {\mathbf {x} ^{\mathsf {H}}M\mathbf {x} }{\mathbf {x} ^{\mathsf {H}}\mathbf {x} }}.$ For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose $\mathbf {x} ^{\mathsf {H}}$ to the usual transpose $\mathbf {x} ^{\mathsf {T}}.$ $R(M,c\mathbf {x} )=R(M,\mathbf {x} )$ for any non-zero real scalar $c.$ Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues. It can be shown[4] that, for a given matrix, the Rayleigh quotient reaches its minimum value $\lambda _{\min }$ (the smallest eigenvalue of M) when $\mathbf {x} $ is $\mathbf {v} _{\min }$ (the corresponding eigenvector). Similarly, $R(M,\mathbf {x} )\leq \lambda _{\max }$ and $R(M,\mathbf {v} _{\max })=\lambda _{\max }.$ The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration. The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, $\lambda _{\max }$ is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to M associates the Rayleigh quotient R(M, x) for a fixed x and M varying through the algebra would be referred to as "vector state" of the algebra. See also • Complex symmetric matrix – Matrix equal to its transposePages displaying short descriptions of redirect targets • Haynsworth inertia additivity formula – Counts positive, negative, and zero eigenvalues of a block partitioned Hermitian matrix • Hermitian form – Generalization of a bilinear formPages displaying short descriptions of redirect targets • Normal matrix – Matrix that commutes with its conjugate transpose • Schur–Horn theorem – Characterizes the diagonal of a Hermitian matrix with given eigenvalues • Self-adjoint operator – Linear operator equal to its own adjoint • Skew-Hermitian matrix – Matrix whose conjugate transpose is its negative (additive inverse) (anti-Hermitian matrix) • Unitary matrix – Complex matrix whose conjugate transpose equals its inverse • Vector space – Algebraic structure in linear algebra References 1. Frankel, Theodore (2004). The Geometry of Physics: an introduction. Cambridge University Press. p. 652. ISBN 0-521-53927-7. 2. Physics 125 Course Notes at California Institute of Technology 3. Trefethan, Lloyd N.; Bau, III, David (1997). Numerical linear algebra. Philadelphia, PA, USA: SIAM. p. 34. ISBN 0-89871-361-7. 4. Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402. 5. Also known as the Rayleigh–Ritz ratio; named after Walther Ritz and Lord Rayleigh. 6. Parlet B. N. The symmetric eigenvalue problem, SIAM, Classics in Applied Mathematics,1998 External links • "Hermitian matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Visualizing Hermitian Matrix as An Ellipse with Dr. Geo, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation. • "Hermitian Matrices". MathPages.com. 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Exclusive or Exclusive or or exclusive disjunction or exclusive alternation, also known as non-equivalence which is the negation of equivalence, is a logical operation that is true if and only if its arguments differ (one is true, the other is false).[1] Exclusive or XOR Truth table$(0110)$ Logic gate Normal forms Disjunctive${\overline {x}}\cdot y+x\cdot {\overline {y}}$ Conjunctive$({\overline {x}}+{\overline {y}})\cdot (x+y)$ Zhegalkin polynomial$x\oplus y$ Post's lattices 0-preservingyes 1-preservingno Monotoneno Affineyes It is symbolized by the prefix operator $J$[2]: 16 and by the infix operators XOR (/ˌɛks ˈɔːr/, /ˌɛks ˈɔː/, /ˈksɔːr/ or /ˈksɔː/), EOR, EXOR, ${\dot {\vee }}$, ${\overline {\vee }}$, ${\underline {\vee }}$, ⩛, $\oplus $, $\nleftrightarrow $ and $\not \equiv $. It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator excludes that case. This is sometimes thought of as "one or the other but not both" or "either one or the other". This could be written as "A or B, but not, A and B". XOR is equivalent to logical inequality (NEQ) since it is true only when the inputs are different (one is true, and one is false). The negation of XOR is the logical biconditional, which yields true if and only if the two inputs are the same, which is equivalent to logical equality (EQ). Since it is associative, it may be considered to be an n-ary operator which is true if and only if an odd number of arguments are true. That is, a XOR b XOR ... may be treated as XOR(a,b,...). Definition The truth table of $A\oplus B$ shows that it outputs true whenever the inputs differ: $A$ $B$ $A\oplus B$ FalseFalseFalse FalseTrueTrue TrueFalseTrue TrueTrueFalse Equivalences, elimination, and introduction Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is true if and only if one is true and the other is false. For example, if two horses are racing, then one of the two will win the race, but not both of them. The exclusive disjunction $p\nleftrightarrow q$, also denoted by $p\operatorname {?} q$ or $Jpq$, can be expressed in terms of the logical conjunction ("logical and", $\wedge $), the disjunction ("logical or", $\lor $), and the negation ($\lnot $) as follows: ${\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land \lnot (p\land q)\end{matrix}}$ The exclusive disjunction $p\nleftrightarrow q$ can also be expressed in the following way: ${\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)\lor (\lnot p\land q)\end{matrix}}$ This representation of XOR may be found useful when constructing a circuit or network, because it has only one $\lnot $ operation and small number of $\wedge $ and $\lor $ operations. A proof of this identity is given below: ${\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\[3pt]&=&((p\land \lnot q)\lor \lnot p)&\land &((p\land \lnot q)\lor q)\\[3pt]&=&((p\lor \lnot p)\land (\lnot q\lor \lnot p))&\land &((p\lor q)\land (\lnot q\lor q))\\[3pt]&=&(\lnot p\lor \lnot q)&\land &(p\lor q)\\[3pt]&=&\lnot (p\land q)&\land &(p\lor q)\end{matrix}}$ It is sometimes useful to write $p\nleftrightarrow q$ in the following way: ${\begin{matrix}p\nleftrightarrow q&=&\lnot ((p\land q)\lor (\lnot p\land \lnot q))\end{matrix}}$ or: ${\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land (\lnot p\lor \lnot q)\end{matrix}}$ This equivalence can be established by applying De Morgan's laws twice to the fourth line of the above proof. The exclusive or is also equivalent to the negation of a logical biconditional, by the rules of material implication (a material conditional is equivalent to the disjunction of the negation of its antecedent and its consequence) and material equivalence. In summary, we have, in mathematical and in engineering notation: ${\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)&=&p{\overline {q}}+{\overline {p}}q\\[3pt]&=&(p\lor q)&\land &(\lnot p\lor \lnot q)&=&(p+q)({\overline {p}}+{\overline {q}})\\[3pt]&=&(p\lor q)&\land &\lnot (p\land q)&=&(p+q)({\overline {pq}})\end{matrix}}$ Negation of the operator The spirit of De Morgan's laws can be applied, we have: $\lnot (p\nleftrightarrow q)\Leftrightarrow \lnot p\nleftrightarrow q\Leftrightarrow p\nleftrightarrow \lnot q.$ Relation to modern algebra Although the operators $\wedge $ (conjunction) and $\lor $ (disjunction) are very useful in logic systems, they fail a more generalizable structure in the following way: The systems $(\{T,F\},\wedge )$ and $(\{T,F\},\lor )$ are monoids, but neither is a group. This unfortunately prevents the combination of these two systems into larger structures, such as a mathematical ring. However, the system using exclusive or $(\{T,F\},\oplus )$ is an abelian group. The combination of operators $\wedge $ and $\oplus $ over elements $\{T,F\}$ produce the well-known two-element field $\mathbb {F} _{2}$. This field can represent any logic obtainable with the system $(\land ,\lor )$ and has the added benefit of the arsenal of algebraic analysis tools for fields. More specifically, if one associates $F$ with 0 and $T$ with 1, one can interpret the logical "AND" operation as multiplication on $\mathbb {F} _{2}$ and the "XOR" operation as addition on $\mathbb {F} _{2}$: ${\begin{matrix}r=p\land q&\Leftrightarrow &r=p\cdot q{\pmod {2}}\\[3pt]r=p\oplus q&\Leftrightarrow &r=p+q{\pmod {2}}\\\end{matrix}}$ The description of a Boolean function as a polynomial in $\mathbb {F} _{2}$, using this basis, is called the function's algebraic normal form.[3] Exclusive or in natural language Disjunction is often understood exclusively in natural languages. In English, the disjunctive word "or" is often understood exclusively, particularly when used with the particle "either". The English example below would normally be understood in conversation as implying that Mary is not both a singer and a poet.[4][5] 1. Mary is a singer or a poet. However, disjunction can also be understood inclusively, even in combination with "either". For instance, the first example below shows that "either" can be felicitously used in combination with an outright statement that both disjuncts are true. The second example shows that the exclusive inference vanishes away under downward entailing contexts. If disjunction were understood as exclusive in this example, it would leave open the possibility that some people ate both rice and beans.[4] 2. Mary is either a singer or a poet or both. 3. Nobody ate either rice or beans. Examples such as the above have motivated analyses of the exclusivity inference as pragmatic conversational implicatures calculated on the basis of an inclusive semantics. Implicatures are typically cancellable and do not arise in downward entailing contexts if their calculation depends on the Maxim of Quantity. However, some researchers have treated exclusivity as a bona fide semantic entailment and proposed nonclassical logics which would validate it.[4] This behavior of English "or" is also found in other languages. However, many languages have disjunctive constructions which are robustly exclusive such as French soit... soit.[4] Alternative symbols The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphasized in a given context of discussion. In addition to the abbreviation "XOR", any of the following symbols may also be seen: • $+$ was used by George Boole in 1847.[6] Although Boole used $+$ mainly on classes, he also considered the case that $x,y$ are propositions in $x+y$, and at the time $+$ is a connective. Furthermore, Boole used it exclusively. Although such use doesn't show the relationship between inclusive disjunction (for which $\vee $ is almost fixedly used nowadays) and exclusive disjunction, and may also bright about confusions with its other uses, some classical and modern textbooks still keep such use.[7][8] • ${\overline {\vee }}$ was used by Christine Ladd-Franklin in 1883.[9] Strictly speaking, Ladd used $A\operatorname {\overline {\vee }} B$ to express "$A$ is-not $B$" or "No $A$ is $B$", i.e., used ${\overline {\vee }}$ as exclusions, while implicitly ${\overline {\vee }}$ has the meaning of exclusive disjunction since the article is titled as "On the Algebra of Logic". • $\not =$, denoting the negation of equivalence, was used by Ernst Schröder in 1890,[10]: 307 Although the usage of $=$ as equivalence could be dated back to George Boole in 1847,[6] during the 40 years after Boole, his followers, such as Charles Sanders Peirce, Hugh MacColl, Giuseppe Peano and so on, didn't use $\not =$ as non-equivalence literally which is possibly because it could be defined from negation and equivalence easily. • $\circ $ was used by Giuseppe Peano in 1894: "$a\circ b=a-b\,\cup \,b-a$. The sign $\circ $ corresponds to Latin aut; the sign $\cup $ to vel."[11]: 10 Note that the Latin word "aut" means "exclusive or" and "vel" means "inclusive or", and that Peano use $\cup $ as inclusive disjunction. • $\vee \vee $ was used by Izrail Solomonovich Gradshtein (Израиль Соломонович Градштейн) in 1936.[12]: 76 • $\oplus $ was used by Claude Shannon in 1938.[13] Shannon borrowed the symbol as exclusive disjunction from Edward Vermilye Huntington in 1904.[14] Huntington borrowed the symbol from Gottfried Wilhelm Leibniz in 1890 (the original date is not definitely known, but almost certainly it's written after 1685; and 1890 is the publishing time).[15] While both Huntington in 1904 and Leibniz in 1890 used the symbol as an algebraic operation. Furthermore, Huntington in 1904 used the symbol as inclusive disjunction (logical sum) too, and in 1933 used $+$ as inclusive disjunction.[16] • $\not \equiv $, also denoting the negation of equivalence, was used by Alonzo Church in 1944.[17] • $J$ (as a prefix operator, $J\phi \psi $) was used by Józef Maria Bocheński in 1949.[2]: 16 Somebody[18] may mistake that it's Jan Łukasiewicz who is the first to use $J$ for exclusive disjunction (it seems that the mistake spreads widely), while neither in 1929[19] nor in other works did Łukasiewicz make such use. In fact, in 1949 Bocheński introduced a system of Polish notation that names all 16 binary connectives of classical logic which is a compatible extension of the notation of Łukasiewicz in 1929, and in which $J$ for exclusive disjunction appeared at the first time. Bocheński's usage of $J$ as exclusive disjunction has no relationship with the Polish "alternatywa rozłączna" of "exclusive or" and is an accident for which see the table on page 16 of the book in 1949. • ^, the caret, has been used in several programming languages to denote the bitwise exclusive or operator, beginning with C[20] and also including C++, C#, D, Java, Perl, Ruby, PHP and Python. • The symmetric difference of two sets $S$ and $T$, which may be interpreted as their elementwise exclusive or, has variously been denoted as $S\ominus T$, $S\mathop {\triangledown } T$, or $S\mathop {\vartriangle } T$.[21] Properties Commutativity: yes $A\oplus B$ $\Leftrightarrow $ $B\oplus A$ $\Leftrightarrow $ Associativity: yes $~A$ $~~~\oplus ~~~$ $(B\oplus C)$ $\Leftrightarrow $ $(A\oplus B)$ $~~~\oplus ~~~$ $~C$ $~~~\oplus ~~~$ $\Leftrightarrow $ $\Leftrightarrow $ $~~~\oplus ~~~$ Distributivity: The exclusive or doesn't distribute over any binary function (not even itself), but logical conjunction distributes over exclusive or. $C\land (A\oplus B)=(C\land A)\oplus (C\land B)$ (Conjunction and exclusive or form the multiplication and addition operations of a field GF(2), and as in any field they obey the distributive law.) Idempotency: no $~A~$ $~\oplus ~$ $~A~$ $\Leftrightarrow $ $~0~$ $\nLeftrightarrow $ $~A~$ $~\oplus ~$ $\Leftrightarrow $ $\nLeftrightarrow $ Monotonicity: no $A\rightarrow B$ $\nRightarrow $ $(A\oplus C)$ $\rightarrow $ $(B\oplus C)$ $\nRightarrow $ $\Leftrightarrow $ $\rightarrow $ Truth-preserving: no When all inputs are true, the output is not true. $A\land B$ $\nRightarrow $ $A\oplus B$ $\nRightarrow $ Falsehood-preserving: yes When all inputs are false, the output is false. $A\oplus B$ $\Rightarrow $ $A\lor B$ $\Rightarrow $ Walsh spectrum: (2,0,0,−2) Non-linearity: 0 The function is linear. If using binary values for true (1) and false (0), then exclusive or works exactly like addition modulo 2. Computer science Bitwise operation Main article: Bitwise operation Exclusive disjunction is often used for bitwise operations. Examples: • 1 XOR 1 = 0 • 1 XOR 0 = 1 • 0 XOR 1 = 1 • 0 XOR 0 = 0 • 11102 XOR 10012 = 01112 (this is equivalent to addition without carry) As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of two n-bit strings is identical to the standard vector of addition in the vector space $(\mathbb {Z} /2\mathbb {Z} )^{n}$. In computer science, exclusive disjunction has several uses: • It tells whether two bits are unequal. • It is an optional bit-flipper (the deciding input chooses whether to invert the data input). • It tells whether there is an odd number of 1 bits ($A\oplus B\oplus C\oplus D\oplus E$ is true if and only if an odd number of the variables are true), which is equal to the parity bit returned by a parity function. In logical circuits, a simple adder can be made with an XOR gate to add the numbers, and a series of AND, OR and NOT gates to create the carry output. On some computer architectures, it is more efficient to store a zero in a register by XOR-ing the register with itself (bits XOR-ed with themselves are always zero) instead of loading and storing the value zero. In simple threshold-activated neural networks, modeling the XOR function requires a second layer because XOR is not a linearly separable function. Exclusive-or is sometimes used as a simple mixing function in cryptography, for example, with one-time pad or Feistel network systems. Exclusive-or is also heavily used in block ciphers such as AES (Rijndael) or Serpent and in block cipher implementation (CBC, CFB, OFB or CTR). Similarly, XOR can be used in generating entropy pools for hardware random number generators. The XOR operation preserves randomness, meaning that a random bit XORed with a non-random bit will result in a random bit. Multiple sources of potentially random data can be combined using XOR, and the unpredictability of the output is guaranteed to be at least as good as the best individual source.[22] XOR is used in RAID 3–6 for creating parity information. For example, RAID can "back up" bytes 100111002 and 011011002 from two (or more) hard drives by XORing the just mentioned bytes, resulting in (111100002) and writing it to another drive. Under this method, if any one of the three hard drives are lost, the lost byte can be re-created by XORing bytes from the remaining drives. For instance, if the drive containing 011011002 is lost, 100111002 and 111100002 can be XORed to recover the lost byte.[23] XOR is also used to detect an overflow in the result of a signed binary arithmetic operation. If the leftmost retained bit of the result is not the same as the infinite number of digits to the left, then that means overflow occurred. XORing those two bits will give a "1" if there is an overflow. XOR can be used to swap two numeric variables in computers, using the XOR swap algorithm; however this is regarded as more of a curiosity and not encouraged in practice. XOR linked lists leverage XOR properties in order to save space to represent doubly linked list data structures. In computer graphics, XOR-based drawing methods are often used to manage such items as bounding boxes and cursors on systems without alpha channels or overlay planes. Encodings It is also called "not left-right arrow" (\nleftrightarrow) in LaTeX-based markdown ($\nleftrightarrow $). Apart from the ASCII codes, the operator is encoded at U+22BB ⊻ XOR (&veebar;) and U+2295 ⊕ CIRCLED PLUS (&CirclePlus;, &oplus;), both in block mathematical operators. See also • Material conditional • (Paradox) • Affirming a disjunct • Ampheck • Controlled NOT gate • Disjunctive syllogism • Inclusive or • Involution • List of Boolean algebra topics • Logical graph • Logical value • Propositional calculus • Rule 90 • XOR cipher • XOR gate • XOR linked list Notes 1. Germundsson, Roger; Weisstein, Eric. "XOR". MathWorld. Wolfram Research. Retrieved 17 June 2015. 2. Bocheński, J. M. (1949). Précis de logique mathématique (PDF) (in French). The Netherlands: F. G. Kroonder, Bussum, Pays-Bas. Translated as Bocheński, J. M. (1959). A Precis of Mathematical Logic. Translated by Bird, O. Dordrecht, Holland: D. Reidel Publishing Company. doi:10.1007/978-94-017-0592-9. 3. Joux, Antoine (2009). "9.2: Algebraic normal forms of Boolean functions". Algorithmic Cryptanalysis. CRC Press. pp. 285–286. ISBN 9781420070033. 4. Aloni, Maria (2016). "Disjunction". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2016 ed.). Metaphysics Research Lab, Stanford University. Retrieved 2020-09-03. 5. Jennings quotes numerous authors saying that the word "or" has an exclusive sense. See Chapter 3, "The First Myth of 'Or'": Jennings, R. E. (1994). The Genealogy of Disjunction. New York: Oxford University Press. 6. Boole, G. (1847). The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning. Cambridge/London: Macmillan, Barclay, & Macmillan/George Bell. p. 17. 7. Enderton, H. (2001) [1972]. A Mathematical Introduction to Logic (2 ed.). San Diego, New York, Boston, London, Toronto, Sydney and Tokyo: A Harcourt Science and Technology Company. p. 51. 8. Rautenberg, W. (2010) [2006]. A Concise Introduction to Mathematical Logic (3 ed.). New York, Dordrecht, Heidelberg and London: Springer. p. 3. 9. Ladd, Christine (1883). "On the Algebra of Logic". In Peirce, C. S. (ed.). Studies in Logic by Members of the Johns Hopkins University. Boston: Little, Brown & Company. pp. 17–71. 10. Schröder, E. (1890). Vorlesungen über die Algebra der Logik (Exakte Logik), Erster Band (in German). Leipzig: Druck und Verlag B. G. Teubner. Reprinted by Thoemmes Press in 2000. 11. Peano, G. (1894). Notations de logique mathématique. Introduction au formulaire de mathématique. Turin: Fratelli Bocca. Reprinted in Peano, G. (1958). Opere Scelte, Volume II. Roma: Edizioni Cremonese. pp. 123–176. 12. ГРАДШТЕЙН, И. С. (1959) [1936]. ПРЯМАЯ И ОБРАТНАЯ ТЕОРЕМЫ: ЭЛЕМЕНТЫ АЛГЕБРЫ ЛОГИКИ (in Russian) (3 ed.). МОСКВА: ГОСУДАРСТВЕННОЕ ИЗДАТЕЛЬСТВО ФИЗИКа-МАТЕМАТИЧЕСКОЙ ЛИТЕРАТУРЫ. Translated as Gradshtein, I. S. (1963). Direct and Converse Theorems: The Elements of Symbolic Logic. Translated by Boddington, T. Oxford, London, New York and Paris: Pergamon Press. 13. Shannon, C. E. (1938). "A Symbolic Analysis of Relay and Switching Circuits" (PDF). Transactions of the American Institute of Electrical Engineers. 57 (12): 713–723. doi:10.1109/T-AIEE.1938.5057767. hdl:1721.1/11173. S2CID 51638483. 14. Huntington, E. V. (1904). "Sets of Independent Postulates for the Algebra of Logic". Transactions of the American Mathematical Society. 5 (3): 288–309. 15. Leibniz, G. W. (1890) [16??/17??]. Gerhardt, C. I. (ed.). Die philosophischen Schriften, Siebter Band (in German). Berlin: Weidmann. p. 237. Retrieved 7 July 2023. 16. Huntington, E. V. (1933). "New Sets of Independent Postulates for the Algebra of Logic, With Special Reference to Whitehead and Russell's Principia Mathematica". Transactions of the American Mathematical Society. 35 (1): 274–304. 17. Church, A. (1996) [1944]. Introduction to Mathematical Logic. New Jersey: Princeton University Press. p. 37. 18. Craig, Edward (1998). Routledge Encyclopedia of Philosophy, Volume 8. Taylor & Francis. p. 496. ISBN 978-0-41507310-3. 19. Łukasiewicz, Jan (1929). Elementy logiki matematycznej [Elements of Mathematical Logic] (in Polish) (1 ed.). Warsaw, Poland: Państwowe Wydawnictwo Naukowe. 20. Kernighan, Brian W.; Ritchie, Dennis M. (1978). "2.9: Bitwise logical operators". The C Programming Language. Prentice-Hall. pp. 44–46. 21. Weisstein, Eric W. "Symmetric Difference". MathWorld. 22. Davies, Robert B (28 February 2002). "Exclusive OR (XOR) and hardware random number generators" (PDF). Retrieved 28 August 2013. 23. Nobel, Rickard (26 July 2011). "How RAID 5 actually works". Retrieved 23 March 2017. External links Wikimedia Commons has media related to Exclusive disjunction. Look up exclusive or or XOR in Wiktionary, the free dictionary. • All About XOR • Proofs of XOR properties and applications of XOR, CS103: Mathematical Foundations of Computing, Stanford University Common logical connectives • Tautology/True $\top $ • Alternative denial (NAND gate) $\uparrow $ • Converse implication $\leftarrow $ • Implication (IMPLY gate) $\rightarrow $ • Disjunction (OR gate) $\lor $ • Negation (NOT gate) $\neg $ • Exclusive or (XOR gate) $\not \leftrightarrow $ • Biconditional (XNOR gate) $\leftrightarrow $ • Statement (Digital buffer) • Joint denial (NOR gate) $\downarrow $ • Nonimplication (NIMPLY gate) $\nrightarrow $ • Converse nonimplication $\nleftarrow $ • Conjunction (AND gate) $\land $ • Contradiction/False $\bot $ Philosophy portal
Logical NOR In Boolean logic, logical NOR or non-disjunction or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form (p NOR q) is true precisely when neither p nor q is true—i.e. when both of p and q are false. It is logically equivalent to $\neg (p\lor q)$ and $\neg p\land \neg q$, where the symbol $\neg $ signifies logical negation, $\lor $ signifies OR, and $\land $ signifies AND. Logical NOR NOR Definition${\overline {x+y}}$ Truth table$(0001)$ Logic gate Normal forms Disjunctive${\overline {x}}\cdot {\overline {y}}$ Conjunctive${\overline {x}}\cdot {\overline {y}}$ Zhegalkin polynomial$1\oplus x\oplus y\oplus xy$ Post's lattices 0-preservingno 1-preservingno Monotoneno Affineno Part of a series on Charles Sanders Peirce • Bibliography Pragmatism in epistemology • Abductive reasoning • Fallibilism • Pragmaticism • as maxim • as theory of truth • Community of inquiry Logic • Continuous predicate • Peirce's law • Entitative graph in Qualitative logic • Existential graph • Functional completeness • Logic gate • Logic of information • Logical graph • Logical NOR • Second-order logic • Trikonic • Type-token distinction Semiotic theory • Indexicality • Interpretant • Semiosis • Sign relation • Universal rhetoric Miscellaneous contributions • Agapism • Bell triangle • Categories • Phaneron • Synechism • Tychism • Classification of sciences • Listing number • Quincuncial projection Biographical • Joseph Morton Ransdell • Allan Marquand • Juliette Peirce • Charles Santiago Sanders Peirce • Roberta Kevelson • Christine Ladd-Franklin • Victoria, Lady Welby • The Metaphysical Club • book • Peirce Geodetic Monument Non-disjunction is usually denoted as $\downarrow $ or ${\overline {\vee }}$ or $X$ (prefix) or $\operatorname {NOR} $. As with its dual, the NAND operator (also known as the Sheffer stroke—symbolized as either $\uparrow $, $\mid $ or $/$), NOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NOR functionally complete). The computer used in the spacecraft that first carried humans to the moon, the Apollo Guidance Computer, was constructed entirely using NOR gates with three inputs.[1] Definition The NOR operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false. In other words, it produces a value of false if and only if at least one operand is true. Truth table The truth table of $P\downarrow Q$ is as follows: $P$ $Q$ $P\downarrow Q$ TrueTrueFalse TrueFalseFalse FalseTrueFalse FalseFalseTrue Logical equivalences The logical NOR $\downarrow $ is the negation of the disjunction: $P\downarrow Q$ $\Leftrightarrow $ $\neg (P\lor Q)$ $\Leftrightarrow $ $\neg $ Alternative notations and names Peirce is the first to show the functional completeness of non-disjunction while he doesn't publish his result.[2][3] Peirce used ${\overline {\curlywedge }}$ for non-conjunction and $\curlywedge $ for non-disjunction (in fact, what Peirce himself used is $\curlywedge $ and he didn't introduce ${\overline {\curlywedge }}$ while Peirce's editors made such disambiguated use).[3] Peirce called $\curlywedge $ as ampheck (from Ancient Greek ἀμφήκης, amphēkēs, "cutting both ways").[3] In 1911, Stamm was the first to publish a description of both non-conjunction (using $\sim $, the Stamm hook), and non-disjunction (using $*$, the Stamm star), and showed their functional completeness.[4]Zach, R. (2023-02-18). "Sheffer stroke before Sheffer: Edward Stamm". Retrieved 2023-07-02.</ref> Note that most uses in logical notation of $\sim $ use this for negation. In 1913, Sheffer described non-disjunction and showed its functional completeness. Sheffer used $\mid $ for non-conjunction, and $\wedge $ for non-disjunction. In 1935, Webb described non-disjunction for $n$-valued logic, and use $\mid $ for the operator. So some people call it Webb operator,[5] Webb operation[6] or Webb function.[7] In 1940, Quine also described non-disjunction and use $\downarrow $ for the operator.[8] So some people call the operator Peirce arrow or Quine dagger. In 1944, Church also described non-disjunction and use ${\overline {\vee }}$ for the operator.[9] In 1954, Bocheński used $X$ in $Xpq$ for non-disjunction in Polish notation.[10] Properties Logical NOR does not possess any of the five qualities (truth-preserving, false-preserving, linear, monotonic, self-dual) required to be absent from at least one member of a set of functionally complete operators. Thus, the set containing only NOR suffices as a complete set. Other Boolean operations in terms of the logical NOR NOR has the interesting feature that all other logical operators can be expressed by interlaced NOR operations. The logical NAND operator also has this ability. Expressed in terms of NOR $\downarrow $, the usual operators of propositional logic are: $\neg P$ $\Leftrightarrow $ $P\downarrow P$ $\neg $ $\Leftrightarrow $ $P\rightarrow Q$ $\Leftrightarrow $ ${\Big (}(P\downarrow P)\downarrow Q{\Big )}$ $\downarrow $ ${\Big (}(P\downarrow P)\downarrow Q{\Big )}$ $\Leftrightarrow $ $\downarrow $ $P\land Q$ $\Leftrightarrow $ $(P\downarrow P)$ $\downarrow $ $(Q\downarrow Q)$ $\Leftrightarrow $ $\downarrow $ $P\lor Q$ $\Leftrightarrow $ $(P\downarrow Q)$ $\downarrow $ $(P\downarrow Q)$ $\Leftrightarrow $ $\downarrow $ See also • Bitwise NOR • Boolean algebra • Boolean domain • Boolean function • Functional completeness • NOR gate • Propositional logic • Sole sufficient operator • Sheffer stroke as symbol for the logical NAND References 1. Hall, Eldon C. (1996). Journey to the Moon: The History of the Apollo Guidance Computer. Reston, Virginia, USA: American Institute of Aeronautics and Astronautics. p. 196. ISBN 1-56347-185-X. 2. Peirce, C. S. (1933) [1880]. "A Boolian Algebra with One Constant". In Hartshorne, C.; Weiss, P. (eds.). Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics. Massachusetts: Harvard University Press. pp. 13–18. 3. Peirce, C. S. (1933) [1902]. "The Simplest Mathematics". In Hartshorne, C.; Weiss, P. (eds.). Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics. Massachusetts: Harvard University Press. pp. 189–262. 4. Stamm, Edward Bronisław [in Polish] (1911). "Beitrag zur Algebra der Logik". Monatshefte für Mathematik und Physik (in German). 22 (1): 137–149. doi:10.1007/BF01742795. S2CID 119816758. 5. Webb, Donald Loomis (May 1935). "Generation of any n-valued logic by one binary operation". Proceedings of the National Academy of Sciences. USA: National Academy of Sciences. 6. Vasyukevich, Vadim O. (2011). "1.10 Venjunctive Properties (Basic Formulae)". Written at Riga, Latvia. Asynchronous Operators of Sequential Logic: Venjunction & Sequention — Digital Circuits Analysis and Design. Lecture Notes in Electrical Engineering (LNEE). Vol. 101 (1st ed.). Berlin / Heidelberg, Germany: Springer-Verlag. p. 20. doi:10.1007/978-3-642-21611-4. ISBN 978-3-642-21610-7. ISSN 1876-1100. LCCN 2011929655. p. 20: Historical background […] Logical operator NOR named Peirce arrow and also known as Webb-operation. (xiii+1+123+7 pages) (NB. The back cover of this book erroneously states volume 4, whereas it actually is volume 101.) 7. Freimann, Michael; Renfro, Dave L.; Webb, Norman (2018-05-24) [2017-02-10]. "Who is Donald L. Webb?". History of Science and Mathematics. Stack Exchange. Archived from the original on 2023-05-18. Retrieved 2023-05-18. 8. Quine, W. V (1981) [1940]. Mathematical Logic (Revised ed.). Cambridge, London, New York, New Rochelle, Melbourne and Sydney: Harvard University Press. p. 45. 9. Church, A. (1996) [1944]. Introduction to Mathematical Logic. New Jersey: Princeton University Press. p. 37. 10. Bocheński, J. M. (1954). Précis de logique mathématique (in French). Netherlands: F. G. Kroonder, Bussum, Pays-Bas. p. 11. External links • Media related to Logical NOR at Wikimedia Commons Common logical connectives • Tautology/True $\top $ • Alternative denial (NAND gate) $\uparrow $ • Converse implication $\leftarrow $ • Implication (IMPLY gate) $\rightarrow $ • Disjunction (OR gate) $\lor $ • Negation (NOT gate) $\neg $ • Exclusive or (XOR gate) $\not \leftrightarrow $ • Biconditional (XNOR gate) $\leftrightarrow $ • Statement (Digital buffer) • Joint denial (NOR gate) $\downarrow $ • Nonimplication (NIMPLY gate) $\nrightarrow $ • Converse nonimplication $\nleftarrow $ • Conjunction (AND gate) $\land $ • Contradiction/False $\bot $ Philosophy portal
Diamond operator In number theory, the diamond operators 〈d〉 are operators acting on the space of modular forms for the group Γ1(N), given by the action of a matrix (a b c δ ) in Γ0(N) where δ ≈ d mod N. The diamond operators form an abelian group and commute with the Hecke operators. Unicode In Unicode, the diamond operator is represented by the character U+22C4 ⋄ DIAMOND OPERATOR.[1] Notes 1. "Mathematical Operators – Unicode" (PDF). Retrieved 2013-04-22. References • Diamond, Fred; Shurman, Jerry (2005), A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Berlin, New York: Springer-Verlag, ISBN 978-0-387-23229-4, MR 2112196
Floor and ceiling functions In mathematics and computer science, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x).[1] Floor and ceiling functions Floor function Ceiling function For example (floor), ⌊2.4⌋ = 2, ⌊−2.4⌋ = −3, and for ceiling; ⌈2.4⌉ = 3, and ⌈−2.4⌉ = −2. Historically, the floor of x has been–and still is–called the integral part or integer part of x, often denoted [x] (as well as a variety of other notations).[2] However, the same term, integer part, is also used for truncation towards zero, which differs from the floor function for negative numbers. For n an integer, ⌊n⌋ = ⌈n⌉ = [n] = n. Examples x Floor ⌊x⌋ Ceiling ⌈x⌉ Fractional part {x} 2 2 2 0 2.4 2 3 0.4 2.9 2 3 0.9 −2.7 −3 −2 0.3 −2 −2 −2 0 Notation The integral part or integer part of a number (partie entière in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl Friedrich Gauss introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808).[3] This remained the standard[4] in mathematics until Kenneth E. Iverson introduced, in his 1962 book A Programming Language, the names "floor" and "ceiling" and the corresponding notations ⌊x⌋ and ⌈x⌉.[5][6] (Iverson used square brackets for a different purpose, the Iverson bracket notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article. In some sources, boldface or double brackets ⟦x⟧ are used for floor, and reversed brackets ⟧x⟦ or ]x[ for ceiling.[7][8] The fractional part is the sawtooth function, denoted by {x} for real x and defined by the formula {x} = x − ⌊x⌋[9] For all x, 0 ≤ {x} < 1. These characters are provided in Unicode: • U+2308 ⌈ LEFT CEILING (&lceil;, &LeftCeiling;) • U+2309 ⌉ RIGHT CEILING (&rceil;, &RightCeiling;) • U+230A ⌊ LEFT FLOOR (&LeftFloor;, &lfloor;) • U+230B ⌋ RIGHT FLOOR (&rfloor;, &RightFloor;) In the LaTeX typesetting system, these symbols can be specified with the \lceil, \rceil, \lfloor, and \rfloor commands in math mode, and extended in size using \left\lceil, \right\rceil, \left\lfloor, and \right\rfloor as needed. Some authors define [x] as the round-toward-zero function, so [2.4] = 2 and [−2.4] = −2, and call it the "integer part". This is truncation to zero significant digits. Definition and properties Given real numbers x and y, integers m and n and the set of integers $\mathbb {Z} $, floor and ceiling may be defined by the equations $\lfloor x\rfloor =\max\{m\in \mathbb {Z} \mid m\leq x\},$ $\lceil x\rceil =\min\{n\in \mathbb {Z} \mid n\geq x\}.$ Since there is exactly one integer in a half-open interval of length one, for any real number x, there are unique integers m and n satisfying the equation $x-1<m\leq x\leq n<x+1.$ where $\lfloor x\rfloor =m$ and $\lceil x\rceil =n$ may also be taken as the definition of floor and ceiling. Equivalences These formulas can be used to simplify expressions involving floors and ceilings.[10] ${\begin{aligned}\lfloor x\rfloor =m&\;\;{\mbox{ if and only if }}&m&\leq x<m+1,\\\lceil x\rceil =n&\;\;{\mbox{ if and only if }}&n-1&<x\leq n,\\\lfloor x\rfloor =m&\;\;{\mbox{ if and only if }}&x-1&<m\leq x,\\\lceil x\rceil =n&\;\;{\mbox{ if and only if }}&x&\leq n<x+1.\end{aligned}}$ In the language of order theory, the floor function is a residuated mapping, that is, part of a Galois connection: it is the upper adjoint of the function that embeds the integers into the reals. ${\begin{aligned}x<n&\;\;{\mbox{ if and only if }}&\lfloor x\rfloor &<n,\\n<x&\;\;{\mbox{ if and only if }}&n&<\lceil x\rceil ,\\x\leq n&\;\;{\mbox{ if and only if }}&\lceil x\rceil &\leq n,\\n\leq x&\;\;{\mbox{ if and only if }}&n&\leq \lfloor x\rfloor .\end{aligned}}$ These formulas show how adding an integer n to the arguments affects the functions: ${\begin{aligned}\lfloor x+n\rfloor &=\lfloor x\rfloor +n,\\\lceil x+n\rceil &=\lceil x\rceil +n,\\\{x+n\}&=\{x\}.\end{aligned}}$ The above are never true if n is not an integer; however, for every x and y, the following inequalities hold: ${\begin{aligned}\lfloor x\rfloor +\lfloor y\rfloor &\leq \lfloor x+y\rfloor \leq \lfloor x\rfloor +\lfloor y\rfloor +1,\\\lceil x\rceil +\lceil y\rceil -1&\leq \lceil x+y\rceil \leq \lceil x\rceil +\lceil y\rceil .\end{aligned}}$ Monotonicity Both floor and ceiling functions are the monotonically non-decreasing function: ${\begin{aligned}x_{1}\leq x_{2}&\Rightarrow \lfloor x_{1}\rfloor \leq \lfloor x_{2}\rfloor ,\\x_{1}\leq x_{2}&\Rightarrow \lceil x_{1}\rceil \leq \lceil x_{2}\rceil .\end{aligned}}$ Relations among the functions It is clear from the definitions that $\lfloor x\rfloor \leq \lceil x\rceil ,$ with equality if and only if x is an integer, i.e. $\lceil x\rceil -\lfloor x\rfloor ={\begin{cases}0&{\mbox{ if }}x\in \mathbb {Z} \\1&{\mbox{ if }}x\not \in \mathbb {Z} \end{cases}}$ In fact, for integers n, both floor and ceiling functions are the identity: $\lfloor n\rfloor =\lceil n\rceil =n.$ Negating the argument switches floor and ceiling and changes the sign: ${\begin{aligned}\lfloor x\rfloor +\lceil -x\rceil &=0\\-\lfloor x\rfloor &=\lceil -x\rceil \\-\lceil x\rceil &=\lfloor -x\rfloor \end{aligned}}$ and: $\lfloor x\rfloor +\lfloor -x\rfloor ={\begin{cases}0&{\text{if }}x\in \mathbb {Z} \\-1&{\text{if }}x\not \in \mathbb {Z} ,\end{cases}}$ $\lceil x\rceil +\lceil -x\rceil ={\begin{cases}0&{\text{if }}x\in \mathbb {Z} \\1&{\text{if }}x\not \in \mathbb {Z} .\end{cases}}$ Negating the argument complements the fractional part: $\{x\}+\{-x\}={\begin{cases}0&{\text{if }}x\in \mathbb {Z} \\1&{\text{if }}x\not \in \mathbb {Z} .\end{cases}}$ The floor, ceiling, and fractional part functions are idempotent: ${\begin{aligned}{\Big \lfloor }\lfloor x\rfloor {\Big \rfloor }&=\lfloor x\rfloor ,\\{\Big \lceil }\lceil x\rceil {\Big \rceil }&=\lceil x\rceil ,\\{\Big \{}\{x\}{\Big \}}&=\{x\}.\end{aligned}}$ The result of nested floor or ceiling functions is the innermost function: ${\begin{aligned}{\Big \lfloor }\lceil x\rceil {\Big \rfloor }&=\lceil x\rceil ,\\{\Big \lceil }\lfloor x\rfloor {\Big \rceil }&=\lfloor x\rfloor \end{aligned}}$ due to the identity property for integers. Quotients If m and n are integers and n ≠ 0, $0\leq \left\{{\frac {m}{n}}\right\}\leq 1-{\frac {1}{\|n\|}}.$ If n is a positive integer[11] $\left\lfloor {\frac {x+m}{n}}\right\rfloor =\left\lfloor {\frac {\lfloor x\rfloor +m}{n}}\right\rfloor ,$ $\left\lceil {\frac {x+m}{n}}\right\rceil =\left\lceil {\frac {\lceil x\rceil +m}{n}}\right\rceil .$ If m is positive[12] $n=\left\lceil {\frac {n}{m}}\right\rceil +\left\lceil {\frac {n-1}{m}}\right\rceil +\dots +\left\lceil {\frac {n-m+1}{m}}\right\rceil ,$ $n=\left\lfloor {\frac {n}{m}}\right\rfloor +\left\lfloor {\frac {n+1}{m}}\right\rfloor +\dots +\left\lfloor {\frac {n+m-1}{m}}\right\rfloor .$ For m = 2 these imply $n=\left\lfloor {\frac {n}{2}}\right\rfloor +\left\lceil {\frac {n}{2}}\right\rceil .$ More generally,[13] for positive m (See Hermite's identity) $\lceil mx\rceil =\left\lceil x\right\rceil +\left\lceil x-{\frac {1}{m}}\right\rceil +\dots +\left\lceil x-{\frac {m-1}{m}}\right\rceil ,$ $\lfloor mx\rfloor =\left\lfloor x\right\rfloor +\left\lfloor x+{\frac {1}{m}}\right\rfloor +\dots +\left\lfloor x+{\frac {m-1}{m}}\right\rfloor .$ The following can be used to convert floors to ceilings and vice versa (m positive)[14] $\left\lceil {\frac {n}{m}}\right\rceil =\left\lfloor {\frac {n+m-1}{m}}\right\rfloor =\left\lfloor {\frac {n-1}{m}}\right\rfloor +1,$ $\left\lfloor {\frac {n}{m}}\right\rfloor =\left\lceil {\frac {n-m+1}{m}}\right\rceil =\left\lceil {\frac {n+1}{m}}\right\rceil -1,$ For all m and n strictly positive integers:[15] $\sum _{k=1}^{n-1}\left\lfloor {\frac {km}{n}}\right\rfloor ={\frac {(m-1)(n-1)+\gcd(m,n)-1}{2}},$ which, for positive and coprime m and n, reduces to $\sum _{k=1}^{n-1}\left\lfloor {\frac {km}{n}}\right\rfloor ={\frac {1}{2}}(m-1)(n-1),$ and similarly for the ceiling and fractional part functions (still for positive and coprime m and n), $\sum _{k=1}^{n-1}\left\lceil {\frac {km}{n}}\right\rceil ={\frac {1}{2}}(m+1)(n-1),$ $\sum _{k=1}^{n-1}\left\{{\frac {km}{n}}\right\}={\frac {1}{2}}(n-1).$ Since the right-hand side of the general case is symmetrical in m and n, this implies that $\left\lfloor {\frac {m}{n}}\right\rfloor +\left\lfloor {\frac {2m}{n}}\right\rfloor +\dots +\left\lfloor {\frac {(n-1)m}{n}}\right\rfloor =\left\lfloor {\frac {n}{m}}\right\rfloor +\left\lfloor {\frac {2n}{m}}\right\rfloor +\dots +\left\lfloor {\frac {(m-1)n}{m}}\right\rfloor .$ More generally, if m and n are positive, ${\begin{aligned}&\left\lfloor {\frac {x}{n}}\right\rfloor +\left\lfloor {\frac {m+x}{n}}\right\rfloor +\left\lfloor {\frac {2m+x}{n}}\right\rfloor +\dots +\left\lfloor {\frac {(n-1)m+x}{n}}\right\rfloor \\=&\left\lfloor {\frac {x}{m}}\right\rfloor +\left\lfloor {\frac {n+x}{m}}\right\rfloor +\left\lfloor {\frac {2n+x}{m}}\right\rfloor +\cdots +\left\lfloor {\frac {(m-1)n+x}{m}}\right\rfloor .\end{aligned}}$ This is sometimes called a reciprocity law.[16] Nested divisions For positive integer n, and arbitrary real numbers m,x:[17] $\left\lfloor {\frac {\lfloor x/m\rfloor }{n}}\right\rfloor =\left\lfloor {\frac {x}{mn}}\right\rfloor $ $\left\lceil {\frac {\lceil x/m\rceil }{n}}\right\rceil =\left\lceil {\frac {x}{mn}}\right\rceil .$ Continuity and series expansions None of the functions discussed in this article are continuous, but all are piecewise linear: the functions $\lfloor x\rfloor $, $\lceil x\rceil $, and $\{x\}$ have discontinuities at the integers. $\lfloor x\rfloor $ is upper semi-continuous and $\lceil x\rceil $ and $\{x\}$ are lower semi-continuous. Since none of the functions discussed in this article are continuous, none of them have a power series expansion. Since floor and ceiling are not periodic, they do not have uniformly convergent Fourier series expansions. The fractional part function has Fourier series expansion[18] $\{x\}={\frac {1}{2}}-{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}$ for x not an integer. At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for y fixed and x a multiple of y the Fourier series given converges to y/2, rather than to x mod y = 0. At points of continuity the series converges to the true value. Using the formula floor(x) = x − {x} gives $\lfloor x\rfloor =x-{\frac {1}{2}}+{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}$ for x not an integer. Applications Mod operator For an integer x and a positive integer y, the modulo operation, denoted by x mod y, gives the value of the remainder when x is divided by y. This definition can be extended to real x and y, y ≠ 0, by the formula $x{\bmod {y}}=x-y\left\lfloor {\frac {x}{y}}\right\rfloor .$ Then it follows from the definition of floor function that this extended operation satisfies many natural properties. Notably, x mod y is always between 0 and y, i.e., if y is positive, $0\leq x{\bmod {y}}<y,$ and if y is negative, $0\geq x{\bmod {y}}>y.$ Quadratic reciprocity Gauss's third proof of quadratic reciprocity, as modified by Eisenstein, has two basic steps.[19][20] Let p and q be distinct positive odd prime numbers, and let $m={\frac {p-1}{2}},$ $n={\frac {q-1}{2}}.$ First, Gauss's lemma is used to show that the Legendre symbols are given by $\left({\frac {q}{p}}\right)=(-1)^{\left\lfloor {\frac {q}{p}}\right\rfloor +\left\lfloor {\frac {2q}{p}}\right\rfloor +\dots +\left\lfloor {\frac {mq}{p}}\right\rfloor }$ and $\left({\frac {p}{q}}\right)=(-1)^{\left\lfloor {\frac {p}{q}}\right\rfloor +\left\lfloor {\frac {2p}{q}}\right\rfloor +\dots +\left\lfloor {\frac {np}{q}}\right\rfloor }.$ The second step is to use a geometric argument to show that $\left\lfloor {\frac {q}{p}}\right\rfloor +\left\lfloor {\frac {2q}{p}}\right\rfloor +\dots +\left\lfloor {\frac {mq}{p}}\right\rfloor +\left\lfloor {\frac {p}{q}}\right\rfloor +\left\lfloor {\frac {2p}{q}}\right\rfloor +\dots +\left\lfloor {\frac {np}{q}}\right\rfloor =mn.$ Combining these formulas gives quadratic reciprocity in the form $\left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{mn}=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.$ There are formulas that use floor to express the quadratic character of small numbers mod odd primes p:[21] $\left({\frac {2}{p}}\right)=(-1)^{\left\lfloor {\frac {p+1}{4}}\right\rfloor },$ $\left({\frac {3}{p}}\right)=(-1)^{\left\lfloor {\frac {p+1}{6}}\right\rfloor }.$ Rounding For an arbitrary real number $x$, rounding $x$ to the nearest integer with tie breaking towards positive infinity is given by ${\text{rpi}}(x)=\left\lfloor x+{\tfrac {1}{2}}\right\rfloor =\left\lceil {\tfrac {\lfloor 2x\rfloor }{2}}\right\rceil $; rounding towards negative infinity is given as ${\text{rni}}(x)=\left\lceil x-{\tfrac {1}{2}}\right\rceil =\left\lfloor {\tfrac {\lceil 2x\rceil }{2}}\right\rfloor $. If tie-breaking is away from 0, then the rounding function is ${\text{ri}}(x)=\operatorname {sgn}(x)\left\lfloor \|x\|+{\tfrac {1}{2}}\right\rfloor $ (see sign function), and rounding towards even can be expressed with the more cumbersome $\lfloor x\rceil =\left\lfloor x+{\tfrac {1}{2}}\right\rfloor +\left\lceil {\tfrac {2x-1}{4}}\right\rceil -\left\lfloor {\tfrac {2x-1}{4}}\right\rfloor -1$, which is the above expression for rounding towards positive infinity ${\text{rpi}}(x)$ minus an integrality indicator for ${\tfrac {2x-1}{4}}$. Number of digits The number of digits in base b of a positive integer k is $\lfloor \log _{b}{k}\rfloor +1=\lceil \log _{b}{(k+1)}\rceil .$ Number of strings without repeated characters The number of possible strings of arbitrary length that doesn't use any character twice is given by[22] $(n)_{0}+\cdots +(n)_{n}=\lfloor en!\rfloor $ where: • n > 0 is the number of letters in the alphabet (e.g., 26 in English) • the falling factorial $(n)_{k}=n(n-1)\cdots (n-k+1)$ denotes the number of strings of length k that don't use any character twice. • n! denotes the factorial of n • e = 2.718… is Euler's number For n = 26, this comes out to 1096259850353149530222034277. Factors of factorials Let n be a positive integer and p a positive prime number. The exponent of the highest power of p that divides n! is given by a version of Legendre's formula[23] $\left\lfloor {\frac {n}{p}}\right\rfloor +\left\lfloor {\frac {n}{p^{2}}}\right\rfloor +\left\lfloor {\frac {n}{p^{3}}}\right\rfloor +\dots ={\frac {n-\sum _{k}a_{k}}{p-1}}$ where $ n=\sum _{k}a_{k}p^{k}$ is the way of writing n in base p. This is a finite sum, since the floors are zero when pk > n. Beatty sequence The Beatty sequence shows how every positive irrational number gives rise to a partition of the natural numbers into two sequences via the floor function.[24] Euler's constant (γ) There are formulas for Euler's constant γ = 0.57721 56649 ... that involve the floor and ceiling, e.g.[25] $\gamma =\int _{1}^{\infty }\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx,$ $\gamma =\lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}\left(\left\lceil {\frac {n}{k}}\right\rceil -{\frac {n}{k}}\right),$ and $\gamma =\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}={\tfrac {1}{2}}-{\tfrac {1}{3}}+2\left({\tfrac {1}{4}}-{\tfrac {1}{5}}+{\tfrac {1}{6}}-{\tfrac {1}{7}}\right)+3\left({\tfrac {1}{8}}-\cdots -{\tfrac {1}{15}}\right)+\cdots $ Riemann zeta function (ζ) The fractional part function also shows up in integral representations of the Riemann zeta function. It is straightforward to prove (using integration by parts)[26] that if $\varphi (x)$ is any function with a continuous derivative in the closed interval [a, b], $\sum _{a<n\leq b}\varphi (n)=\int _{a}^{b}\varphi (x)\,dx+\int _{a}^{b}\left(\{x\}-{\tfrac {1}{2}}\right)\varphi '(x)\,dx+\left(\{a\}-{\tfrac {1}{2}}\right)\varphi (a)-\left(\{b\}-{\tfrac {1}{2}}\right)\varphi (b).$ Letting $\varphi (n)=n^{-s}$ for real part of s greater than 1 and letting a and b be integers, and letting b approach infinity gives $\zeta (s)=s\int _{1}^{\infty }{\frac {{\frac {1}{2}}-\{x\}}{x^{s+1}}}\,dx+{\frac {1}{s-1}}+{\frac {1}{2}}.$ This formula is valid for all s with real part greater than −1, (except s = 1, where there is a pole) and combined with the Fourier expansion for {x} can be used to extend the zeta function to the entire complex plane and to prove its functional equation.[27] For s = σ + it in the critical strip 0 < σ < 1, $\zeta (s)=s\int _{-\infty }^{\infty }e^{-\sigma \omega }(\lfloor e^{\omega }\rfloor -e^{\omega })e^{-it\omega }\,d\omega .$ In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zeta function.[28] Formulas for prime numbers The floor function appears in several formulas characterizing prime numbers. For example, since $\left\lfloor {\frac {n}{m}}\right\rfloor -\left\lfloor {\frac {n-1}{m}}\right\rfloor $ is equal to 1 if m divides n, and to 0 otherwise, it follows that a positive integer n is a prime if and only if[29] $\sum _{m=1}^{\infty }\left(\left\lfloor {\frac {n}{m}}\right\rfloor -\left\lfloor {\frac {n-1}{m}}\right\rfloor \right)=2.$ One may also give formulas for producing the prime numbers. For example, let pn be the n-th prime, and for any integer r > 1, define the real number α by the sum $\alpha =\sum _{m=1}^{\infty }p_{m}r^{-m^{2}}.$ Then[30] $p_{n}=\left\lfloor r^{n^{2}}\alpha \right\rfloor -r^{2n-1}\left\lfloor r^{(n-1)^{2}}\alpha \right\rfloor .$ A similar result is that there is a number θ = 1.3064... (Mills' constant) with the property that $\left\lfloor \theta ^{3}\right\rfloor ,\left\lfloor \theta ^{9}\right\rfloor ,\left\lfloor \theta ^{27}\right\rfloor ,\dots $ are all prime.[31] There is also a number ω = 1.9287800... with the property that $\left\lfloor 2^{\omega }\right\rfloor ,\left\lfloor 2^{2^{\omega }}\right\rfloor ,\left\lfloor 2^{2^{2^{\omega }}}\right\rfloor ,\dots $ are all prime.[31] Let π(x) be the number of primes less than or equal to x. It is a straightforward deduction from Wilson's theorem that[32] $\pi (n)=\sum _{j=2}^{n}\left\lfloor {\frac {(j-1)!+1}{j}}-\left\lfloor {\frac {(j-1)!}{j}}\right\rfloor \right\rfloor .$ Also, if n ≥ 2,[33] $\pi (n)=\sum _{j=2}^{n}\left\lfloor {\frac {1}{\sum _{k=2}^{j}\left\lfloor \left\lfloor {\frac {j}{k}}\right\rfloor {\frac {k}{j}}\right\rfloor }}\right\rfloor .$ None of the formulas in this section are of any practical use.[34][35] Solved problems Ramanujan submitted these problems to the Journal of the Indian Mathematical Society.[36] If n is a positive integer, prove that 1. $\left\lfloor {\tfrac {n}{3}}\right\rfloor +\left\lfloor {\tfrac {n+2}{6}}\right\rfloor +\left\lfloor {\tfrac {n+4}{6}}\right\rfloor =\left\lfloor {\tfrac {n}{2}}\right\rfloor +\left\lfloor {\tfrac {n+3}{6}}\right\rfloor ,$ 2. $\left\lfloor {\tfrac {1}{2}}+{\sqrt {n+{\tfrac {1}{2}}}}\right\rfloor =\left\lfloor {\tfrac {1}{2}}+{\sqrt {n+{\tfrac {1}{4}}}}\right\rfloor ,$ 3. $\left\lfloor {\sqrt {n}}+{\sqrt {n+1}}\right\rfloor =\left\lfloor {\sqrt {4n+2}}\right\rfloor .$ Some generalizations to the above floor function identities have been proven.[37] Unsolved problem The study of Waring's problem has led to an unsolved problem: Are there any positive integers k ≥ 6 such that[38] $3^{k}-2^{k}\left\lfloor \left({\tfrac {3}{2}}\right)^{k}\right\rfloor >2^{k}-\left\lfloor \left({\tfrac {3}{2}}\right)^{k}\right\rfloor -2$ ? Mahler[39] has proved there can only be a finite number of such k; none are known. Computer implementations In most programming languages, the simplest method to convert a floating point number to an integer does not do floor or ceiling, but truncation. The reason for this is historical, as the first machines used ones' complement and truncation was simpler to implement (floor is simpler in two's complement). FORTRAN was defined to require this behavior and thus almost all processors implement conversion this way. Some consider this to be an unfortunate historical design decision that has led to bugs handling negative offsets and graphics on the negative side of the origin. A bit-wise right-shift of a signed integer $x$ by $n$ is the same as $\left\lfloor {\frac {x}{2^{n}}}\right\rfloor $. Division by a power of 2 is often written as a right-shift, not for optimization as might be assumed, but because the floor of negative results is required. Assuming such shifts are "premature optimization" and replacing them with division can break software. Many programming languages (including C, C++,[40][41] C#,[42][43] Java,[44][45] PHP,[46][47] R,[48] and Python[49]) provide standard functions for floor and ceiling, usually called floor and ceil, or less commonly ceiling.[50] The language APL uses ⌊x for floor. The J Programming Language, a follow-on to APL that is designed to use standard keyboard symbols, uses <. for floor and >. for ceiling.[51] ALGOL usesentier for floor. In Microsoft Excel the floor function is implemented as INT (which rounds down rather than toward zero).[52] The command FLOOR in earlier versions would round toward zero, effectively the opposite of what "int" and "floor" do in other languages. Since 2010 FLOOR has been fixed to round down, with extra arguments that can reproduce previous behavior.[53] The OpenDocument file format, as used by OpenOffice.org, Libreoffice and others, uses the same function names; INT does floor[54] and FLOOR has a third argument that can make it round toward zero.[55] See also • Bracket (mathematics) • Integer-valued function • Step function • Modulo operation Citations 1. Graham, Knuth, & Patashnik, Ch. 3.1 2. 1) Luke Heaton, A Brief History of Mathematical Thought, 2015, ISBN 1472117158 (n.p.) 2) Albert A. Blank et al., Calculus: Differential Calculus, 1968, p. 259 3) John W. Warris, Horst Stocker, Handbook of mathematics and computational science, 1998, ISBN 0387947469, p. 151 3. Lemmermeyer, pp. 10, 23. 4. e.g. Cassels, Hardy & Wright, and Ribenboim use Gauss's notation. Graham, Knuth & Patashnik, and Crandall & Pomerance use Iverson's. 5. Iverson, p. 12. 6. Higham, p. 25. 7. Mathwords: Floor Function. 8. Mathwords: Ceiling Function 9. Graham, Knuth, & Patashnik, p. 70. 10. Graham, Knuth, & Patashink, Ch. 3 11. Graham, Knuth, & Patashnik, p. 73 12. Graham, Knuth, & Patashnik, p. 85 13. Graham, Knuth, & Patashnik, p. 85 and Ex. 3.15 14. Graham, Knuth, & Patashnik, Ex. 3.12 15. Graham, Knuth, & Patashnik, p. 94. 16. Graham, Knuth, & Patashnik, p. 94 17. Graham, Knuth, & Patashnik, p. 71, apply theorem 3.10 with x/m as input and the division by n as function 18. Titchmarsh, p. 15, Eq. 2.1.7 19. Lemmermeyer, § 1.4, Ex. 1.32–1.33 20. Hardy & Wright, §§ 6.11–6.13 21. Lemmermeyer, p. 25 22. OEIS sequence A000522 (Total number of arrangements of a set with n elements: a(n) = Sum_{k=0..n} n!/k!.) (See Formulas.) 23. Hardy & Wright, Th. 416 24. Graham, Knuth, & Patashnik, pp. 77–78 25. These formulas are from the Wikipedia article Euler's constant, which has many more. 26. Titchmarsh, p. 13 27. Titchmarsh, pp.14–15 28. Crandall & Pomerance, p. 391 29. Crandall & Pomerance, Ex. 1.3, p. 46. The infinite upper limit of the sum can be replaced with n. An equivalent condition is n > 1 is prime if and only if $\sum _{m=1}^{\lfloor {\sqrt {n}}\rfloor }\left(\left\lfloor {\frac {n}{m}}\right\rfloor -\left\lfloor {\frac {n-1}{m}}\right\rfloor \right)=1$ . 30. Hardy & Wright, § 22.3 31. Ribenboim, p. 186 32. Ribenboim, p. 181 33. Crandall & Pomerance, Ex. 1.4, p. 46 34. Ribenboim, p. 180 says that "Despite the nil practical value of the formulas ... [they] may have some relevance to logicians who wish to understand clearly how various parts of arithmetic may be deduced from different axiomatzations ... " 35. Hardy & Wright, pp. 344—345 "Any one of these formulas (or any similar one) would attain a different status if the exact value of the number α ... could be expressed independently of the primes. There seems no likelihood of this, but it cannot be ruled out as entirely impossible." 36. Ramanujan, Question 723, Papers p. 332 37. Somu, Sai Teja; Kukla, Andrzej (2022). "On some generalizations to floor function identities of Ramanujan" (PDF). Integers. 22. arXiv:2109.03680. 38. Hardy & Wright, p. 337 39. Mahler, Kurt (1957). "On the fractional parts of the powers of a rational number II". Mathematika. 4 (2): 122–124. doi:10.1112/S0025579300001170. 40. "C++ reference of floor function". Retrieved 5 December 2010. 41. "C++ reference of ceil function". Retrieved 5 December 2010. 42. dotnet-bot. "Math.Floor Method (System)". docs.microsoft.com. Retrieved 28 November 2019. 43. dotnet-bot. "Math.Ceiling Method (System)". docs.microsoft.com. Retrieved 28 November 2019. 44. "Math (Java SE 9 & JDK 9 )". docs.oracle.com. Retrieved 20 November 2018. 45. "Math (Java SE 9 & JDK 9 )". docs.oracle.com. Retrieved 20 November 2018. 46. "PHP manual for ceil function". Retrieved 18 July 2013. 47. "PHP manual for floor function". Retrieved 18 July 2013. 48. "R: Rounding of Numbers". 49. "Python manual for math module". Retrieved 18 July 2013. 50. Sullivan, p. 86. 51. "Vocabulary". J Language. Retrieved 6 September 2011. 52. "INT function". Retrieved 29 October 2021. 53. "FLOOR function". Retrieved 29 October 2021. 54. "Documentation/How Tos/Calc: INT function". Retrieved 29 October 2021. 55. "Documentation/How Tos/Calc: FLOOR function". Retrieved 29 October 2021. References • J.W.S. Cassels (1957), An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 45, Cambridge University Press • Crandall, Richard; Pomerance, Carl (2001), Prime Numbers: A Computational Perspective, New York: Springer, ISBN 0-387-94777-9 • Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994), Concrete Mathematics, Reading Ma.: Addison-Wesley, ISBN 0-201-55802-5 • Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 • Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, p. 25 • ISO/IEC. ISO/IEC 9899::1999(E): Programming languages — C (2nd ed), 1999; Section 6.3.1.4, p. 43. • Iverson, Kenneth E. (1962), A Programming Language, Wiley • Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3-540-66957-4 • Ramanujan, Srinivasa (2000), Collected Papers, Providence RI: AMS / Chelsea, ISBN 978-0-8218-2076-6 • Ribenboim, Paulo (1996), The New Book of Prime Number Records, New York: Springer, ISBN 0-387-94457-5 • Michael Sullivan. Precalculus, 8th edition, p. 86 • Titchmarsh, Edward Charles; Heath-Brown, David Rodney ("Roger") (1986), The Theory of the Riemann Zeta-function (2nd ed.), Oxford: Oxford U. P., ISBN 0-19-853369-1 External links Wikimedia Commons has media related to Floor and ceiling functions. • "Floor function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Štefan Porubský, "Integer rounding functions", Interactive Information Portal for Algorithmic Mathematics, Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic, retrieved 24 October 2008 • Weisstein, Eric W. "Floor Function". MathWorld. • Weisstein, Eric W. "Ceiling Function". MathWorld.
Origin (mathematics) In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. Cartesian coordinates In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.[1] The origin divides each of these axes into two halves, a positive and a negative semiaxis.[2] Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three.[1] Other coordinate systems In a polar coordinate system, the origin may also be called the pole. It does not itself have well-defined polar coordinates, because the polar coordinates of a point include the angle made by the positive x-axis and the ray from the origin to the point, and this ray is not well-defined for the origin itself.[3] In Euclidean geometry, the origin may be chosen freely as any convenient point of reference.[4] The origin of the complex plane can be referred as the point where real axis and imaginary axis intersect each other. In other words, it is the complex number zero.[5] See also • Null vector, an analogous point of a vector space • Distance from a point to a plane • Pointed space, a topological space with a distinguished point • Radial basis function, a function depending only on the distance from the origin References 1. Madsen, David A. (2001), Engineering Drawing and Design, Delmar drafting series, Thompson Learning, p. 120, ISBN 9780766816343. 2. Pontrjagin, Lev S. (1984), Learning higher mathematics, Springer series in Soviet mathematics, Springer-Verlag, p. 73, ISBN 9783540123514. 3. Tanton, James Stuart (2005), Encyclopedia of Mathematics, Infobase Publishing, ISBN 9780816051243. 4. Lee, John M. (2013), Axiomatic Geometry, Pure and Applied Undergraduate Texts, vol. 21, American Mathematical Society, p. 134, ISBN 9780821884782. 5. Gonzalez, Mario (1991), Classical Complex Analysis, Chapman & Hall Pure and Applied Mathematics, CRC Press, ISBN 9780824784157.
Flatness (mathematics) In mathematics, the flatness (symbol: ⏥) of a surface is the degree to which it approximates a mathematical plane. The term is often generalized for higher-dimensional manifolds to describe the degree to which they approximate the Euclidean space of the same dimensionality. (See curvature.)[1] Flatness in homological algebra and algebraic geometry means, of an object $A$ in an abelian category, that $-\otimes A$ is an exact functor. See flat module or, for more generality, flat morphism.[2] Character encodings Character information Preview⏥ Unicode name FLATNESS Encodingsdecimalhex Unicode9189U+23E5 UTF-8226 143 165E2 8F A5 Numeric character reference⏥⏥ See also • Developable surface • Flat (mathematics) References 1. Committee 117, A. C. I. (November 3, 2006). Specifications for Tolerances for Concrete Construction and Materials and Commentary. American Concrete Institute. ISBN 9780870312212 – via Google Books. 2. Ballast, David Kent (March 16, 2007). Handbook of Construction Tolerances. John Wiley & Sons. ISBN 9780471931515 – via Google Books.
Semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, π radians, or a half-turn). It has only one line of symmetry (reflection symmetry). Semicircle Areaπr2/2 Perimeter(π+2)r In non-technical usage, the term "semicircle" is sometimes used to refer to either a closed curve that also includes the diameter segment from one end of the arc to the other or to the half-disk, which is a two-dimensional geometric region that further includes all the interior points. By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a right triangle, with a right angle at the third vertex. All lines intersecting the semicircle perpendicularly are concurrent at the center of the circle containing the given semicircle. Uses A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass. For a semicircle with a diameter of a + b, the length of its radius is the arithmetic mean of a and b (since the radius is half of the diameter). The geometric mean can be found by dividing the diameter into two segments of lengths a and b, and then connecting their common endpoint to the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean. This can be proven by applying the Pythagorean theorem to three similar right triangles, each having as vertices the point where the perpendicular touches the semicircle and two of the three endpoints of the segments of lengths a and b.[1] The construction of the geometric mean can be used to transform any rectangle into a square of the same area, a problem called the quadrature of a rectangle. The side length of the square is the geometric mean of the side lengths of the rectangle. More generally, it is used as a lemma in a general method for transforming any polygonal shape into a similar copy of itself with the area of any other given polygonal shape.[2] Equation The equation of a semicircle with midpoint $(x_{0},y_{0})$ on the diameter between its endpoints and which is entirely concave from below is $y=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}}$ If it is entirely concave from above, the equation is $y=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}}}$ Arbelos An arbelos is a region in the plane bounded by three semicircles connected at the corners, all on the same side of a straight line (the baseline) that contains their diameters. See also • Amphitheater • Archimedes' twin circles • Archimedes' quadruplets • Salinon • Wigner semicircle distribution References 1. Euclid's Elements, Book VI, Proposition 13 2. Euclid's Elements, Book VI, Proposition 25 External links • Weisstein, Eric W. "Semicircle". MathWorld.
Pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around the five points creates a similar symbol referred to as the pentacle,[1] which is used widely by Wiccans and in paganism, or as a sign of life and connections. The word "pentagram" refers only to the five-pointed star, not the surrounding circle of a pentacle. Pentagrams were used symbolically in ancient Greece and Babylonia. Christians once commonly used the pentagram to represent the five wounds of Jesus. The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon),[2] from πέντε (pente), "five" + γραμμή (grammē), "line".[3] Pentagram refers to just the star and pentacle refers to the star within the circle specifically although there is some overlap in usage.[4] The word pentalpha is a 17th-century revival of a post-classical Greek name of the shape.[5] History Early history Early pentagrams have been found on Sumerian pottery from Ur circa 3500 BCE, and the five-pointed star was at various times the symbol of Ishtar or Marduk.[6][7] Pentagram symbols from about 5,000 years ago were found in the Liangzhu culture of China.[9] The pentagram was known to the ancient Greeks, with a depiction on a vase possibly dating back to the 7th century BCE.[10] Pythagoreanism originated in the 6th century BCE and used the pentagram as a symbol of mutual recognition, of wellbeing, and to recognize good deeds and charity.[11] From around 300-150 BCE the pentagram stood as the symbol of Jerusalem, marked by the 5 Hebrew letters ירשלם spelling its name.[12] The word Pentemychos (πεντέμυχος lit. "five corners" or "five recesses")[13] was the title of the cosmogony of Pherecydes of Syros.[14] Here, the "five corners" are where the seeds of Chronos are placed within the Earth in order for the cosmos to appear.[15] In Neoplatonism, the pentagram was said to have been used as a symbol or sign of recognition by the Pythagoreans, who called the pentagram ὑγιεία hugieia "health"[16] Middle Ages The pentagram was used in ancient times as a Christian symbol for the five senses,[17] or of the five wounds of Christ. The pentagram plays an important symbolic role in the 14th-century English poem Sir Gawain and the Green Knight, in which the symbol decorates the shield of the hero, Gawain. The unnamed poet credits the symbol's origin to King Solomon, and explains that each of the five interconnected points represents a virtue tied to a group of five: Gawain is perfect in his five senses and five fingers, faithful to the Five Wounds of Christ, takes courage from the five joys that Mary had of Jesus, and exemplifies the five virtues of knighthood,[18] which are generosity, friendship, chastity, chivalry, and piety.[19] The North rose of Amiens Cathedral (built in the 13th century) exhibits a pentagram-based motif. Some sources interpret the unusual downward-pointing star as symbolizing the Holy Spirit descending on people. Renaissance Heinrich Cornelius Agrippa and others perpetuated the popularity of the pentagram as a magic symbol, attributing the five neoplatonic elements to the five points, in typical Renaissance fashion. Romanticism By the mid-19th century, a further distinction had developed amongst occultists regarding the pentagram's orientation. With a single point upwards it depicted spirit presiding over the four elements of matter, and was essentially "good". However, the influential but controversial writer Éliphas Lévi, known for believing that magic was a real science, had called it evil whenever the symbol appeared the other way up. • "A reversed pentagram, with two points projecting upwards, is a symbol of evil and attracts sinister forces because it overturns the proper order of things and demonstrates the triumph of matter over spirit. It is the goat of lust attacking the heavens with its horns, a sign execrated by initiates."[20] • "The flaming star, which, when turned upside down, is the heirolgyphic [sic] sign of the goat of black magic, whose head may be drawn in the star, the two horns at the top, the ears to the right and left, the beard at the bottom. It is a sign of antagonism and fatality. It is the goat of lust attacking the heavens with its horns."[21] • "Let us keep the figure of the Five-pointed Star always upright, with the topmost triangle pointing to heaven, for it is the seat of wisdom, and if the figure is reversed, perversion and evil will be the result."[22] • Man inscribed in a pentagram, from Heinrich Cornelius Agrippa's De occulta philosophia libri tres. The five signs at the pentagram's vertices are astrological. • Another pentagram from Agrippa's book. This one has the Pythagorean letters inscribed around the circle. • The occultist and magician Éliphas Lévi's pentagram, which he considered to be a symbol of the microcosm, or human Star polygons • pentagram • hexagram • heptagram • octagram • enneagram • decagram • hendecagram • dodecagram The apotropaic (protective) use in German folklore of the pentagram symbol (called Drudenfuss in German) is referred to by Goethe in Faust (1808), where a pentagram prevents Mephistopheles from leaving a room (but did not prevent him from entering by the same way, as the outward pointing corner of the diagram happened to be imperfectly drawn): Mephistopheles: I must confess, I'm prevented though By a little thing that hinders me, The Druid's-foot on your doorsill– Faust: The Pentagram gives you pain? Then tell me, you Son of Hell, If that's the case, how did you gain Entry? Are spirits like you cheated? Mephistopheles: Look carefully! It's not completed: One angle, if you inspect it closely Has, as you see, been left a little open.[23] Also protective is the use in Icelandic folklore of a gestured or carved rather than painted pentagram (called smèrhnút in Icelandic), according to 19th century folklorist Jón Árnason:[24] A butter that comes from the fake vomit is called a fake butter; it looks like something else; but if one makes a sign of a cross over it, or carves a cross on it, or a figure called a buttermilk-knot,* it all explodes into small pieces and becomes like a grain of dross, so that nothing remains of it, except only particles, or it subsides like foam. Therefore it seems more prudent, if a person is offered a horrible butter to eat, or as a fee,[25] to make either mark on it, because a fake butter cannot withstand either a cross mark or a butter-knot. * The butter-knot is shaped like this: East Asian symbolism Wu Xing (Chinese: 五行; pinyin: Wǔ Xíng) are the five phases, or five elements in Taoists Chinese tradition. They are differentiated from the formative ancient Japanese or Greek elements, due to their emphasis on cyclic transformations and change. The five phases are: Fire (火 huǒ), Earth (土 tǔ), Metal (金 jīn), Water (水 shuǐ), and Wood (木 mù). The Wuxing is the fundamental philosophy and doctrine of traditional Chinese Medicine and Acupuncture.[26] Uses in modern occultism Based on Renaissance-era occultism, the pentagram found its way into the symbolism of modern occultists. Its major use is a continuation of the ancient Babylonian use of the pentagram as an apotropaic charm to protect against evil forces.[27] Éliphas Lévi claimed that "The Pentagram expresses the mind's domination over the elements and it is by this sign that we bind the demons of the air, the spirits of fire, the spectres of water, and the ghosts of earth."[28] In this spirit, the Hermetic Order of the Golden Dawn developed the use of the pentagram in the lesser banishing ritual of the pentagram, which is still used to this day by those who practice Golden Dawn-type magic. Aleister Crowley made use of the pentagram in his Thelemic system of magick: an adverse or inverted pentagram represents the descent of spirit into matter, according to the interpretation of Lon Milo DuQuette.[29] Crowley contradicted his old comrades in the Hermetic Order of the Golden Dawn, who, following Levi, considered this orientation of the symbol evil and associated it with the triumph of matter over spirit. Baháʼí Faith The five-pointed star is a symbol of the Baháʼí Faith.[30][31] In the Baháʼí Faith, the star is known as the Haykal (Arabic: "temple"), and it was initiated and established by the Báb. The Báb and Bahá'u'lláh wrote various works in the form of a pentagram.[32][33] The Church of Jesus Christ of Latter-day Saints The Church of Jesus Christ of Latter-day Saints is theorized to have begun using both upright and inverted five-pointed stars in Temple architecture, dating from the Nauvoo Illinois Temple dedicated on 30 April 1846.[34] Other temples decorated with five-pointed stars in both orientations include the Salt Lake Temple and the Logan Utah Temple. These usages come from the symbolism found in Revelation chapter 12: "And there appeared a great wonder in heaven; a woman clothed with the sun, and the moon under her feet, and upon her head a crown of twelve stars."[35] Wicca Typical Neopagan pentagram (circumscribed) USVA headstone emblem 37 Because of a perceived association with Satanism and occultism, many United States schools in the late 1990s sought to prevent students from displaying the pentagram on clothing or jewelry.[36] In public schools, such actions by administrators were determined in 2000 to be in violation of students' First Amendment right to free exercise of religion.[37] The encircled pentagram (referred to as a pentacle by the plaintiffs) was added to the list of 38 approved religious symbols to be placed on the tombstones of fallen service members at Arlington National Cemetery on 24 April 2007. The decision was made following ten applications from families of fallen soldiers who practiced Wicca. The government paid the families US$225,000 to settle their pending lawsuits.[38][39] Satanism The inverted pentagram is the most notable and widespread symbol of Satanism. The Sigil of Baphomet, the official insignia of the Church of Satan and LaVeyan Satanism The inverted pentagram is the symbol used for Satanism, sometimes depicted with the goat's head of Baphomet within it, which originated from the Church of Satan. In some depictions the devil is depicted, like Baphomet, as a goat, therefore the goat and goat's head is a significant symbol throughout Satanism. The inverted pentagram is also used as the logo for The Satanic Temple, which also featured a depiction of Baphomet's head. The Sigil of Baphomet is also adopted by the Joy of Satan Ministries who instead incorporate cuneiform script, attributing it to the earliest use of the pentagram in Sumeria. Serer religion The five-pointed star is a symbol of the Serer religion and the Serer people of West Africa. Called Yoonir in their language, it symbolizes the universe in the Serer creation myth, and also represents the star Sirius.[40][41] Judaism The pentagram has been used in Judaism since at least 300BCE when it first was used as the stamp of Jerusalem. It is used to represent justice, mercy, and wisdom. Other modern use • The pentagram is featured on the national flags of Morocco (adopted 1915) and Ethiopia (adopted 1996 and readopted 2009) • Morocco's flag • Ethiopia's flag • The Order of the Eastern Star, an organization (established 1850) associated with Freemasonry, uses a pentagram as its symbol, with the five isosceles triangles of the points colored blue, yellow, white, green, and red. In most Grand Chapters the pentagram is used pointing down, but in a few, it is pointing up. Grand Chapter officers often have a pentagon inscribed around the star[42](the emblem shown here is from the Prince Hall Association). • Order of the Eastern Star emblem • A pentagram is featured on the flag of the Dutch city of Haaksbergen, as well on its coat of arms. • Flag of Haaksbergen • A pentagram is featured on the flag of the Japanese city of Nagasaki, as well on its emblem. • Flag of Nagasaki Geometry The pentagram is the simplest regular star polygon. The pentagram contains ten points (the five points of the star, and the five vertices of the inner pentagon) and fifteen line segments. It is represented by the Schläfli symbol {5/2}. Like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as its symmetry group the dihedral group of order 10. It can be seen as a net of a pentagonal pyramid although with isosceles triangles. Construction The pentagram can be constructed by connecting alternate vertices of a pentagon; see details of the construction. It can also be constructed as a stellation of a pentagon, by extending the edges of a pentagon until the lines intersect. Golden ratio The golden ratio, φ = (1 + √5) / 2 ≈ 1.618, satisfying $\varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }\,$ $\varphi =1/(2\sin(\pi /10))=1/(2\sin 18^{\circ })\,$ $\varphi =2\cos(\pi /5)=2\cos 36^{\circ }\,$ plays an important role in regular pentagons and pentagrams. Each intersection of edges sections the edges in the golden ratio: the ratio of the length of the edge to the longer segment is φ, as is the length of the longer segment to the shorter. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is φ. As the four-color illustration shows: ${\frac {\mathrm {red} }{\mathrm {green} }}={\frac {\mathrm {green} }{\mathrm {blue} }}={\frac {\mathrm {blue} }{\mathrm {magenta} }}=\varphi .$ The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon. Trigonometric values Main article: Exact trigonometric values ${\begin{aligned}\sin {\frac {\pi }{10}}&=\sin 18^{\circ }={\frac {{\sqrt {5}}-1}{4}}={\frac {\varphi -1}{2}}={\frac {1}{2\varphi }}\\[5pt]\cos {\frac {\pi }{10}}&=\cos 18^{\circ }={\frac {\sqrt {2(5+{\sqrt {5}})}}{4}}\\[5pt]\tan {\frac {\pi }{10}}&=\tan 18^{\circ }={\frac {\sqrt {5(5-2{\sqrt {5}})}}{5}}\\[5pt]\cot {\frac {\pi }{10}}&=\cot 18^{\circ }={\sqrt {5+2{\sqrt {5}}}}\\[5pt]\sin {\frac {\pi }{5}}&=\sin 36^{\circ }={\frac {\sqrt {2(5-{\sqrt {5}})}}{4}}\\[5pt]\cos {\frac {\pi }{5}}&=\cos 36^{\circ }={\frac {{\sqrt {5}}+1}{4}}={\frac {\varphi }{2}}\\[5pt]\tan {\frac {\pi }{5}}&=\tan 36^{\circ }={\sqrt {5-2{\sqrt {5}}}}\\[5pt]\cot {\frac {\pi }{5}}&=\cot 36^{\circ }={\frac {\sqrt {5(5+2{\sqrt {5}})}}{5}}\end{aligned}}$ As a result, in an isosceles triangle with one or two angles of 36°, the longer of the two side lengths is φ times that of the shorter of the two, both in the case of the acute as in the case of the obtuse triangle. Spherical pentagram Further information: Pentagramma mirificum A pentagram can be drawn as a star polygon on a sphere, composed of five great circle arcs, whose all internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio (Description of the wonderful rule of logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). It was studied later by Carl Friedrich Gauss. Three-dimensional figures Further information: Uniform polyhedron: Icosahedral symmetry Several polyhedra incorporate pentagrams: • Pentagrammic prism • Pentagrammic antiprism • Pentagrammic crossed-antiprism • Small stellated dodecahedron • Great stellated dodecahedron • Small ditrigonal icosidodecahedron • Dodecadodecahedron Higher dimensions Orthogonal projections of higher dimensional polytopes can also create pentagrammic figures: 4D 5D The regular 5-cell (4-simplex) has five vertices and 10 edges. The rectified 5-cell has 10 vertices and 30 edges. The rectified 5-simplex has 15 vertices, seen in this orthogonal projection as three nested pentagrams. The birectified 5-simplex has 20 vertices, seen in this orthogonal projection as four overlapping pentagrams. All ten 4-dimensional Schläfli–Hess 4-polytopes have either pentagrammic faces or vertex figure elements. Pentagram of Venus The pentagram of Venus is the apparent path of the planet Venus as observed from Earth. Successive inferior conjunctions of Venus repeat with an orbital resonance of approximately 13:8—that is, Venus orbits the Sun approximately 13 times for every eight orbits of Earth—shifting 144° at each inferior conjunction.[44] The tips of the five loops at the center of the figure have the same geometric relationship to one another as the five vertices, or points, of a pentagram, and each group of five intersections equidistant from the figure's center have the same geometric relationship. In computer systems The pentagram has these Unicode code points that enable them to be included in documents: • U+26E4 ⛤ PENTAGRAM • U+26E5 ⛥ RIGHT-HANDED INTERLACED PENTAGRAM • U+26E6 ⛦ LEFT-HANDED INTERLACED PENTAGRAM • U+26E7 ⛧ INVERTED PENTAGRAM See also • Abe no Seimei – Japanese painter • Christian symbolism – Use of symbols, including archetypes, acts, artwork or events, by Christianity • Command at Sea insignia • Enneagram (geometry) – Nine-pointed star polygon • Five-pointed star – Geometrically a regular concave decagon, is a common ideogram in modern culture • Heptagram – Star polygon • Hexagram – Six-pointed star polygon • Lute of Pythagoras – Self-similar geometric figure • Medal of Honor – Highest award in the United States Armed Forces • Pentachoron – the 4-simplex • Pentagram map – Discrete dynamical system on the moduli space of polygons in the projective plane • Pentalpha – Puzzle involving stones and a pentagram • Petersen graph – Cubic graph with 10 vertices and 15 edges • Ptolemy's theorem – Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle • Seal of Solomon – Signet ring attributed to the Israelite king Solomon • Star polygons in art and culture – Polygons as symbolic elements • Star (heraldry) – In heraldry, any pierced or unpierced star-shaped charge with any number of straight or wavy rays • Stellated polygons – Extending the elements of a polytope to form a new figure References 1. Gene Brown (n.d.). "Difference Between Pentagram and Pentacle". Difference Between. Retrieved 29 June 2023. 2. πεντάγραμμον, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus; a noun form of adjectival πεντάγραμμος (pentagrammos) or πεντέγραμμος (pentegrammos), a word meaning roughly "five-lined" or "five lines" 3. πέντε, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus; Satan all 3 names mentioned before daylight full γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus 4. this usage is borne out by the Oxford English Dictionary, although that work specifies that a circumscription makes the form of a five-pointed star and its etymon post-classical Latin pentaculum [...] A pentagram, esp. one enclosed in a circle; a talisman or magical symbol in the shape of or inscribed with a pentagram. Also, in extended use: any similar magical symbol (freq. applied to a hexagram formed by two intersecting or interlaced equilateral triangles)." 5. πένταλφα, "five Alphas", interpreting the shape as five Α shapes overlapping at 72-degree angles. 6. Budge, Sir E. A. Wallis (1968). Amulets and Talismans. p. 433. 7. Scott, Dustin Jon (2006). "History of the Pentagram". Retrieved 18 May 2021. 8. Allman, G. J., Greek Geometry From Thales to Euclid (1889), p.26. 9. 馬愛平 (23 September 2019). "距今5000年！良渚文物中發現最古老五角星圖案" (in Chinese). China Daily. 10. Coxeter, H.S.M.; Regular Polytopes, 3rd edn, Dover, 1973, p. 114. 11. Ball, W. W. Rouse and Coxeter, H. S. M.; Mathematical Recreations and Essays, 13th Edn., Dover, 1987, p. 176. 12. "Star of David vs. Pentagram: Everything You Need to Know". 17 July 2020. 13. πεντέμυχος, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus 14. This is a lost book, but its contents are preserved in Damascius, De principiis, quoted in Kirk and Raven, (1983) [1956], p. 55. 15. "the divine products of Chronos' seed, when disposed in five recesses, were called πεντέμυχος (Pentemychos)" Kirk, Geoffrey Stephen; Raven, John Earle; Schofield, Malcolm (1983) [1957]. The Presocratic Philosophers: A Critical History with a Selection of Texts (2nd, illustrated, revised, reprint ed.). Cambridge University Press. pp. 51–52, 55. ISBN 978-0-521-27455-5. the only other place in Homer where Ortygie [sic] is mentioned is Odyssey V, 123, where Orion, having been carried off by Eos [the dawn], is slain... by Artemis... since solstices would normally be observed at sunrise in summer, and so in the north-east-by-east direction, that is what the phrase might suggest... the dwelling-place of Eos... Aia.. 16. Allman, G. J., Greek Geometry From Thales to Euclid, part I (1877), in Hermathena 3.5, pp. 183, 197, citing Iamblichus and the Scholiast on Aristophanes. The pentagram was said to have been so called from Pythagoras himself having written the letters Υ, Γ, Ι, Θ (= /ei/), Α on its vertices. 17. Christian Symbols Ancient and Modern, Child, Heather and Dorothy Colles. New York: Charles Scribner's Sons, 1971, ISBN 0-7135-1960-6. 18. Morgan, Gerald (1979). "The Significance of the Pentangle Symbolism in "Sir Gawain and the Green Knight"". The Modern Language Review. 74 (4): 769–790. doi:10.2307/3728227. JSTOR 3728227. 19. Sir Gawain and the Green Knight, lines 619–665 20. Lévi, Éliphas (1999) [1896 (translated), 1854 (first published)]. Transcendental Magic, its Doctrine and Ritual [Dogme et rituel de la haute magie]. Trans. by A. E. Waite. York Beach: Weiser. OCLC 263626874. 21. Lévi, Éliphas (2002) [1939 (translated), 1859 (first published)]. The Key of the Mysteries [la Clef des grands mystères suivant Hénoch, Abraham, Hermès Trismégiste et Salomon]. Trans. by Aleister Crowley. Boston: Weiser. p. 69. OCLC 49053462. 22. Hartmann, Franz (1895) [1886]. Magic, White and Black (5th ed.). New York: The Path. OCLC 476635673. 23. "Goethe, Johann Wolfgang von (1749–1832) - Faust, Part I: Scenes I to III". www.poetryintranslation.com. Retrieved 25 May 2021. 24. Árnason, Jón (1862). "Töfrabrogð [Magic trick]". Íslenzkar Þjoðsögur og Æfintýri [Icelandic Folktales and Legends] (in Icelandic). Vol. 1. Leipzig: J. C. Hinrich's Bookstore. p. 432. Smèr það, er verður af tilberaspýunni, er kallað tilberasmèr; er það útlits sem annað smèr; en gjöri maður krossmark yfir því, eða risti á það kross, eða mynd þá, er smèrhnútur heitir,* springur það alt í smámola og verður eins og draflakyrníngur, svo ekki sèst eptir af því, nema agnir einar, eða það hjaðnar niður sem froða. Þykir það því varlegra, ef manni er boðið óhrjálegt smèr að borða, eða í gjöld, að gjóra annaðhvort þetta mark á það, því tilberasmèr þolir hvorki krossmark né smjörhnút. / * Smèrhnútur er svo í lögun: 25. In the Middle Ages, butter was used for payment, e.g. rent. See: • Sexton, Regina (2003). "The Role and Function of Butter in the Diet of the Monk and Penitent in Early Medieval Ireland". In Walker, Harlan (ed.). The Fat of the Land: Proceedings of the Oxford Symposium on Food and Cooking 2002. Bristol: Footwork. pp. 253–269. 26. Chen, Yuan Julian (2014). "Legitimation Discourse and the Theory of the Five Elements in Imperial China". Journal of Song-Yuan Studies. 44 (1): 325–364. doi:10.1353/sys.2014.0000. S2CID 147099574. 27. Schouten, Jan (1968). The Pentagram as Medical Symbol: An Iconological Study. Hes & De Graaf. p. 18. ISBN 978-90-6004-166-6. 28. Waite, Arthur Edward (1886). The Mysteries of Magic: A Digest of the Writings of Eliphas Lévi. London: George Redway. p. 136. 29. DuQuette, Lon Milo (2003). The Magick of Aleister Crowley: A Handbook of the Rituals of Thelema. Weiser Books. pp. 93, 247. ISBN 978-1-57863-299-2. 30. "Bahá'í Reference Library - Directives from the Guardian, Pages 51-52". reference.bahai.org. 31. "The Nine-Pointed Star". bahai-library.com. 32. Moojan Momen (2019). The Star Tablet of the Bab. British Library Blog. 33. Bayat, Mohamad Ghasem (2001). An Introduction to the Súratu'l-Haykal (Discourse of The Temple) in Lights of Irfan, Book 2. 34. See the Nauvoo Temple Archived 17 May 2020 at the Wayback Machine website discussing its architecture, and particularly the page on Nauvoo Temple exterior symbolism Archived 17 May 2020 at the Wayback Machine. Retrieved 16 December 2006. 35. Brown, Matthew B (2002). "Inverted Stars on LDS Temples" (PDF). FAIRLDS.org. Archived from the original (PDF) on 29 February 2008. 36. "Religious Clothing in School", Robinson, B.A., Ontario Consultants on Religious Tolerance, 20 August 1999, updated 29 April 2005. Retrieved 10 February 2006. "ACLU Defends Honor Student Witch Pentacle" (Press release). American Civil Liberties Union of Michigan. 10 February 1999. Archived from the original on 8 November 2003. Retrieved 10 February 2006.{{cite press release}}: CS1 maint: bot: original URL status unknown (link) "Witches and wardrobes: Boy says he was suspended from school for wearing magical symbol" Rouvalis, Cristina; Pittsburgh Post-Gazette, 27 September 2000. Retrieved 10 February 2006. 37. "Federal judge upholds Indiana students' right to wear Wiccan symbols". Associated Press. 1 May 2000. Archived from the original on 30 March 2014. Retrieved 21 September 2007. 38. "Wiccan symbol OK for soldiers' graves". CNN.com. Associated Press. 23 April 2007. Archived from the original on 26 April 2007. 39. "Burial and Memorials: Available Emblems of Belief for Placement on Government Headstones and Markers". United States Department of Veterans Affairs. 3 July 2013. Retrieved 13 January 2014. 40. Gravrand 1990, p. 20. 41. Madiya, Clémentine Faïk-Nzuji (1996). Tracing Memory: A Glossary of Graphic Signs and Symbols in African Art and Culture. Mercury series, no. 71. Hull, Québec: Canadian Museum of Civilization. pp. 27, 155. ISBN 0-660-15965-1. 42. Ritual of the Order of the Eastern Star, 1976 43. Pietrocola, Giorgio (2005). "Tartapelago. Exposure of fractals". Maecla. 44. Baez, John (4 January 2014). "The Pentagram of Venus". Azimuth. Archived from the original on 14 December 2015. Retrieved 7 January 2016. Bibliography • Becker, Udo (1994). "Pentagram". The Continuum Encyclopedia of Symbols. Translated by Garmer, Lance W. New York City: Continuum Books. p. 230. ISBN 978-0-8264-0644-6. • Conway, John Horton; Burgiel, Heidi; Goodman-Strauss, Chaim (April 2008). "Chapter 26, Higher Still: Regular Star-Polytopes". The Symmetries of Things. Wellesley, Massachusetts: A. K. Peters. p. 404. ISBN 978-1-56881-220-5. • Ferguson, George Wells (1966) [1954]. Signs and Symbols in Christian Art. New York City: Oxford University Press. p. 59. OCLC 65081051. • Gravrand, Henry (January 1990). La civilisation Sereer, Volume II: Pangool. Nouvelles éditions Africaines du Sénégal (in French). Dakar, Senegal. ISBN 2-7236-1055-1.{{cite book}}: CS1 maint: location missing publisher (link) • Grünbaum, Branko; Shephard, Geoffrey Colin (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3. • Grünbaum, Branko (1994). "Polyhedra with Hollow Faces". In Bisztriczky, T.; McMullen, P.; Schneider, A.; Weiss, A. Ivić (eds.). Polytopes: Abstract, Convex and Computational. NATO ASI Series C: Mathematical and Physical Sciences. Vol. 440. Dordrecht: Springer Netherlands. pp. 43–70. doi:10.1007/978-94-011-0924-6_3. ISBN 978-94-010-4398-4. External links Wikimedia Commons has media related to Pentagrams. • Weisstein, Eric W. "Pentagram". MathWorld. • The Pythagorean Pentacle from the Biblioteca Arcana. Polygons (List) Triangles • Acute • Equilateral • Ideal • Isosceles • Kepler • Obtuse • Right Quadrilaterals • Antiparallelogram • Bicentric • Crossed • Cyclic • Equidiagonal • Ex-tangential • Harmonic • Isosceles trapezoid • Kite • Orthodiagonal • Parallelogram • Rectangle • Right kite • Right trapezoid • Rhombus • Square • Tangential • Tangential trapezoid • Trapezoid By number of sides 1–10 sides • Monogon (1) • Digon (2) • Triangle (3) • Quadrilateral (4) • Pentagon (5) • Hexagon (6) • Heptagon (7) • Octagon (8) • Nonagon (Enneagon, 9) • Decagon (10) 11–20 sides • Hendecagon (11) • Dodecagon (12) • Tridecagon (13) • Tetradecagon (14) • Pentadecagon (15) • Hexadecagon (16) • Heptadecagon (17) • Octadecagon (18) • Icosagon (20) >20 sides • Icositrigon (23) • Icositetragon (24) • Triacontagon (30) • 257-gon • Chiliagon (1000) • Myriagon (10,000) • 65537-gon • Megagon (1,000,000) • Apeirogon (∞) Star polygons • Pentagram • Hexagram • Heptagram • Octagram • Enneagram • Decagram • Hendecagram • Dodecagram Classes • Concave • Convex • Cyclic • Equiangular • Equilateral • Infinite skew • Isogonal • Isotoxal • Magic • Pseudotriangle • Rectilinear • Regular • Reinhardt • Simple • Skew • Star-shaped • Tangential • Weakly simple Serer topics Peoples • Laalaa • Ndut • Niominka • Noon • Palor • Saafi • Seex Religion Key topics • Ciiɗ • Classical Ndut teachings • Creation myth • Criticism • Festivals • Jaaniiw • Junjung • Lamane • Pangool • Religion • Sadax • Saltigue • Symbolism • Women • Xooy Supreme deities • Kokh Kox • Koox • Kopé Tiatie Cac • Roog (main) Other deities • Kumba Njaay • Takhar • Tiurakh Sacred sites • Fatick • Sine River • Sine-Saloum • Somb • Point of Sangomar • Tattaguine • Tukar • Yaboyabo History • Amar Godomat • Cekeen Tumuli • Khasso • Kingdom of Baol • Kingdom of Biffeche • kingdom of Saloum • Kingdom of Sine • Serer prehistory • Serer history • States headed by Serer Lamanes • Battle of Fandane-Thiouthioune • Battle of Logandème • Timeline of Serer history • Western Sahara Demographics By region • Gambia • Mauritania • Senegal • Serer country Languages • Cangin • Lehar/Laalaa • Ndut • Noon • Palor • Safen • Serer Culture • Birth • Chere (or saay) • Death • Inheritance • Marriage • Mbalax • Njuup • Sabar • Tama • Tassu • Njom Royalty Kings (Maad) and Lamanes (ancient kings / landowners) • Lamane Jegan Joof • Maad a Sinig Kumba Ndoffene Famak Joof • Maad a Sinig Kumba Ndoffene Famak Joof • Maad a Sinig Kumba Ndoffene Fa Ndeb Joof • Maad a Sinig Mahecor Joof • Maad a Sinig Maysa Wali Jaxateh Manneh • Maad a Sinig Ama Joof Gnilane Faye Joof • Maad Ndaah Njemeh Joof • Maad Semou Njekeh Joof Queens & Queen Mothers • Lingeer Fatim Beye • Lingeer Ndoye Demba • Lingeer Ngoné Dièye • Lingeer Selbeh Ndoffene Joof • Serer maternal clans Dynasties and royal houses • Faye family • Guelowar • Joof family • Joos Maternal Dynasty • The Royal House of Boureh Gnilane Joof • The Royal House of Jogo Siga Joof • The Royal House of Semou Njekeh Joof Families and royal titles • Buumi • Faye family • Joof family • Lamane • Lingeer • Loul • Maad Saloum • Maad a Sinig • Njie family • Sarr family • Sene family • Teigne • Thilas Related people • Jola people • Lebu people • Toucouleur people • Wolof people Authority control: National • Germany • Israel • United States
Spherical angle A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs of great circles on a sphere. It is measured by the angle between the planes containing the arcs (which naturally also contain the centre of the sphere).[1] Not to be confused with Solid angle. See also • Spherical coordinate system • Spherical trigonometry • Transcendent angle References 1. Green, Robin Michael (1985), Spherical Astronomy, Cambridge University Press, p. 3, ISBN 9780521317795.
Tiny and miny In mathematics, tiny and miny are operators that yield infinitesimal values when applied to numbers in combinatorial game theory. Given a positive number G, tiny G (denoted by ⧾G in many texts) is equal to {0\|{0\|-G}} for any game G, whereas miny G (analogously denoted ⧿G) is tiny G's negative, or {{G\|0}\|0}. Tiny and miny aren't just abstract mathematical operators on combinatorial games: tiny and miny games do occur "naturally" in such games as toppling dominoes. Specifically, tiny n, where n is a natural number, can be generated by placing two black dominoes outside n + 2 white dominoes. Tiny games and up have certain curious relational characteristics. Specifically, though ⧾G is infinitesimal with respect to ↑ for all positive values of x, ⧾⧾⧾G is equal to up. Expansion of ⧾⧾⧾G into its canonical form yields {0\|{{0\|{{0\|{0\|-G}}\|0}}\|0}}. While the expression appears daunting, some careful and persistent expansion of the game tree of ⧾⧾⧾G + ↓ will show that it is a second player win, and that, consequently, ⧾⧾⧾G = ↑. Similarly curious, mathematician John Horton Conway noted, calling it "amusing," that "↑ is the unique solution of ⧾G = G." Conway's assertion is also easily verifiable with canonical forms and game trees. References • Albert, Michael H.; Nowakowski, Richard J.; Wolfe, David (2007). Lessons in Play: An Introduction to Combinatorial Game Theory. A K Peters, Ltd. ISBN 1-56881-277-9. • Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (2003). Winning Ways for Your Mathematical Plays. A K Peters, Ltd.
Integral symbol The integral symbol: ∫ (Unicode), $\displaystyle \int $ (LaTeX) ∫ Integral symbol In UnicodeU+222B ∫ INTEGRAL (∫, &Integral;) Graphical variants $\displaystyle \int $ Different from Different fromU+017F ſ LONG S U+0283 ʃ ESH is used to denote integrals and antiderivatives in mathematics, especially in calculus. History Main article: Leibniz's notation The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1675 in his private writings;[1][2] it first appeared publicly in the article "De Geometria Recondita et analysi indivisibilium atque infinitorum" (On a hidden geometry and analysis of indivisibles and infinites), published in Acta Eruditorum in June 1686.[3][4] The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands. Typography in Unicode and LaTeX Fundamental symbol Main article: Integral calculus The integral symbol is U+222B ∫ INTEGRAL in Unicode[5] and \int in LaTeX. In HTML, it is written as ∫ (hexadecimal), ∫ (decimal) and ∫ (named entity). The original IBM PC code page 437 character set included a couple of characters ⌠ and ⌡ (codes 244 and 245 respectively) to build the integral symbol. These were deprecated in subsequent MS-DOS code pages, but they still remain in Unicode (U+2320 and U+2321 respectively) for compatibility. The ∫ symbol is very similar to, but not to be confused with, the letter ʃ ("esh"). Extensions of the symbol See also: Multiple integral Related symbols include:[5][6] Meaning Unicode LaTeX Double integral ∬ U+222C $\iint $ \iint Triple integral ∭ U+222D $\iiint $ \iiint Quadruple integral ⨌ U+2A0C $\iiiint $ \iiiint Contour integral ∮ U+222E $\oint $ \oint Clockwise integral ∱ U+2231 Counterclockwise integral ⨑ U+2A11 Clockwise contour integral ∲ U+2232 \varointclockwise Counterclockwise contour integral ∳ U+2233 \ointctrclockwise Closed surface integral ∯ U+222F \oiint Closed volume integral ∰ U+2230 \oiiint Typography in other languages In other languages, the shape of the integral symbol differs slightly from the shape commonly seen in English-language textbooks. While the English integral symbol leans to the right, the German symbol (used throughout Central Europe) is upright, and the Russian variant leans slightly to the left to occupy less horizontal space.[7] Another difference is in the placement of limits for definite integrals. Generally, in English-language books, limits go to the right of the integral symbol: $\int _{0}^{5}f(t)\,\mathrm {d} t,\quad \int _{g(t)=a}^{g(t)=b}f(t)\,\mathrm {d} t.$ By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing, but is more compact horizontally, especially when longer expressions are used in the limits: $\int \limits _{0}^{T}f(t)\,\mathrm {d} t,\quad \int \limits _{\!\!\!\!\!g(t)=a\!\!\!\!\!}^{\!\!\!\!\!g(t)=b\!\!\!\!\!}f(t)\,\mathrm {d} t.$ See also • Capital sigma notation • Capital pi notation Notes 1. Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674–1676, Berlin: Akademie Verlag, 2008, pp. 288–295 Archived 2021-10-09 at the Wayback Machine ("Analyseos tetragonisticae pars secunda", October 29, 1675) and 321–331 Archived 2016-10-03 at the Wayback Machine ("Methodi tangentium inversae exempla", November 11, 1675). 2. Aldrich, John. "Earliest Uses of Symbols of Calculus". Retrieved 20 April 2017. 3. Swetz, Frank J., Mathematical Treasure: Leibniz's Papers on Calculus – Integral Calculus, Convergence, Mathematical Association of America, retrieved February 11, 2017 4. Stillwell, John (1989). Mathematics and its History. Springer. p. 110. 5. "Mathematical Operators – Unicode" (PDF). Retrieved 2013-04-26. 6. "Supplemental Mathematical Operators – Unicode" (PDF). Retrieved 2013-05-05. 7. "Russian Typographical Traditions in Mathematical Literature" (PDF). giftbot.toolforge.org. Archived from the original (PDF) on 28 September 2012. Retrieved 11 October 2021. References • Stewart, James (2003). "Integrals". Single Variable Calculus: Early Transcendentals (5th ed.). Belmont, CA: Brooks/Cole. p. 381. ISBN 0-534-39330-6. • Zaitcev, V.; Janishewsky, A.; Berdnikov, A. (1999), "Russian Typographical Traditions in Mathematical Literature" (PDF), Russian Typographical Traditions in Mathematical Literature, EuroTeX'99 Proceedings External links • Fileformat.info Infinitesimals History • Adequality • Leibniz's notation • Integral symbol • Criticism of nonstandard analysis • The Analyst • The Method of Mechanical Theorems • Cavalieri's principle Related branches • Nonstandard analysis • Nonstandard calculus • Internal set theory • Synthetic differential geometry • Smooth infinitesimal analysis • Constructive nonstandard analysis • Infinitesimal strain theory (physics) Formalizations • Differentials • Hyperreal numbers • Dual numbers • Surreal numbers Individual concepts • Standard part function • Transfer principle • Hyperinteger • Increment theorem • Monad • Internal set • Levi-Civita field • Hyperfinite set • Law of continuity • Overspill • Microcontinuity • Transcendental law of homogeneity Mathematicians • Gottfried Wilhelm Leibniz • Abraham Robinson • Pierre de Fermat • Augustin-Louis Cauchy • Leonhard Euler Textbooks • Analyse des Infiniment Petits • Elementary Calculus • Cours d'Analyse
Nimber In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication. Not to be confused with Number. Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering. The nimber addition and multiplication operations are associative and commutative. Each nimber is its own negative. In particular for some pairs of ordinals, their nimber sum is smaller than either addend.[1] The minimum excludant operation is applied to sets of nimbers. Uses Nim Main article: Nim Nim is a game in which two players take turns removing objects from distinct heaps. As moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric, Nim is an impartial game. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap. The goal of the game is to be the player who removes the last object. The nimber of a heap is simply the number of objects in that heap. Using nim addition, one can calculate the nimber of the game as a whole. The winning strategy is to force the nimber of the game to 0 for the opponent's turn.[2] Cram Main article: Cram (game) Cram is a game often played on a rectangular board in which players take turns placing dominoes either horizontally or vertically until no more dominoes can be placed. The first player that cannot make a move loses. As the possible moves for both players are the same, it is an impartial game and can have a nimber value. For example, any board that is an even size by an even size will have a nimber of 0. Any board that is even by odd will have a non-zero nimber. Any 2×n board will have a nimber of 0 for all even n and a nimber of 1 for all odd n. Northcott's game In Northcott's game, pegs for each player are placed along a column with a finite number of spaces. Each turn each player must move the piece up or down the column, but may not move past the other player's piece. Several columns are stacked together to add complexity. The player that can no longer make any moves loses. Unlike many other nimber related games, the number of spaces between the two tokens on each row are the sizes of the Nim heaps. If your opponent increases the number of spaces between two tokens, just decrease it on your next move. Else, play the game of Nim and make the Nim-sum of the number of spaces between the tokens on each row be 0.[3] Hackenbush Hackenbush is a game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line. Players take turns removing line segments. An impartial game version, thereby a game able to be analyzed using nimbers, can be found by removing distinction from the lines, allowing either player to cut any branch. Any segments reliant on the newly removed segment in order to connect to the ground line are removed as well. In this way, each connection to the ground can be considered a nim heap with a nimber value. Additionally, all the separate connections to the ground line can also be summed for a nimber of the game state. Addition Nimber addition (also known as nim-addition) can be used to calculate the size of a single nim heap equivalent to a collection of nim heaps. It is defined recursively by α ⊕ β = mex({α′ ⊕ β : α' < α} ∪ {α ⊕ β′ : β′ < β}), where the minimum excludant mex(S) of a set S of ordinals is defined to be the smallest ordinal that is not an element of S. For finite ordinals, the nim-sum is easily evaluated on a computer by taking the bitwise exclusive or (XOR, denoted by ⊕) of the corresponding numbers. For example, the nim-sum of 7 and 14 can be found by writing 7 as 111 and 14 as 1110; the ones place adds to 1; the twos place adds to 2, which we replace with 0; the fours place adds to 2, which we replace with 0; the eights place adds to 1. So the nim-sum is written in binary as 1001, or in decimal as 9. This property of addition follows from the fact that both mex and XOR yield a winning strategy for Nim and there can be only one such strategy; or it can be shown directly by induction: Let α and β be two finite ordinals, and assume that the nim-sum of all pairs with one of them reduced is already defined. The only number whose XOR with α is α ⊕ β is β, and vice versa; thus α ⊕ β is excluded. On the other hand, for any ordinal γ < α ⊕ β, XORing ξ ≔ α ⊕ β ⊕ γ with all of α, β and γ must lead to a reduction for one of them (since the leading 1 in ξ must be present in at least one of the three); since ξ ⊕ γ = α ⊕ β > γ, we must have α > ξ ⊕ α = β ⊕ γ or β > ξ ⊕ β = α ⊕ γ; thus γ is included as (β ⊕ γ) ⊕ β or as α ⊕ (α ⊕ γ), and hence α ⊕ β is the minimum excluded ordinal. Nimber addition is associative and commutative, with 0 as the additive identity element. Moreover, a nimber is its own additive inverse.[4] It follows that α ⊕ β = 0 if and only if α = β. Multiplication Nimber multiplication (nim-multiplication) is defined recursively by α β = mex({α′ β ⊕ α β′ ⊕ α' β′ : α′ < α, β′ < β}). Nimber multiplication is associative and commutative, with the ordinal 1 as the multiplicative identity element. Moreover, nimber multiplication distributes over nimber addition.[4] Thus, except for the fact that nimbers form a proper class and not a set, the class of nimbers forms a ring. In fact, it even determines an algebraically closed field of characteristic 2, with the nimber multiplicative inverse of a nonzero ordinal α given by α−1 = mex(S), where S is the smallest set of ordinals (nimbers) such that 1. 0 is an element of S; 2. if 0 < α′ < α and β′ is an element of S, then (1 + (α′ − α) β′) / α′−1 is also an element of S. For all natural numbers n, the set of nimbers less than 22n form the Galois field GF(22n) of order 22n. Therefore, the set of finite nimbers is isomorphic to the direct limit as n → ∞ of the fields GF(22n). This subfield is not algebraically closed, since no field GF(2k) with k not a power of 2 is contained in any of those fields, and therefore not in their direct limit; for instance the polynomial x3 + x + 1, which has a root in GF(23), does not have a root in the set of finite nimbers. Just as in the case of nimber addition, there is a means of computing the nimber product of finite ordinals. This is determined by the rules that 1. The nimber product of a Fermat 2-power (numbers of the form 22n) with a smaller number is equal to their ordinary product; 2. The nimber square of a Fermat 2-power x is equal to 3x/2 as evaluated under the ordinary multiplication of natural numbers. The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal ωωω, where ω is the smallest infinite ordinal. It follows that as a nimber, ωωω is transcendental over the field.[5] Addition and multiplication tables The following tables exhibit addition and multiplication among the first 16 nimbers. This subset is closed under both operations, since 16 is of the form 22n. (If you prefer simple text tables, they are here.) See also • Surreal number Notes 1. Advances in computer games : 14th International Conference, ACG 2015, Leiden, the Netherlands, July 1-3, 2015, Revised selected papers. Herik, Jaap van den,, Plaat, Aske,, Kosters, Walter. Cham. 2015-12-24. ISBN 978-3319279923. OCLC 933627646.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link) 2. Anany., Levitin (2012). Introduction to the design & analysis of algorithms (3rd ed.). Boston: Pearson. ISBN 9780132316811. OCLC 743298766. 3. "Theory of Impartial Games" (PDF). Feb 3, 2009. 4. Brown, Ezra; Guy, Richard K. (2021). "2.5 Nim arithmetic and Nim algebra". The Unity of Combinatorics. Vol. 36 of The Carus Mathematical Monographs (reprint ed.). American Mathematical Society. p. 35. ISBN 978-1-4704-6509-4. 5. Conway 1976, p. 61. References • Conway, John Horton (1976). On Numbers and Games. Academic Press Inc. (London) Ltd. • Lenstra, H. W. (1978). Nim multiplication. Report IHES/M/78/211. Institut des hautes études scientifiques. hdl:1887/2125. • Schleicher, Dierk; Stoll, Michael (2004). "An Introduction to Conway's Games and Numbers". arXiv:math.DO/0410026. which discusses games, surreal numbers, and nimbers.
Smash product In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (X, x0) and (Y, y0) is the quotient of the product space X × Y under the identifications (x, y0) ∼ (x0, y) for all x in X and y in Y. The smash product is itself a pointed space, with basepoint being the equivalence class of (x0, y0). The smash product is usually denoted X ∧ Y or X ⨳ Y. The smash product depends on the choice of basepoints (unless both X and Y are homogeneous). For the smash product in the theory of Hopf algebras, see Hopf smash product. One can think of X and Y as sitting inside X × Y as the subspaces X × {y0} and {x0} × Y. These subspaces intersect at a single point: (x0, y0), the basepoint of X × Y. So the union of these subspaces can be identified with the wedge sum $X\vee Y=(X\amalg Y)\;/{\sim }$. In particular, {x0} × Y in X × Y is identified with Y in $X\vee Y$, ditto for X × {y0} and X. In $X\vee Y$, subspaces X and Y intersect in the single point $x_{0}\sim y_{0}$. The smash product is then the quotient $X\wedge Y=(X\times Y)/(X\vee Y).$ The smash product shows up in homotopy theory, a branch of algebraic topology. In homotopy theory, one often works with a different category of spaces than the category of all topological spaces. In some of these categories the definition of the smash product must be modified slightly. For example, the smash product of two CW complexes is a CW complex if one uses the product of CW complexes in the definition rather than the product topology. Similar modifications are necessary in other categories. Examples • The smash product of any pointed space X with a 0-sphere (a discrete space with two points) is homeomorphic to X. • The smash product of two circles is a quotient of the torus homeomorphic to the 2-sphere. • More generally, the smash product of two spheres Sm and Sn is homeomorphic to the sphere Sm+n. • The smash product of a space X with a circle is homeomorphic to the reduced suspension of X: $\Sigma X\cong X\wedge S^{1}.$ • The k-fold iterated reduced suspension of X is homeomorphic to the smash product of X and a k-sphere $\Sigma ^{k}X\cong X\wedge S^{k}.$ • In domain theory, taking the product of two domains (so that the product is strict on its arguments). As a symmetric monoidal product For any pointed spaces X, Y, and Z in an appropriate "convenient" category (e.g., that of compactly generated spaces), there are natural (basepoint preserving) homeomorphisms ${\begin{aligned}X\wedge Y&\cong Y\wedge X,\\(X\wedge Y)\wedge Z&\cong X\wedge (Y\wedge Z).\end{aligned}}$ However, for the naive category of pointed spaces, this fails, as shown by the counterexample $X=Y=\mathbb {Q} $ and $Z=\mathbb {N} $ found by Dieter Puppe.[1] A proof due to Kathleen Lewis that Puppe's counterexample is indeed a counterexample can be found in the book of Johann Sigurdsson and J. Peter May.[2] These isomorphisms make the appropriate category of pointed spaces into a symmetric monoidal category with the smash product as the monoidal product and the pointed 0-sphere (a two-point discrete space) as the unit object. One can therefore think of the smash product as a kind of tensor product in an appropriate category of pointed spaces. Adjoint relationship Adjoint functors make the analogy between the tensor product and the smash product more precise. In the category of R-modules over a commutative ring R, the tensor functor $(-\otimes _{R}A)$ is left adjoint to the internal Hom functor $\mathrm {Hom} (A,-)$, so that $\mathrm {Hom} (X\otimes A,Y)\cong \mathrm {Hom} (X,\mathrm {Hom} (A,Y)).$ In the category of pointed spaces, the smash product plays the role of the tensor product in this formula: if $A,X$ are compact Hausdorff then we have an adjunction $\mathrm {Maps_{}} (X\wedge A,Y)\cong \mathrm {Maps_{}} (X,\mathrm {Maps_{}} (A,Y))$ where $\operatorname {Maps_{}} $ denotes continuous maps that send basepoint to basepoint, and $\mathrm {Maps_{}} (A,Y)$ carries the compact-open topology.[3] In particular, taking $A$ to be the unit circle $S^{1}$, we see that the reduced suspension functor $\Sigma $ is left adjoint to the loop space functor $\Omega $: $\mathrm {Maps_{}} (\Sigma X,Y)\cong \mathrm {Maps_{*}} (X,\Omega Y).$ Notes 1. Puppe, Dieter (1958). "Homotopiemengen und ihre induzierten Abbildungen. I.". Mathematische Zeitschrift. 69: 299–344. doi:10.1007/BF01187411. MR 0100265. S2CID 121402726. (p. 336) 2. May, J. Peter; Sigurdsson, Johann (2006). Parametrized Homotopy Theory. Mathematical Surveys and Monographs. Vol. 132. Providence, RI: American Mathematical Society. section 1.5. ISBN 978-0-8218-3922-5. MR 2271789. 3. "Algebraic Topology", Maunder, Theorem 6.2.38c References • Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0.
Hexagon In geometry, a hexagon (from Greek ἕξ, hex, meaning "six", and γωνία, gonía, meaning "corner, angle") is a six-sided polygon.[1] The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. For the crystal system, see Hexagonal crystal family. Regular hexagon A regular hexagon TypeRegular polygon Edges and vertices6 Schläfli symbol{6}, t{3} Coxeter–Dynkin diagrams Symmetry groupDihedral (D6), order 2×6 Internal angle (degrees)120° PropertiesConvex, cyclic, equilateral, isogonal, isotoxal Dual polygonSelf Regular hexagon A regular hexagon has Schläfli symbol {6}[2] and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges. A step-by-step animation of the construction of a regular hexagon using compass and straightedge, given by Euclid's Elements, Book IV, Proposition 15: this is possible as 6 $=$ 2 × 3, a product of a power of two and distinct Fermat primes. When the side length AB is given, drawing a circular arc from point A and point B gives the intersection M, the center of the circumscribed circle. Transfer the line segment AB four times on the circumscribed circle and connect the corner points. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals ${\tfrac {2}{\sqrt {3}}}$ times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (rotational symmetry of order six) and six reflection symmetries (six lines of symmetry), making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral. Parameters The maximal diameter (which corresponds to the long diagonal of the hexagon), D, is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor: ${\frac {1}{2}}d=r=\cos(30^{\circ })R={\frac {\sqrt {3}}{2}}R={\frac {\sqrt {3}}{2}}t$ and, similarly, $d={\frac {\sqrt {3}}{2}}D.$ The area of a regular hexagon ${\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}R^{2}=3Rr=2{\sqrt {3}}r^{2}\\[3pt]&={\frac {3{\sqrt {3}}}{8}}D^{2}={\frac {3}{4}}Dd={\frac {\sqrt {3}}{2}}d^{2}\\[3pt]&\approx 2.598R^{2}\approx 3.464r^{2}\\&\approx 0.6495D^{2}\approx 0.866d^{2}.\end{aligned}}$ For any regular polygon, the area can also be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, and p${}=6R=4r{\sqrt {3}}$, so ${\begin{aligned}A&={\frac {ap}{2}}\\&={\frac {r\cdot 4r{\sqrt {3}}}{2}}=2r^{2}{\sqrt {3}}\\&\approx 3.464r^{2}.\end{aligned}}$ The regular hexagon fills the fraction ${\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270$ of its circumscribed circle. If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then PE + PF = PA + PB + PC + PD. It follows from the ratio of circumradius to inradius that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long diagonal of 1.0000000 will have a distance of 0.8660254 between parallel sides. Point in plane For an arbitrary point in the plane of a regular hexagon with circumradius $R$, whose distances to the centroid of the regular hexagon and its six vertices are $L$ and $d_{i}$ respectively, we have[3] $d_{1}^{2}+d_{4}^{2}=d_{2}^{2}+d_{5}^{2}=d_{3}^{2}+d_{6}^{2}=2\left(R^{2}+L^{2}\right),$ $d_{1}^{2}+d_{3}^{2}+d_{5}^{2}=d_{2}^{2}+d_{4}^{2}+d_{6}^{2}=3\left(R^{2}+L^{2}\right),$ $d_{1}^{4}+d_{3}^{4}+d_{5}^{4}=d_{2}^{4}+d_{4}^{4}+d_{6}^{4}=3\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right).$ If $d_{i}$ are the distances from the vertices of a regular hexagon to any point on its circumcircle, then [3] $\left(\sum _{i=1}^{6}d_{i}^{2}\right)^{2}=4\sum _{i=1}^{6}d_{i}^{4}.$ Symmetry Example hexagons by symmetry r12 regular i4 d6 isotoxal g6 directed p6 isogonal d2 g2 general parallelogon p2 g3 a1 The regular hexagon has D6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D6), 2 dihedral: (D3, D2), 4 cyclic: (Z6, Z3, Z2, Z1) and the trivial (e) These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order.[4] r12 is full symmetry, and a1 is no symmetry. p6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, and d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations. p6m (632) cmm (222) p2 (2222) p31m (33) pmg (22) pg (××) r12 i4 g2 d2 d2 p2 a1 Dih6 Dih2 Z2 Dih1 Z1 A2 and G2 groups A2 group roots G2 group roots The 6 roots of the simple Lie group A2, represented by a Dynkin diagram , are in a regular hexagonal pattern. The two simple roots have a 120° angle between them. The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them. Dissection 6-cube projection 12 rhomb dissection Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into 1⁄2m(m − 1) parallelograms.[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a Petrie polygon projection of a cube, with 3 of 6 square faces. Other parallelogons and projective directions of the cube are dissected within rectangular cuboids. Dissection of hexagons into three rhombs and parallelograms 2D Rhombs Parallelograms Regular {6} Hexagonal parallelogons 3D Square faces Rectangular faces Cube Rectangular cuboid Related polygons and tilings A regular hexagon has Schläfli symbol {6}. A regular hexagon is a part of the regular hexagonal tiling, {6,3}, with three hexagonal faces around each vertex. A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D3 symmetry. A truncated hexagon, t{6}, is a dodecagon, {12}, alternating two types (colors) of edges. An alternated hexagon, h{6}, is an equilateral triangle, {3}. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. A regular hexagon can be dissected into six equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling. A regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling. Regular {6} Truncated t{3} = {6} Hypertruncated triangles Stellated Star figure 2{3} Truncated t{6} = {12} Alternated h{6} = {3} Crossed hexagon A concave hexagon A self-intersecting hexagon (star polygon) Extended Central {6} in {12} A skew hexagon, within cube Dissected {6} projection octahedron Complete graph Self-crossing hexagons There are six self-crossing hexagons with the vertex arrangement of the regular hexagon: Self-intersecting hexagons with regular vertices Dih2 Dih1 Dih3 Figure-eight Center-flip Unicursal Fish-tail Double-tail Triple-tail Hexagonal structures From bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest hexagons. This means that honeycombs require less wax to construct and gain much strength under compression. Irregular hexagons with parallel opposite edges are called parallelogons and can also tile the plane by translation. In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3-space by translation. Hexagonal prism tessellations Form Hexagonal tiling Hexagonal prismatic honeycomb Regular Parallelogonal Tesselations by hexagons Main article: Hexagonal tiling In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane. Hexagon inscribed in a conic section Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration. Cyclic hexagon The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. If the successive sides of a cyclic hexagon are a, b, c, d, e, f, then the three main diagonals intersect in a single point if and only if ace = bdf.[6] If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent.[7] If a hexagon has vertices on the circumcircle of an acute triangle at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle.[8]: p. 179 Hexagon tangential to a conic section Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point. In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f,[9] $a+c+e=b+d+f.$ Equilateral triangles on the sides of an arbitrary hexagon If an equilateral triangle is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the centroids of opposite triangles form another equilateral triangle.[10]: Thm. 1 Skew hexagon A skew hexagon is a skew polygon with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. A skew zig-zag hexagon has vertices alternating between two parallel planes. A regular skew hexagon is vertex-transitive with equal edge lengths. In three dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D3d, [2+,6] symmetry, order 12. The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons. Skew hexagons on 3-fold axes Cube Octahedron Petrie polygons The regular skew hexagon is the Petrie polygon for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections: 4D 5D 3-3 duoprism 3-3 duopyramid 5-simplex Convex equilateral hexagon A principal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a, there exists[11]: p.184, #286.3 a principal diagonal d1 such that ${\frac {d_{1}}{a}}\leq 2$ and a principal diagonal d2 such that ${\frac {d_{2}}{a}}>{\sqrt {3}}.$ Polyhedra with hexagons There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron. These hexagons can be considered truncated triangles, with Coxeter diagrams of the form and . Hexagons in Archimedean solids Tetrahedral Octahedral Icosahedral truncated tetrahedron truncated octahedron truncated cuboctahedron truncated icosahedron truncated icosidodecahedron There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G(2,0): Hexagons in Goldberg polyhedra Tetrahedral Octahedral Icosahedral Chamfered tetrahedron Chamfered cube Chamfered dodecahedron There are also 9 Johnson solids with regular hexagons: Johnson solids with hexagons triangular cupola elongated triangular cupola gyroelongated triangular cupola augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism augmented truncated tetrahedron triangular hebesphenorotunda Truncated triakis tetrahedron Prismoids with hexagons Hexagonal prism Hexagonal antiprism Hexagonal pyramid Tilings with regular hexagons Regular 1-uniform {6,3} r{6,3} rr{6,3} tr{6,3} 2-uniform tilings Gallery of natural and artificial hexagons • The ideal crystalline structure of graphene is a hexagonal grid. • Assembled E-ELT mirror segments • A beehive honeycomb • The scutes of a turtle's carapace • Saturn's hexagon, a hexagonal cloud pattern around the north pole of the planet • Micrograph of a snowflake • Benzene, the simplest aromatic compound with hexagonal shape. • Hexagonal order of bubbles in a foam. • Crystal structure of a molecular hexagon composed of hexagonal aromatic rings. • Naturally formed basalt columns from Giant's Causeway in Northern Ireland; large masses must cool slowly to form a polygonal fracture pattern • An aerial view of Fort Jefferson in Dry Tortugas National Park • The James Webb Space Telescope mirror is composed of 18 hexagonal segments. • In French, l'Hexagone refers to Metropolitan France for its vaguely hexagonal shape. • Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals • Hexagonal barn • The Hexagon, a hexagonal theatre in Reading, Berkshire • Władysław Gliński's hexagonal chess • Pavilion in the Taiwan Botanical Gardens • Hexagonal window See also • 24-cell: a four-dimensional figure which, like the hexagon, has orthoplex facets, is self-dual and tessellates Euclidean space • Hexagonal crystal system • Hexagonal number • Hexagonal tiling: a regular tiling of hexagons in a plane • Hexagram: six-sided star within a regular hexagon • Unicursal hexagram: single path, six-sided star, within a hexagon • Honeycomb conjecture • Havannah: abstract board game played on a six-sided hexagonal grid References 1. Cube picture 2. Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595, archived from the original on 2016-01-02, retrieved 2015-11-06. 3. Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:2010.12340. doi:10.26713/cma.v11i3.1420 (inactive 1 August 2023).{{cite journal}}: CS1 maint: DOI inactive as of August 2023 (link) 4. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) 5. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141 6. Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40. 7. Dergiades, Nikolaos (2014). "Dao's theorem on six circumcenters associated with a cyclic hexagon". Forum Geometricorum. 14: 243–246. Archived from the original on 2014-12-05. Retrieved 2014-11-17. 8. Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). 9. Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", Archived 2012-05-11 at the Wayback Machine, Accessed 2012-04-17. 10. Dao Thanh Oai (2015). "Equilateral triangles and Kiepert perspectors in complex numbers". Forum Geometricorum. 15: 105–114. Archived from the original on 2015-07-05. Retrieved 2015-04-12. 11. Inequalities proposed in "Crux Mathematicorum", Archived 2017-08-30 at the Wayback Machine. External links Look up hexagon in Wiktionary, the free dictionary. • Weisstein, Eric W. "Hexagon". MathWorld. • Definition and properties of a hexagon with interactive animation and construction with compass and straightedge. • An Introduction to Hexagonal Geometry on Hexnet a website devoted to hexagon mathematics. • Hexagons are the Bestagons on YouTube – an animated internet video about hexagons by CGP Grey. Polygons (List) Triangles • Acute • Equilateral • Ideal • Isosceles • Kepler • Obtuse • Right Quadrilaterals • Antiparallelogram • Bicentric • Crossed • Cyclic • Equidiagonal • Ex-tangential • Harmonic • Isosceles trapezoid • Kite • Orthodiagonal • Parallelogram • Rectangle • Right kite • Right trapezoid • Rhombus • Square • Tangential • Tangential trapezoid • Trapezoid By number of sides 1–10 sides • Monogon (1) • Digon (2) • Triangle (3) • Quadrilateral (4) • Pentagon (5) • Hexagon (6) • Heptagon (7) • Octagon (8) • Nonagon (Enneagon, 9) • Decagon (10) 11–20 sides • Hendecagon (11) • Dodecagon (12) • Tridecagon (13) • Tetradecagon (14) • Pentadecagon (15) • Hexadecagon (16) • Heptadecagon (17) • Octadecagon (18) • Icosagon (20) >20 sides • Icositrigon (23) • Icositetragon (24) • Triacontagon (30) • 257-gon • Chiliagon (1000) • Myriagon (10,000) • 65537-gon • Megagon (1,000,000) • Apeirogon (∞) Star polygons • Pentagram • Hexagram • Heptagram • Octagram • Enneagram • Decagram • Hendecagram • Dodecagram Classes • Concave • Convex • Cyclic • Equiangular • Equilateral • Infinite skew • Isogonal • Isotoxal • Magic • Pseudotriangle • Rectilinear • Regular • Reinhardt • Simple • Skew • Star-shaped • Tangential • Weakly simple Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
Soroban The soroban (算盤, そろばん, counting tray) is an abacus developed in Japan. It is derived from the ancient Chinese suanpan, imported to Japan in the 14th century.[1][nb 1] Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocket electronic calculators. Construction The soroban is composed of an odd number of columns or rods, each having beads: one separate bead having a value of five, called go-dama (五玉, ごだま, "five-bead") and four beads each having a value of one, called ichi-dama (一玉, いちだま, "one-bead"). Each set of beads of each rod is divided by a bar known as a reckoning bar. The number and size of beads in each rod make a standard-sized 13-rod soroban much less bulky than a standard-sized suanpan of similar expressive power. The number of rods in a soroban is always odd and never fewer than seven. Basic models usually have thirteen rods, but the number of rods on practical or standard models often increases to 21, 23, 27 or even 31, thus allowing calculation of more digits or representations of several different numbers at the same time. Each rod represents a digit, and a larger number of rods allows the representation of more digits, either in singular form or during operations. The beads and rods are made of a variety of different materials. Most soroban made in Japan are made of wood and have wood, metal, rattan, or bamboo rods for the beads to slide on. The beads themselves are usually biconal (shaped like a double-cone). They are normally made of wood, although the beads of some soroban, especially those made outside Japan, can be marble, stone, or even plastic. The cost of a soroban is commensurate with the materials used in its construction. One unique feature that sets the soroban apart from its Chinese cousin is a dot marking every third rod in a soroban. These are unit rods and any one of them is designated to denote the last digit of the whole number part of the calculation answer. Any number that is represented on rods to the right of this designated rod is part of the decimal part of the answer, unless the number is part of a division or multiplication calculation. Unit rods to the left of the designated one also aid in place value by denoting the groups in the number (such as thousands, millions, etc.). Suanpan usually do not have this feature. Usage Representation of numbers The soroban uses a bi-quinary coded decimal system, where each of the rods can represent a single digit from 0 to 9. By moving beads towards the reckoning bar, they are put in the "on" position; i.e., they assume value. For the "five bead" this means it is moved downwards, while "one beads" are moved upwards. In this manner, all digits from 0 to 9 can be represented by different configurations of beads, as shown below: Representation of digits 0 - 9 on the soroban 0123456789 These digits can subsequently be used to represent multiple-digit numbers. This is done in the same way as in Western, decimal notation: the rightmost digit represents units, the one to the left of it represents tens, etc. The number 8036, for instance, is represented by the following configuration: 8036 The soroban user is free to choose which rod is used for the units; typically this will be one of the rods marked with a dot (see the 6 in the example above). Any digits to the right of the units represent decimals: tenths, hundredths, etc. In order to change 8036 into 80.36, for instance, the user places the digits in such a way that the 0 falls on a rod marked with a dot: 80.36 Methods of operation The methods of addition and subtraction on a soroban are basically the same as the equivalent operations on a suanpan, with basic addition and subtraction making use of a complementary number to add or subtract ten in carrying over. There are many methods to perform both multiplication and division on a soroban, especially Chinese methods that came with the importation of the suanpan. The authority in Japan on the soroban, the Japan Abacus Committee, has recommended so-called standard methods for both multiplication and division which require only the use of the multiplication table. These methods were chosen for efficiency and speed in calculation. Because the soroban developed through a reduction in the number of beads from seven, to six, and then to the present five, these methods can be used on the suanpan as well as on soroban produced before the 1930s, which have five "one" beads and one "five" bead. Modern use The Japanese abacus has been taught in school for over 500 years, deeply rooted in the value of learning the fundamentals as a form of art.[3] However, the introduction of the West during the Meiji period and then again after World War II has gradually altered the Japanese education system. Now, the strive is for speed and turning out deliverables rather than understanding the subtle intricacies of the concepts behind the product. Calculators have since replaced sorobans, and elementary schools are no longer required to teach students how to use the soroban, though some do so by choice. The growing popularity of calculators within the context of Japanese modernization has driven the study of soroban from public schools to private after school classrooms. Where once it was an institutionally required subject in school for children grades 2 to 6, current laws have made keeping this art form and perspective on math practiced amongst the younger generations more lenient.[4] Today, it shifted from a given to a game where one can take The Japanese Chamber of Commerce and Industry's examination in order to obtain a certificate and license.[5] There are six levels of mastery, starting from sixth-grade (very skilled) all the way up to first-grade (for those who have completely mastered the use of the soroban). Those obtaining at least a third-grade certificate/license are qualified to work in public corporations. The soroban is still taught in some primary schools as a way to visualize and grapple with mathematical concepts. The practice of soroban includes the teacher reciting a string of numbers (addition, subtraction, multiplication, and division) in a song-like manner where at the end, the answer is given by the teacher. This helps train the ability to follow the tempo given by the teacher while remaining calm and accurate. In this way, it reflects on a fundamental aspect of Japanese culture of practicing meditative repetition in every aspect of life.[3] Primary school students often bring two soroban to class, one with the modern configuration and the one having the older configuration of one heavenly bead and five earth beads. Shortly after the beginning of one's soroban studies, drills to enhance mental calculation, known as anzan (暗算, "blind calculation") in Japanese are incorporated. Students are asked to solve problems mentally by visualizing the soroban and working out the solution by moving the beads theoretically in one's mind. The mastery of anzan is one reason why, despite the access to handheld calculators, some parents still send their children to private tutors to learn the soroban. The soroban is also the basis for two kinds of abaci developed for the use of blind people. One is the toggle-type abacus wherein flip switches are used instead of beads. The second is the Cranmer abacus which has circular beads, longer rods, and a leather backcover so the beads do not slide around when in use. Brief history The soroban's physical resemblance is derived from the suanpan but the number of beads is identical to the Roman abacus, which had four beads below and one at the top. Most historians on the soroban agree that it has its roots on the suanpan's importation to Japan via the Korean peninsula around the 14th century.[1][nb 1] When the suanpan first became native to Japan as the soroban (with its beads modified for ease of use), it had two heavenly beads and five earth beads. But the soroban was not widely used until the 17th century, although it was in use by Japanese merchants since its introduction.[6] Once the soroban became popularly known, several Japanese mathematicians, including Seki Kōwa, studied it extensively. These studies became evident on the improvements on the soroban itself and the operations used on it. In the construction of the soroban itself, the number of beads had begun to decrease. In around 1850, one heavenly bead was removed from the suanpan configuration of two heavenly beads and five earth beads. This new Japanese configuration existed concurrently with the suanpan until the start of the Meiji era, after which the suanpan fell completely out of use. In 1891, Irie Garyū further removed one earth bead, forming the modern configuration of one heavenly bead and four earth beads.[7] This configuration was later reintroduced in 1930 and became popular in the 1940s. Also, when the suanpan was imported to Japan, it came along with its division table. The method of using the table was called kyūkihō (九帰法, "nine returning method") in Japanese, while the table itself was called the hassan (八算, "eight calculation"). The division table used along with the suanpan was more popular because of the original hexadecimal configuration of Japanese currency . But because using the division table was complicated and it should be remembered along with the multiplication table, it soon fell out in 1935 (soon after the soroban's present form was reintroduced in 1930), with a so-called standard method replacing the use of the division table. This standard method of division, recommended today by the Japan Abacus Committee, is in fact an old method which used counting rods, first suggested by mathematician Momokawa Chubei in 1645,[8] and therefore had to compete with the division table during the latter's heyday Comparison with the electric calculator On November 12, 1946, a contest was held in Tokyo between the Japanese soroban, used by Kiyoshi Matsuzaki, and an electric calculator, operated by US Army Private Thomas Nathan Wood. The basis for scoring in the contest was speed and accuracy of results in all four basic arithmetic operations and a problem which combines all four. The soroban won 4 to 1, with the electric calculator prevailing in multiplication.[9] About the event, the Nippon Times newspaper reported that "Civilization ... tottered" that day, while the Stars and Stripes newspaper described the soroban's "decisive" victory as an event in which "the machine age took a step backward....". The breakdown of results is as follows: • Five additions problems for each heat, each problem consisting of 50 three- to six-digit numbers. The soroban won in two successive heats. • Five subtraction problems for each heat, each problem having six- to eight-digit minuends and subtrahends. The soroban won in the first and third heats; the second heat was a no contest. • Five multiplication problems, each problem having five- to 12-digit factors. The calculator won in the first and third heats; the soroban won on the second. • Five division problems, each problem having five- to 12-digit dividends and divisors. The soroban won in the first and third heats; the calculator won on the second. • A composite problem which the soroban answered correctly and won on this round. It consisted of: • An addition problem involving 30 six-digit numbers • Three subtraction problems, each with two six-digit numbers • Three multiplication problems, each with two figures containing a total of five to twelve digits • Three division problems, each with two figures containing a total of five to twelve digits Even with the improvement of technology involving calculators, this event has yet to be replicated officially. See also • Abacus • Suanpan • Chisanbop • Pental system • Bi-quinary coded decimal Notes 1. Some sources give a date of introduction of around 1600.[2] Footnotes Wikimedia Commons has media related to Soroban. 1. Gullberg 1997, p. 169 harvnb error: no target: CITEREFGullberg1997 (help) 2. Fernandes 2013 3. Suzuki, Daisetz T. (1959). Zen and the Japanese Culture. Princeton University Press. 4. "Soroban in Education and Modern Japanese Society". History of Soroban. Retrieved 21 November 2018. 5. Kojima, Takashi (1954). The Japanese Abacus: its Use and Theory. Tokyo: Charles E. Tuttle. ISBN 0-8048-0278-5. 6. "そろばんの歴史　ー　西欧、中国、そして日本へ", "トモエそろばん", Retrieved 2017-10-19. 7. Frédéric, Louis (2002). Japan encyclopedia. Translated by Roth, Käthe. Harvard University Press. pp. 303, 903. ISBN 9780674017535. 8. Smith, David Eugene; Mikami, Yoshio (1914). "Chapter III: The Development of the Soroban.". A History of Japanese Mathematics. The Open Court Publishing. pp. 43–44. digital copy Archived 2010-12-03 at the Wayback Machine 9. Stoddard, Edward (1994). Speed Mathematics Simplified. Dover. p. 12. References • Kojima, Takashi (1963). Advanced Abacus: Japanese Theory and Practice. Tokyo: Charles E. Tuttle. • Soroban. Japan: The Japan Chamber of Commerce and Industry. 1989. • Bernazzani, David (March 2, 2005). Soroban Abacus Handbook (PDF) (Rev 1.05 ed.). • Fernandes, Luis (2013). "The Abacus: A Brief History". ee.ryerson.ca. Archived from the original on March 3, 2000. Retrieved July 31, 2014. • Heffelfinger, Totton; Flom, Gary (2004). Abacus: Mystery of the Bead. • Knott, Cargill Gilston (1886). "The Abacus, in Its Historic and Scientific Aspects" (PDF). The Transactions of the Asiatic Society of Japan. xiv: 18–72. External links • Japanese Soroban Association (in English)
Tangram The tangram (Chinese: 七巧板; pinyin: qīqiǎobǎn; lit. 'seven boards of skill') is a dissection puzzle consisting of seven flat polygons, called tans, which are put together to form shapes. The objective is to replicate a pattern (given only an outline) generally found in a puzzle book using all seven pieces without overlap. Alternatively the tans can be used to create original minimalist designs that are either appreciated for their inherent aesthetic merits or as the basis for challenging others to replicate its outline. It is reputed to have been invented in China sometime around the late 18th century and then carried over to America and Europe by trading ships shortly after.[1] It became very popular in Europe for a time, and then again during World War I. It is one of the most widely recognized dissection puzzles in the world and has been used for various purposes including amusement, art, and education. [2][3][4] Etymology The origin of the English word 'tangram' is unclear. One conjecture holds that it is a compound of the Greek element '-gram' derived from γράμμα ('written character, letter, that which is drawn') with the 'tan-' element being variously conjectured to be Chinese t'an 'to extend' or Cantonese t'ang 'Chinese'.[5] Alternatively, the word may be derivative of the archaic English 'tangram' meaning "an odd, intricately contrived thing".[6] In either case, the first known use of the word is believed to be found in the 1848 book Geometrical Puzzle for the Young by mathematician and future Harvard University president Thomas Hill.[7] Hill likely coined the term in the same work, and vigorously promoted the word in numerous articles advocating for the puzzle's use in education, and in 1864 the word received official recognition in the English language when it was included in Noah Webster's American Dictionary.[8] History Origins Despite its relatively recent emergence in the West, there is a much older tradition of dissection amusements in China which likely played a role in its inspiration. In particular, the modular banquet tables of the Song dynasty bear an uncanny resemblance to the playing pieces of the Tangram and there were books dedicated to arranging them together to form pleasing patterns.[9] Several Chinese sources broadly report a well-known Song dynasty polymath Huang Bosi 黄伯思 who developed a form of entertainment for his dinner guests based on creative arrangements of six small tables called 宴几 or 燕几（feast tables or swallow tables respectively). One diagram shows these as oblong rectangles, and other reports suggest a seventh table being added later, perhaps by a later inventor. According to Western sources, however, the tangram's historical Chinese inventor is unknown except through the pen name Yang-cho-chu-shih (Dim-witted (?) recluse, recluse = 处士). It is believed that the puzzle was originally introduced in a book titled Ch'i chi'iao t'u which was already being reported as lost in 1815 by Shan-chiao in his book New Figures of the Tangram. Nevertheless, it is generally reputed that the puzzle's origins would have been around 20 years earlier than this.[10] The prominent third-century mathematician Liu Hui made use of construction proofs in his works and some bear a striking resemblance to the subsequently developed Banquet tables which in turn seem to anticipate the Tangram. While there is no reason to suspect that tangrams were used in the proof of the Pythagorean theorem, as is sometimes reported, it is likely that this style of geometric reasoning went on to exert an influence on Chinese cultural life that lead directly to the puzzle.[11] The early years of attempting to date the Tangram were confused by the popular but fraudulently written history by famed puzzle maker Samuel Loyd in his 1908 The Eighth Book Of Tan. This work contains many whimsical features that aroused both interest and suspicion amongst contemporary scholars who attempted to verify the account. By 1910 it was clear that it was a hoax. A letter dated from this year from the Oxford Dictionary editor Sir James Murray on behalf of a number of Chinese scholars to the prominent puzzlist Henry Dudeney reads "The result has been to show that the man Tan, the god Tan, and the Book of Tan are entirely unknown to Chinese literature, history or tradition."[6] Along with its many strange details The Eighth Book of Tan's date of creation for the puzzle of 4000 years in antiquity had to be regarded as entirely baseless and false. Reaching the Western world (1815–1820s) The earliest extant tangram was given to the Philadelphia shipping magnate and congressman Francis Waln in 1802 but it was not until over a decade later that Western audiences, at large, would be exposed to the puzzle.[1] In 1815, American Captain M. Donnaldson was given a pair of author Sang-Hsia-koi's books on the subject (one problem and one solution book) when his ship, Trader docked there. They were then brought with the ship to Philadelphia, in February 1816. The first tangram book to be published in America was based on the pair brought by Donnaldson.[12] The puzzle eventually reached England, where it became very fashionable. The craze quickly spread to other European countries. This was mostly due to a pair of British tangram books, The Fashionable Chinese Puzzle, and the accompanying solution book, Key.[13] Soon, tangram sets were being exported in great number from China, made of various materials, from glass, to wood, to tortoise shell.[14] Many of these unusual and exquisite tangram sets made their way to Denmark. Danish interest in tangrams skyrocketed around 1818, when two books on the puzzle were published, to much enthusiasm.[15] The first of these was Mandarinen (About the Chinese Game). This was written by a student at Copenhagen University, which was a non-fictional work about the history and popularity of tangrams. The second, Det nye chinesiske Gaadespil (The new Chinese Puzzle Game), consisted of 339 puzzles copied from The Eighth Book of Tan, as well as one original.[15] One contributing factor in the popularity of the game in Europe was that although the Catholic Church forbade many forms of recreation on the sabbath, they made no objection to puzzle games such as the tangram.[16] Second craze in Germany (1891–1920s) Tangrams were first introduced to the German public by industrialist Friedrich Adolf Richter around 1891.[17] The sets were made out of stone or false earthenware,[18] and marketed under the name "The Anchor Puzzle".[17] More internationally, the First World War saw a great resurgence of interest in tangrams, on the homefront and trenches of both sides. During this time, it occasionally went under the name of "The Sphinx" an alternative title for the "Anchor Puzzle" sets.[19][20] Paradoxes A tangram paradox is a dissection fallacy: Two figures composed with the same set of pieces, one of which seems to be a proper subset of the other.[21] One famous paradox is that of the two monks, attributed to Henry Dudeney, which consists of two similar shapes, one with and the other missing a foot.[22] In reality, the area of the foot is compensated for in the second figure by a subtly larger body. The two-monks paradox – two similar shapes but one missing a foot: The Magic Dice Cup tangram paradox – from Sam Loyd's book The 8th Book of Tan (1903).[23] Each of these cups was composed using the same seven geometric shapes. But the first cup is whole, and the others contain vacancies of different sizes. (Notice that the one on the left is slightly shorter than the other two. The one in the middle is ever-so-slightly wider than the one on the right, and the one on the left is narrower still.)[24] Clipped square tangram paradox – from Loyd's book The Eighth Book of Tan (1903):[23] The seventh and eighth figures represent the mysterious square, built with seven pieces: then with a corner clipped off, and still the same seven pieces employed.[25] Number of configurations Over 6500 different tangram problems have been created from 19th century texts alone, and the current number is ever-growing.[26] Fu Traing Wang and Chuan-Chih Hsiung proved in 1942 that there are only thirteen convex tangram configurations (config segment drawn between any two points on the configuration's edge always pass through the configuration's interior, i.e., configurations with no recesses in the outline).[27][28] Pieces Choosing a unit of measurement so that the seven pieces can be assembled to form a square of side one unit and having area one square unit, the seven pieces are:[29] • 2 large right triangles (hypotenuse 1, sides √2/2, area 1/4) • 1 medium right triangle (hypotenuse √2/2, sides 1/2, area 1/8) • 2 small right triangles (hypotenuse 1/2, sides √2/4, area 1/16) • 1 square (sides √2/4, area 1/8) • 1 parallelogram (sides of 1/2 and √2/4, height of 1/4, area 1/8) Of these seven pieces, the parallelogram is unique in that it has no reflection symmetry but only rotational symmetry, and so its mirror image can be obtained only by flipping it over. Thus, it is the only piece that may need to be flipped when forming certain shapes. See also • Tangram (video game) • Egg of Columbus (tangram puzzle) • Mathematical puzzle • Ostomachion • Tiling puzzle References 1. Slocum (2003), p. 21. 2. Campillo-Robles, Jose M.; Alonso, Ibon; Gondra, Ane; Gondra, Nerea (1 September 2022). "Calculation and measurement of center of mass: An all-in-one activity using Tangram puzzles". American Journal of Physics. 90 (9): 652. Bibcode:2022AmJPh..90..652C. doi:10.1119/5.0061884. ISSN 0002-9505. S2CID 251917733. 3. Slocum (2001), p. 9. 4. Forbrush, William Byron (1914). Manual of Play. Jacobs. p. 315. Retrieved 2010-10-13. 5. Oxford English Dictionary, 1910, s.v. 6. Slocum (2003), p. 23. 7. Hill, Thomas (1848). Puzzles to teach geometry : in seventeen cards numbered from the first to the seventeenth inclusive. Boston : Wm. Crosby & H.P. Nichols. 8. Slocum (2003), p. 25. 9. Slocum (2003), p. 16. 10. Slocum (2003), pp. 16–19. 11. Slocum (2003), p. 15. 12. Slocum (2003), p. 30. 13. Slocum (2003), p. 31. 14. Slocum (2003), p. 49. 15. Slocum (2003), pp. 99–100. 16. Slocum (2003), p. 51. 17. "Tangram the incredible timeless 'Chinese' puzzle". www.archimedes-lab.org. 18. Treasury Decisions Under customs and other laws, Volume 25. United States Department Of The Treasury. 1890–1926. p. 1421. Retrieved 2010-09-16. 19. Wyatt (26 April 2006). "Tangram – The Chinese Puzzle". h2g2. BBC. Archived from the original on 2011-10-02. Retrieved 2010-10-03. 20. Braman, Arlette (2002). Kids Around The World Play!. John Wiley and Sons. p. 10. ISBN 978-0-471-40984-7. Retrieved 2010-09-05. 21. Tangram Paradox, by Barile, Margherita, From MathWorld – A Wolfram Web Resource, created by Eric W. Weisstein. 22. Dudeney, H. (1958). Amusements in Mathematics. New York: Dover Publications. 23. The 8th Book of Tan by Sam Loyd. 1903 – via Tangram Channel. 24. "The Magic Dice Cup". 2 April 2011. 25. Loyd, Sam (1968). The eighth book of Tan – 700 Tangrams by Sam Loyd with an introduction and solutions by Peter Van Note. New York: Dover Publications. p. 25. 26. Slocum 2001, p. 37. 27. Fu Traing Wang; Chuan-Chih Hsiung (November 1942). "A Theorem on the Tangram". The American Mathematical Monthly. 49 (9): 596–599. doi:10.2307/2303340. JSTOR 2303340. 28. Read, Ronald C. (1965). Tangrams : 330 Puzzles. New York: Dover Publications. p. 53. ISBN 0-486-21483-4. 29. Brooks, David J. (1 December 2018). "How to Make a Classic Tangram Puzzle". Boys' Life magazine. Retrieved 2020-03-10. Sources • Slocum, Jerry (2001). The Tao of Tangram. Barnes & Noble. ISBN 978-1-4351-0156-2. • Slocum, Jerry (2003). The Tangram Book. Sterling. ISBN 978-1-4027-0413-0. Further reading • Anno, Mitsumasa. Anno's Math Games (three volumes). New York: Philomel Books, 1987. ISBN 0-399-21151-9 (v. 1), ISBN 0-698-11672-0 (v. 2), ISBN 0-399-22274-X (v. 3). • Botermans, Jack, et al. The World of Games: Their Origins and History, How to Play Them, and How to Make Them (translation of Wereld vol spelletjes). New York: Facts on File, 1989. ISBN 0-8160-2184-8. • Dudeney, H. E. Amusements in Mathematics. New York: Dover Publications, 1958. • Gardner, Martin. "Mathematical Games—on the Fanciful History and the Creative Challenges of the Puzzle Game of Tangrams", Scientific American Aug. 1974, p. 98–103. • Gardner, Martin. "More on Tangrams", Scientific American Sep. 1974, p. 187–191. • Gardner, Martin. The 2nd Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon & Schuster, 1961. ISBN 0-671-24559-7. • Loyd, Sam. Sam Loyd's Book of Tangram Puzzles (The 8th Book of Tan Part I). Mineola, New York: Dover Publications, 1968. • Slocum, Jerry, et al. Puzzles of Old and New: How to Make and Solve Them. De Meern, Netherlands: Plenary Publications International (Europe); Amsterdam, Netherlands: ADM International; Seattle: Distributed by University of Washington Press, 1986. ISBN 0-295-96350-6. External links Wikimedia Commons has media related to Tangrams. • Past & Future: The Roots of Tangram and Its Developments • Turning Your Set of Tangram Into A Magic Math Puzzle by puzzle designer G. Sarcone Polyforms Polyominoes • Domino • Tromino • Tetromino • Pentomino • Hexomino • Heptomino • Octomino • Nonomino • Decomino Higher dimensions • Polyominoid • Polycube Others • Polyabolo • Polydrafter • Polyhex • Polyiamond • Pseudo-polyomino • Polystick Games and puzzles • Blokus • Soma cube • Snake cube • Tangram • Hexastix • Tantrix • Tetris WikiProject Portal Authority control: National • Germany • Israel • United States • Czech Republic
Shinichi Mochizuki Shinichi Mochizuki (望月新一, Mochizuki Shin'ichi, born March 29, 1969) is a Japanese mathematician working in number theory and arithmetic geometry. He is one of the main contributors to anabelian geometry. His contributions include his solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. Mochizuki has also worked in Hodge–Arakelov theory and p-adic Teichmüller theory. Mochizuki developed inter-universal Teichmüller theory,[2][3][4][5] which has attracted attention from non-mathematicians due to claims it provides a resolution of the abc conjecture.[6] Shinichi Mochizuki Born (1969-03-29) March 29, 1969[1] Tokyo, Japan[1] NationalityJapanese Alma materPrinceton University Known forAnabelian geometry Inter-universal Teichmüller theory AwardsJSPS Prize, Japan Academy Medal[1] Scientific career FieldsMathematics InstitutionsKyoto University Doctoral advisorGerd Faltings Biography Early life Shinichi Mochizuki was born to parents Kiichi and Anne Mochizuki.[7] When he was five years old, Shinichi Mochizuki and his family left Japan to live in the United States. His father was Fellow of the Center for International Affairs and Center for Middle Eastern Studies at Harvard University (1974–76).[8] Mochizuki attended Phillips Exeter Academy and graduated in 1985.[9] Mochizuki entered Princeton University as an undergraduate student at the age of 16 and graduated as salutatorian with an A.B. in mathematics in 1988.[9] He completed his senior thesis, titled "Curves and their deformations," under the supervision of Gerd Faltings.[10] He remained at Princeton for graduate studies and received his Ph.D. in mathematics in 1992 after completing his doctoral dissertation, titled "The geometry of the compactification of the Hurwitz scheme," also under the supervision of Faltings.[11] After his graduate studies, Mochizuki spent two years at Harvard University and then in 1994 moved back to Japan to join the Research Institute for Mathematical Sciences (RIMS) at Kyoto University in 1992, and was promoted to professor in 2002.[1][12] Career Mochizuki proved Grothendieck's conjecture on anabelian geometry in 1996. He was an invited speaker at the International Congress of Mathematicians in 1998.[13] In 2000–2008 he discovered several new theories including the theory of frobenioids, mono-anabelian geometry and the etale theta theory for line bundles over tempered covers of the Tate curve. On August 30, 2012 Mochizuki released four preprints, whose total size was about 500 pages, that developped inter-universal Teichmüller theory and applied it in an attempt to prove several very famous problems in Diophantine geometry.[14] These include the strong Szpiro conjecture, the hyperbolic Vojta conjecture and the abc conjecture over every number field. In September 2018, Mochizuki posted a report on his work by Peter Scholze and Jakob Stix asserting that the third preprint contains an irreparable flaw; he also posted several documents containing his rebuttal of their criticism.[15] The majority of number theorists have found Mochizuki's preprints very difficult to follow and have not accepted the conjectures as settled, although there are a few prominent exceptions, including Go Yamashita, Ivan Fesenko, and Yuichiro Hoshi, who vouch for the work and have written expositions of the theory.[16][17] On April 3, 2020, two Japanese mathematicians, Masaki Kashiwara and Akio Tamagawa, announced that Mochizuki's claimed proof of the abc conjecture would be published in Publications of the Research Institute for Mathematical Sciences, a journal of which Mochizuki is chief editor.[18] The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by Nature as "unlikely to move many researchers over to Mochizuki's camp".[18] The special issue containing Mochizuki's articles was published on March 5, 2021.[2][3][4][5] Publications • Mochizuki, Shinichi (1997), "A Version of the Grothendieck Conjecture for p-adic Local Fields" (PDF), International Journal of Mathematics, Singapore: World Scientific Pub. Co., 8 (3): 499–506, CiteSeerX 10.1.1.161.7778, doi:10.1142/S0129167X97000251, ISSN 0129-167X • Mochizuki, Shinichi (1998), "The intrinsic Hodge theory of p-adic hyperbolic curves, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)", Documenta Mathematica: 187–196, ISSN 1431-0635, MR 1648069 • Mochizuki, Shinichi (1999), Foundations of p-adic Teichmüller theory, AMS/IP Studies in Advanced Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1190-0, MR 1700772 Inter-universal Teichmüller theory • Mochizuki, Shinichi (2011), "Inter-universal Teichmüller Theory: A Progress Report" (PDF), Development of Galois–Teichmüller Theory and Anabelian Geometry, The 3rd Mathematical Society of Japan, Seasonal Institute. • Mochizuki, Shinichi (2012a), Inter-universal Teichmuller Theory I: Construction of Hodge Theaters (PDF). • Mochizuki, Shinichi (2012b), Inter-universal Teichmuller Theory II: Hodge–Arakelov-theoretic Evaluation (PDF). • Mochizuki, Shinichi (2012c), Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice (PDF). • Mochizuki, Shinichi (2012d), Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations (PDF). References 1. Mochizuki, Shinichi. "Curriculum Vitae" (PDF). Retrieved 14 September 2012. 2. Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory I: Construction of Hodge Theaters" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 3–207. doi:10.4171/PRIMS/57-1-1. S2CID 233829305. 3. Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 209–401. doi:10.4171/PRIMS/57-1-2. S2CID 233794971. 4. Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory III: Canonical Splittings of the Log-Theta-Lattice" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 403–626. doi:10.4171/PRIMS/57-1-3. S2CID 233777314. 5. Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 627–723. doi:10.4171/PRIMS/57-1-4. S2CID 3135393. 6. Crowell 2017. 7. Leah P. (Edelman) Rauch Philly.com on Mar. 6, 2005 8. MOCHIZUKI, Kiichi Dr. National Association of Japan-America Societies, Inc. 9. "Seniors address commencement crowd". Princeton Weekly Bulletin. Vol. 77. 20 June 1988. p. 4. Archived from the original on 3 April 2013.{{cite news}}: CS1 maint: bot: original URL status unknown (link) 10. Mochizuki, Shinichi (1988). Curves and their deformations. Princeton, NJ: Department of Mathematics. 11. Mochizuki, Shinichi (1992). The geometry of the compactification of the Hurwitz scheme. 12. Castelvecchi 2015. 13. "International Congress of Mathematicians 1998". Archived from the original on 2015-12-19. 14. Inter-universal Teichmüller theory IV: log-volume computations and set-theoretic foundations, Shinichi Mochizuki, August 2012 15. Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture". Quanta Magazine. 16. Fesenko, Ivan (2016), "Fukugen", Inference: International Review of Science, 2 (3), doi:10.37282/991819.16.25 17. Roberts, David Michael (2019), "A crisis of identification", Inference: International Review of Science, 4 (3), doi:10.37282/991819.19.2, S2CID 232514600 18. Castelvecchi, Davide (April 3, 2020). "Mathematical proof that rocked number theory will be published". Nature. 580 (7802): 177. Bibcode:2020Natur.580..177C. doi:10.1038/d41586-020-00998-2. PMID 32246118. S2CID 214786566. Retrieved April 4, 2020. Sources • Castelvecchi, Davide (7 October 2015), "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof", Nature, 526 (7572): 178–181, Bibcode:2015Natur.526..178C, doi:10.1038/526178a, PMID 26450038 • Crowell, Rachel (19 September 2017). "On a summary of Shinichi Mochizuki's proof for the abc conjecture". American Mathematical Society. • Ishikura, Tetsuya (16 December 2017). "Mathematician in Kyoto cracks formidable brainteaser". The Asahi Shimbun. External links Wikiquote has quotations related to Shinichi Mochizuki • Shinichi Mochizuki at the Mathematics Genealogy Project • Personal website • Papers of Shinichi Mochizuki • A brief introduction to inter-universal geometry • On inter-universal Teichmüller theory of Shinichi Mochizuki, colloquium talk by Ivan Fesenko • Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki by Ivan Fesenko • Introduction to inter-universal Teichmüller theory (in Japanese), a survey by Yuichiro Hoshi • RIMS Joint Research Workshop: On the verification and further development of inter-universal Teichmuller theory, March 2015, Kyoto* • CMI workshop on IUT theory of Shinichi Mochizuki, December 2015, Oxford* Authority control International • ISNI • VIAF National • Germany • Israel • United States • Japan • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
Tally marks Tally marks, also called hash marks, are a form of numeral used for counting. They can be thought of as a unary numeral system. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. However, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were also historically used for this purpose. Early history Counting aids other than body parts appear in the Upper Paleolithic. The oldest tally sticks date to between 35,000 and 25,000 years ago, in the form of notched bones found in the context of the European Aurignacian to Gravettian and in Africa's Late Stone Age. The so-called Wolf bone is a prehistoric artifact discovered in 1937 in Czechoslovakia during excavations at Vestonice, Moravia, led by Karl Absolon. Dated to the Aurignacian, approximately 30,000 years ago, the bone is marked with 55 marks which may be tally marks. The head of an ivory Venus figurine was excavated close to the bone.[1] The Ishango bone, found in the Ishango region of the present-day Democratic Republic of Congo, is dated to over 20,000 years old. Upon discovery, it was thought to portray a series of prime numbers. In the book How Mathematics Happened: The First 50,000 Years, Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."[2] Alexander Marshack examined the Ishango bone microscopically, and concluded that it may represent a six-month lunar calendar.[3] Clustering Tally marks are typically clustered in groups of five for legibility. The cluster size 5 has the advantages of (a) easy conversion into decimal for higher arithmetic operations and (b) avoiding error, as humans can far more easily correctly identify a cluster of 5 than one of 10. • Tally marks representing (from left to right) the numbers 1, 2, 3, 4 and 5 that was used in most of Europe, the Anglosphere, and Southern Africa. In some variants, the diagonal/horizontal slash is used on its own when five or more units are added at once. • Cultures using Chinese characters tally by forming the character 正,[lower-alpha 1] which consists of five strokes.[4][5] • Tally marks used in France, Portugal, Spain, and their former colonies, including Latin America. 1 to 5 and so on. These are most commonly used for registering scores in card games, like Truco. • In the dot and line (or dot-dash) tally, dots represent counts from 1 to 4, lines 5 to 8, and diagonal lines 9 and 10. This method is commonly used in forestry and related fields.[6] Writing systems Part of a series on Numeral systems Place-value notation Hindu-Arabic numerals • Western Arabic • Eastern Arabic • Bengali • Devanagari • Gujarati • Gurmukhi • Odia • Sinhala • Tamil • Malayalam • Telugu • Kannada • Dzongkha • Tibetan • Balinese • Burmese • Javanese • Khmer • Lao • Mongolian • Sundanese • Thai East Asian systems Contemporary • Chinese • Suzhou • Hokkien • Japanese • Korean • Vietnamese Historic • Counting rods • Tangut Other systems • History Ancient • Babylonian Post-classical • Cistercian • Mayan • Muisca • Pentadic • Quipu • Rumi Contemporary • Cherokee • Kaktovik (Iñupiaq) By radix/base Common radices/bases • 2 • 3 • 4 • 5 • 6 • 8 • 10 • 12 • 16 • 20 • 60 • (table) Non-standard radices/bases • Bijective (1) • Signed-digit (balanced ternary) • Mixed (factorial) • Negative • Complex (2i) • Non-integer (φ) • Asymmetric Sign-value notation Non-alphabetic • Aegean • Attic • Aztec • Brahmi • Chuvash • Egyptian • Etruscan • Kharosthi • Prehistoric counting • Proto-cuneiform • Roman • Tally marks Alphabetic • Abjad • Armenian • Alphasyllabic • Akṣarapallī • Āryabhaṭa • Kaṭapayādi • Coptic • Cyrillic • Geʽez • Georgian • Glagolitic • Greek • Hebrew List of numeral systems Roman numerals, the Brahmi and Chinese numerals for one through three (一二三), and rod numerals were derived from tally marks, as possibly was the ogham script.[7] Base 1 arithmetic notation system is a unary positional system similar to tally marks. It is rarely used as a practical base for counting due to its difficult readability. The numbers 1, 2, 3, 4, 5, 6 ... would be represented in this system as[8] 0, 00, 000, 0000, 00000, 000000 ... Base 1 notation is widely used in type numbers of flour; the higher number represents a higher grind. Unicode In 2015, Ken Lunde and Daisuke Miura submitted a proposal to encode various systems of tally marks in the Unicode Standard.[9] However, the box tally and dot-and-dash tally characters were not accepted for encoding, and only the five ideographic tally marks (正 scheme) and two Western tally digits were added to the Unicode Standard in the Counting Rod Numerals block in Unicode version 11.0 (June 2018). Only the tally marks for the numbers 1 and 5 are encoded, and tally marks for the numbers 2, 3 and 4 are intended to be composed from sequences of tally mark 1 at the font level. Counting Rod Numerals[1][2] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+1D36x 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 𝍪 𝍫 𝍬 𝍭 𝍮 𝍯 U+1D37x 𝍰 𝍱 𝍲 𝍳 𝍴 𝍵 𝍶 𝍷 𝍸 Notes 1.^ As of Unicode version 15.0 2.^ Grey areas indicate non-assigned code points See also • History of writing ancient numbers • Abacus • Australian Aboriginal enumeration • Carpenters' marks • Cherty i rezy • Chuvash numerals • Counting rods • Finger counting • Hangman (game) • History of communication • History of mathematics • Lebombo bone • List of international common standards • Paleolithic tally sticks • Prehistoric numerals • Quipu • Roman numerals • Tally stick Wikimedia Commons has media related to Tally marks. Notes 1. This character was apparently chosen purely due to appropriateness of the physical process of writing it using the conventional stroke-order system—i.e., the physical movements of the strokes have a distinct alternation right-down-right-down-right working down the character, but the semantics of the character have no particular relation to the concept of "5" (neither in the character etymology nor the word etymology, which in languages using Chinese characters are two originally-separate-but-historically-complexly-interacting things). By contrast, the character for "five", 五, which looks like it also has 5 distinct lines, has only 4 strokes when written using conventional stroke-order.) References • Graham Flegg, Numbers: their history and meaning, Courier Dover Publications, 2002 ISBN 978-0-486-42165-0, pp. 41-42. 1. Rudman, Peter Strom (2007). How Mathematics Happened: The First 50,000 Years. Prometheus Books. p. 64. ISBN 978-1-59102-477-4. 2. Marshack, Alexander (1991): The Roots of Civilization, Colonial Hill, Mount Kisco, NY. 3. Hsieh, Hui-Kuang (1981) "Chinese tally mark", The American Statistician, 35 (3), p. 174, doi:10.2307/2683999 4. Ken Lunde, Daisuke Miura, L2/16-046: Proposal to encode five ideographic tally marks, 2016 5. Schenck, Carl A. (1898) Forest mensuration. The University Press. (Note: The linked reference appears to actually be "Bulletin of the Ohio Agricultural Experiment Station", Number 302, August 1916) 6. Macalister, R. A. S., Corpus Inscriptionum Insularum Celticarum Vol. I and II, Dublin: Stationery Office (1945). 7. Hext, Jan (1990), Programming Structures: Machines and programs, Programming Structures, vol. 1, Prentice Hall, p. 33, ISBN 9780724809400. 8. Lunde, Ken; Miura, Daisuke (30 November 2015). "Proposal to encode tally marks" (PDF). Unicode Consortium.
Chinese numerals Chinese numerals are words and characters used to denote numbers in Chinese. Part of a series on Numeral systems Place-value notation Hindu-Arabic numerals • Western Arabic • Eastern Arabic • Bengali • Devanagari • Gujarati • Gurmukhi • Odia • Sinhala • Tamil • Malayalam • Telugu • Kannada • Dzongkha • Tibetan • Balinese • Burmese • Javanese • Khmer • Lao • Mongolian • Sundanese • Thai East Asian systems Contemporary • Chinese • Suzhou • Hokkien • Japanese • Korean • Vietnamese Historic • Counting rods • Tangut Other systems • History Ancient • Babylonian Post-classical • Cistercian • Mayan • Muisca • Pentadic • Quipu • Rumi Contemporary • Cherokee • Kaktovik (Iñupiaq) By radix/base Common radices/bases • 2 • 3 • 4 • 5 • 6 • 8 • 10 • 12 • 16 • 20 • 60 • (table) Non-standard radices/bases • Bijective (1) • Signed-digit (balanced ternary) • Mixed (factorial) • Negative • Complex (2i) • Non-integer (φ) • Asymmetric Sign-value notation Non-alphabetic • Aegean • Attic • Aztec • Brahmi • Chuvash • Egyptian • Etruscan • Kharosthi • Prehistoric counting • Proto-cuneiform • Roman • Tally marks Alphabetic • Abjad • Armenian • Alphasyllabic • Akṣarapallī • Āryabhaṭa • Kaṭapayādi • Coptic • Cyrillic • Geʽez • Georgian • Glagolitic • Greek • Hebrew List of numeral systems Today, speakers of Chinese languages use three written numeral systems: the system of Arabic numerals used worldwide, and two indigenous systems. The more familiar indigenous system is based on Chinese characters that correspond to numerals in the spoken language. These may be shared with other languages of the Chinese cultural sphere such as Korean, Japanese, and Vietnamese. Most people and institutions in China primarily use the Arabic or mixed Arabic-Chinese systems for convenience, with traditional Chinese numerals used in finance, mainly for writing amounts on cheques, banknotes, some ceremonial occasions, some boxes, and on commercials. The other indigenous system consists of the Suzhou numerals, or huama, a positional system, the only surviving form of the rod numerals. These were once used by Chinese mathematicians, and later by merchants in Chinese markets, such as those in Hong Kong until the 1990s, but were gradually supplanted by Arabic numerals. Characters used to represent numbers The Chinese character numeral system consists of the Chinese characters used by the Chinese written language to write spoken numerals. Similar to spelling-out numbers in English (e.g., "one thousand nine hundred forty-five"), it is not an independent system per se. Since it reflects spoken language, it does not use the positional system as in Arabic numerals, in the same way that spelling out numbers in English does not. Standard numbers There are characters representing the numbers zero through nine, and other characters representing larger numbers such as tens, hundreds, thousands, ten thousands and hundred millions. There are two sets of characters for Chinese numerals: one for everyday writing, known as xiǎoxiě (traditional Chinese: 小寫; simplified Chinese: 小写; lit. 'small writing'), and one for use in commercial, accounting or financial contexts, known as dàxiě (traditional Chinese: 大寫; simplified Chinese: 大写; lit. 'big writing'). The latter arose because the characters used for writing numerals are geometrically simple, so simply using those numerals cannot prevent forgeries in the same way spelling numbers out in English would.[1] A forger could easily change the everyday characters 三十 (30) to 五千 (5000) just by adding a few strokes. That would not be possible when writing using the financial characters 參拾 (30) and 伍仟 (5000). They are also referred to as "banker's numerals", "anti-fraud numerals", or "banker's anti-fraud numerals". For the same reason, rod numerals were never used in commercial records. T denotes Traditional Chinese characters, while S denotes Simplified Chinese characters. Financial Normal Value Pīnyīn (Mandarin) Jyutping (Cantonese) Pe̍h-ōe-jī (Hokkien) Wugniu (Shanghainese) Notes Character (T)Character (S)Character (T)Character (S) 零零〇0 língling4khòng/lênglin Usually 零 is preferred, but in some areas, 〇 may be a more common informal way to represent zero. The original Chinese character is 空 or 〇, 零 is referred as remainder something less than 1 yet not nil [說文] referred. The traditional 零 is more often used in schools. In Unicode, 〇 is treated as a Chinese symbol or punctuation, rather than a Chinese ideograph. 壹一1 yījat1it/chi̍tiq Also 弌 (obsolete financial), can be easily manipulated into 弍 (two) or 弎 (three). 貳贰二2 èrji6jī/nn̄ggni/er/lian Also 弍 (obsolete financial), can be easily manipulated into 弌 (one) or 弎 (three). Also 兩 (T) or 两 (S), see Characters with regional usage section. 參参三3 sānsaam1sam/saⁿsé Also 弎 (obsolete financial), which can be easily manipulated into 弌 (one) or 弍 (two). 肆四4 sìsei3sù/sìsy Also 䦉 (obsolete financial)[nb 1] 伍五5 wǔng5ngó͘/gō͘ng 陸陆六6 liùluk6lio̍k/la̍kloq 柒七7 qīcat1chhitchiq 捌八8 bābaat3pat/pehpaq 玖九9 jiǔgau2kiú/káucieu 拾十10 shísap6si̍p/cha̍pzeq Although some people use 什 as financial, it is not ideal because it can be easily manipulated into 伍 (five) or 仟 (thousand). 佰百100 bǎibaak3pek/pahpaq 仟千1,000 qiāncin1chhian/chhengchi 萬萬万104 wànmaan6bānve Chinese numbers group by ten-thousands; see Reading and transcribing numbers below. 億億亿105/108 yìjik1eki For variant meanings and words for higher values, see Large numbers below and ja:大字 (数字). Characters with regional usage Financial Normal Value Pinyin (Mandarin) Standard alternative Notes 空 0 kòng 零 Historically, the use of 空 for "zero" predates 零. This is now archaic in most varieties of Chinese, but it is still used in Southern Min. 洞 0 dòng 零 Literally means "a hole" and is analogous to the shape of "0" and "〇", it is used to unambiguously pronounce "#0" in radio communication. [2][3] 幺 1 yāo 一 Literally means "the smallest", it is used to unambiguously pronounce "#1" in radio communication. [2][3] This usage is not observed in Cantonese except for 十三幺 (a special winning hand) in Mahjong. 蜀 1 shǔ 一 In most Min varieties, there are two words meaning "one". For example, in Hokkien, chi̍t is used before a classifier: "one person" is chi̍t ê lâng, not it ê lâng. In written Hokkien, 一 is often used for both chi̍t and it, but some authors differentiate, writing 蜀 for chi̍t and 一 for it. 兩(T) or 两(S) 2 liǎng 二 Used instead of 二 before a classifier. For example, "two people" is "两个人", not "二个人". However, in some lects, such as Shanghainese, 兩 is the generic term used for two in most contexts, such as "四十兩" and not "四十二". It appears where "a pair of" would in English, but 两 is always used in such cases. It is also used for numbers, with usage varying from dialect to dialect, even person to person. For example, "2222" can be read as "二千二百二十二", "兩千二百二十二" or even "兩千兩百二十二" in Mandarin. It is used to unambiguously pronounce "#2" in radio communication. [2][3] 倆(T) or 俩(S) 2 liǎ 兩 In regional dialects of Northeastern Mandarin, 倆 represents a "lazy" pronunciation of 兩 within the local dialect. It can be used as an alternative for 兩个 "two of" (e.g. 我们倆 Wǒmen liǎ, "the two of us", as opposed to 我们兩个 Wǒmen liǎng gè). A measure word (such as 个) never follows after 倆. 仨 3 sā 三 In regional dialects of Northeastern Mandarin, 仨 represents a "lazy" pronunciation of three within the local dialect. It can be used as a general number to represent "three" (e.g.第仨号 dì sā hào, "number three"; 星期仨 xīngqīsā, "Wednesday"), or as an alternative for 三个 "three of" (e.g. 我们仨 Wǒmen sā, "the three of us", as opposed to 我们三个 Wǒmen sān gè). Regardless of usage, a measure word (such as 个) never follows after 仨. 拐 7 guǎi 七 Literally means "a turn" or "a walking stick" and is analogous to the shape of "7" and "七", it is used to unambiguously pronounce "#7" in radio communication. [2][3] 勾 9 gōu 九 Literally means "a hook" and is analogous to the shape of "9", it is used to unambiguously pronounce "#9" in radio communication. [2][3] 呀 10 yà 十 In spoken Cantonese, 呀 (aa6) can be used in place of 十 when it is used in the middle of a number, preceded by a multiplier and followed by a ones digit, e.g. 六呀三, 63; it is not used by itself to mean 10. This usage is not observed in Mandarin. 念廿 20 niàn 二十 A contraction of 二十. The written form is still used to refer to dates, especially Chinese calendar dates. Spoken form is still used in various dialects of Chinese. See Reading and transcribing numbers section below. In spoken Cantonese, 廿 (jaa6) can be used in place of 二十 when followed by another digit such as in numbers 21-29 (e.g. 廿三, 23), a measure word (e.g. 廿個), a noun, or in a phrase like 廿幾 ("twenty-something"); it is not used by itself to mean 20. 卄 is a rare variant. 卅 30 sà 三十 A contraction of 三十. The written form is still used to abbreviate date references in Chinese. For example, May 30 Movement (五卅運動). Spoken form is still used in various dialects of Chinese. In spoken Cantonese, 卅 (saa1) can be used in place of 三十 when followed by another digit such as in numbers 31–39, a measure word (e.g. 卅個), a noun, or in phrases like 卅幾 ("thirty-something"); it is not used by itself to mean 30. When spoken 卅 is pronounced as 卅呀 (saa1 aa6). Thus 卅一 (31), is pronounced as saa1 aa6 jat1. 卌 40 xì 四十 A contraction of 四十. Found in historical writings written in Classical Chinese. Spoken form is still used in various dialects of Chinese, albeit very rare. See Reading and transcribing numbers section below. In spoken Cantonese 卌 (sei3) can be used in place of 四十 when followed by another digit such as in numbers 41–49, a measure word (e.g. 卌個), a noun, or in phrases like 卌幾 ("forty-something"); it is not used by itself to mean 40. When spoken, 卌 is pronounced as 卌呀 (sei3 aa6). Thus 卌一 (41), is pronounced as sei3 aa6 jat1. 皕 200 bì 二百 Very rarely used; one example is in the name of a library in Huzhou, 皕宋樓 (Bìsòng Lóu). Large numbers For numbers larger than 10,000, similarly to the long and short scales in the West, there have been four systems in ancient and modern usage. The original one, with unique names for all powers of ten up to the 14th, is ascribed to the Yellow Emperor in the 6th century book by Zhen Luan, Wujing suanshu (Arithmetic in Five Classics). In modern Chinese only the second system is used, in which the same ancient names are used, but each represents a number 10,000 (myriad, 萬 wàn) times the previous: Character (T) 萬億兆京垓秭穰溝澗正載 Factor of increase Character (S) 万亿兆京垓秭穰沟涧正载 Pinyin wàn yì zhào jīng gāi zǐ ráng gōu jiàn zhèng zǎi Jyutping maan6 jik1 siu6 ging1 goi1 zi2 joeng4 kau1 gaan3 zing3 zoi2 Hokkien POJ bān ek tiāu keng kai chí jiông ko͘ kàn chèng cháiⁿ Shanghainese ve i zau cín ké tsy gnian kéu ké tsen tse Alternative 經/经𥝱壤 Rank 1 2 3 4 5 6 7 8 9 10 11 =n "short scale" (下數) 104 105 106 107 108 109 1010 1011 1012 1013 1014 =10n+3 Each numeral is 10 (十 shí) times the previous. "myriad scale" (萬進, current usage) 104 108 1012 1016 1020 1024 1028 1032 1036 1040 1044 =104n Each numeral is 10,000 (萬 (T) or 万 (S) wàn) times the previous. "mid-scale" (中數) 104 108 1016 1024 1032 1040 1048 1056 1064 1072 1080 =108(n-1) Starting with 亿, each numeral is 108 (萬乘以萬 (T) or 万乘以万 (S) wàn chéng yǐ wàn, 10000 times 10000) times the previous. "long scale" (上數) 104 108 1016 1032 1064 10128 10256 10512 101024 102048 104096 =102n+1 Each numeral is the square of the previous. This is similar to the -yllion system. In practice, this situation does not lead to ambiguity, with the exception of 兆 (zhào), which means 1012 according to the system in common usage throughout the Chinese communities as well as in Japan and Korea, but has also been used for 106 in recent years (especially in mainland China for megabyte). To avoid problems arising from the ambiguity, the PRC government never uses this character in official documents, but uses 万亿 (wànyì) or 太 (tài, as the translation for tera) instead. Partly due to this, combinations of 万 and 亿 are often used instead of the larger units of the traditional system as well, for example 亿亿 (yìyì) instead of 京. The ROC government in Taiwan uses 兆 (zhào) to mean 1012 in official documents. Large numbers from Buddhism Numerals beyond 載 zǎi come from Buddhist texts in Sanskrit, but are mostly found in ancient texts. Some of the following words are still being used today, but may have transferred meanings. Character (T) Character (S) Pinyin Jyutping Hokkien POJ Shanghainese Value Notes 極极 jí gik1 ke̍k jiq5 1048 Literally means "Extreme". 恆河沙恒河沙 héng hé shā hang4 ho4 sa1 hêng-hô-soa ghen3-wu-so 1052 Literally means "Sands of the Ganges"; a metaphor used in a number of Buddhist texts referring to the grains of sand in the Ganges River. 阿僧祇 ā sēng qí aa1 zang1 kei4 a-seng-kî a1-sen-ji 1056 From Sanskrit Asaṃkhyeya असंख्येय, meaning "incalculable, innumerable, infinite". 那由他 nà yóu tā naa5 jau4 taa1 ná-iû-thaⁿ na1-yeu-tha 1060 From Sanskrit nayuta नियुत, meaning "myriad". 不可思議不可思议 bùkě sīyì bat1 ho2 si1 ji3 put-khó-su-gī peq4-khu sy1-gni 1064 Literally translated as "unfathomable". This word is commonly used in Chinese as a chengyu, meaning "unimaginable", instead of its original meaning of the number 1064. 無量大數无量大数 wú liàng dà shù mou4 loeng6 daai6 sou3 bû-liōng tāi-siàu m3-lian du3-su 1068 "无量" literally translated as "without measure", and can mean 1068. This word is also commonly used in Chinese as a commendatory term, means "no upper limit". E.g.: 前途无量 lit. front journey no limit, which means "a great future". "大数" literally translated as "a large number; the great number", and can mean 1072. Small numbers The following are characters used to denote small order of magnitude in Chinese historically. With the introduction of SI units, some of them have been incorporated as SI prefixes, while the rest have fallen into disuse. Character(s) (T) Character(s) (S) Pinyin Value Notes 漠 mò 10−12 (Ancient Chinese) 皮 corresponds to the SI prefix pico-. 渺 miǎo 10−11 (Ancient Chinese) 埃 āi 10−10 (Ancient Chinese) 塵尘 chén 10−9 Literally, "Dust" 奈 (T) or 纳 (S) corresponds to the SI prefix nano-. 沙 shā 10−8 Literally, "Sand" 纖纤 xiān 10−7 Literally, "Fiber" 微 wēi 10−6 still in use, corresponds to the SI prefix micro-. 忽 hū 10−5 (Ancient Chinese) 絲丝 sī 10−4 also 秒. Literally, "Thread" 毫 háo 10−3 also 毛. still in use, corresponds to the SI prefix milli-. 厘 lí 10−2 also 釐. still in use, corresponds to the SI prefix centi-. 分 fēn 10−1 still in use, corresponds to the SI prefix deci-. Small numbers from Buddhism Character(s) (T) Character(s) (S) Pinyin Value Notes 涅槃寂靜涅槃寂静 niè pán jì jìng 10−24 Literally, "Nirvana's Tranquility" 攸 (T) or 幺 (S) corresponds to the SI prefix yocto-. 阿摩羅阿摩罗 ā mó luó 10−23 (Ancient Chinese, from Sanskrit अमल amala) 阿頼耶阿赖耶 ā lài yē 10−22 (Ancient Chinese, from Sanskrit आलय ālaya) 清靜清净 qīng jìng 10−21 Literally, "Quiet" 介 (T) or 仄 (S) corresponds to the SI prefix zepto-. 虛空虚空 xū kōng 10−20 Literally, "Void" 六德 liù dé 10−19 (Ancient Chinese) 剎那刹那 chà nà 10−18 Literally, "Brevity", from Sanskrit क्षण ksaṇa 阿 corresponds to the SI prefix atto-. 彈指弹指 tán zhǐ 10−17 Literally, "Flick of a finger". Still commonly used in the phrase "弹指一瞬间" (A very short time) 瞬息 shùn xī 10−16 Literally, "Moment of Breath". Still commonly used in Chengyu "瞬息万变" (Many things changed in a very short time) 須臾须臾 xū yú 10−15 (Ancient Chinese, rarely used in Modern Chinese as "a very short time") 飛 (T) or 飞 (S) corresponds to the SI prefix femto-. 逡巡 qūn xún 10−14 (Ancient Chinese) 模糊 mó hu 10−13 Literally, "Blurred" SI prefixes In the People's Republic of China, the early translation for the SI prefixes in 1981 was different from those used today. The larger (兆, 京, 垓, 秭, 穰) and smaller Chinese numerals (微, 纖, 沙, 塵, 渺) were defined as translation for the SI prefixes as mega, giga, tera, peta, exa, micro, nano, pico, femto, atto, resulting in the creation of yet more values for each numeral.[4] The Republic of China (Taiwan) defined 百萬 as the translation for mega and 兆 as the translation for tera. This translation is widely used in official documents, academic communities, informational industries, etc. However, the civil broadcasting industries sometimes use 兆赫 to represent "megahertz". Today, the governments of both China and Taiwan use phonetic transliterations for the SI prefixes. However, the governments have each chosen different Chinese characters for certain prefixes. The following table lists the two different standards together with the early translation. SI Prefixes Value Symbol English Early translation PRC standard ROC standard 1024Yyotta- 尧 yáo 佑 yòu 1021Zzetta- 泽 zé 皆 jiē 1018Eexa- 穰[4] ráng艾 ài 艾 ài 1015Ppeta- 秭[4] zǐ拍 pāi 拍 pāi 1012Ttera- 垓[4] gāi太 tài 兆 zhào 109Ggiga- 京[4] jīng吉 jí 吉 jí 106Mmega- 兆[4] zhào兆 zhào 百萬 bǎiwàn 103kkilo- 千 qiān千 qiān 千 qiān 102hhecto- 百 bǎi百 bǎi百 bǎi 101dadeca- 十 shí十 shí 十 shí 100(base)one 一 yī 一 yī 10−1ddeci- 分 fēn分 fēn 分 fēn 10−2ccenti- 厘 lí厘 lí 厘 lí 10−3mmilli- 毫 háo毫 háo 毫 háo 10−6µmicro- 微[4] wēi 微 wēi 微 wēi 10−9nnano- 纖[4] xiān 纳 nà 奈 nài 10−12ppico- 沙[4] shā皮 pí 皮 pí 10−15ffemto- 塵[4] chén飞 fēi 飛 fēi 10−18aatto- 渺[4] miǎo 阿 à 阿 à 10−21zzepto- 仄 zè 介 jiè 10−24yyocto- 幺 yāo 攸 yōu Reading and transcribing numbers Whole numbers Multiple-digit numbers are constructed using a multiplicative principle; first the digit itself (from 1 to 9), then the place (such as 10 or 100); then the next digit. In Mandarin, the multiplier 兩 (liǎng) is often used rather than 二 (èr) for all numbers 200 and greater with the "2" numeral (although as noted earlier this varies from dialect to dialect and person to person). Use of both 兩 (liǎng) or 二 (èr) are acceptable for the number 200. When writing in the Cantonese dialect, 二 (yi6) is used to represent the "2" numeral for all numbers. In the southern Min dialect of Chaozhou (Teochew), 兩 (no6) is used to represent the "2" numeral in all numbers from 200 onwards. Thus: Number Structure Characters Mandarin Cantonese Chaozhou Shanghainese 60[6] [10]六十六十六十六十 20[2] [10] or [20]二十二十 or 廿二十廿 200[2] (èr or liǎng) [100]二百 or 兩百二百 or 兩百兩百兩百 2000[2] (èr or liǎng) [1000]二千 or 兩千二千 or 兩千兩千兩千 45[4] [10] [5]四十五四十五 or 卌五四十五四十五 2,362[2] [1000] [3] [100] [6] [10] [2]兩千三百六十二二千三百六十二兩千三百六十二兩千三百六十二 For the numbers 11 through 19, the leading "one" (一; yī) is usually omitted. In some dialects, like Shanghainese, when there are only two significant digits in the number, the leading "one" and the trailing zeroes are omitted. Sometimes, the one before "ten" in the middle of a number, such as 213, is omitted. Thus: Number Strict Putonghua Colloquial or dialect usage Structure Characters Structure Characters 14[10] [4]十四 12000[1] [10000] [2] [1000]一萬兩千[1] [10000] [2]一萬二 or 萬二 114[1] [100] [1] [10] [4]一百一十四[1] [100] [10] [4]一百十四 1158[1] [1000] [1] [100] [5] [10] [8]一千一百五十八See note 1 below Notes: 1. Nothing is ever omitted in large and more complicated numbers such as this. In certain older texts like the Protestant Bible or in poetic usage, numbers such as 114 may be written as [100] [10] [4] (百十四). Outside of Taiwan, digits are sometimes grouped by myriads instead of thousands. Hence it is more convenient to think of numbers here as in groups of four, thus 1,234,567,890 is regrouped here as 12,3456,7890. Larger than a myriad, each number is therefore four zeroes longer than the one before it, thus 10000 × wàn (萬) = yì (億). If one of the numbers is between 10 and 19, the leading "one" is omitted as per the above point. Hence (numbers in parentheses indicate that the number has been written as one number rather than expanded): Number Structure Taiwan Mainland China 12,345,678,902,345 (12,3456,7890,2345) (12) [1,0000,0000,0000] (3456) [1,0000,0000] (7890) [1,0000] (2345)十二兆三千四百五十六億七千八百九十萬兩千三百四十五十二兆三千四百五十六亿七千八百九十万二千三百四十五 In Taiwan, pure Arabic numerals are officially always and only grouped by thousands.[5] Unofficially, they are often not grouped, particularly for numbers below 100,000. Mixed Arabic-Chinese numerals are often used in order to denote myriads. This is used both officially and unofficially, and come in a variety of styles: Number Structure Mixed numerals 12,345,000(1234) [1,0000] (5) [1,000]1,234萬5千[6] 123,450,000 (1) [1,0000,0000] (2345) [1,0000] 1億2345萬[7] 12,345 (1) [1,0000] (2345) 1萬2345[8] Interior zeroes before the unit position (as in 1002) must be spelt explicitly. The reason for this is that trailing zeroes (as in 1200) are often omitted as shorthand, so ambiguity occurs. One zero is sufficient to resolve the ambiguity. Where the zero is before a digit other than the units digit, the explicit zero is not ambiguous and is therefore optional, but preferred. Thus: Number Structure Characters 205[2] [100] [0] [5]二百零五 100,004 (10,0004) [10] [10,000] [0] [4]十萬零四 10,050,026 (1005,0026) (1005) [10,000] (026) or (1005) [10,000] (26) 一千零五萬零二十六 or 一千零五萬二十六 Fractional values To construct a fraction, the denominator is written first, followed by 分; fēn; 'parts', then the literary possessive particle 之; zhī; 'of this', and lastly the numerator. This is the opposite of how fractions are read in English, which is numerator first. Each half of the fraction is written the same as a whole number. For example, to express "two thirds", the structure "three parts of-this two" is used. Mixed numbers are written with the whole-number part first, followed by 又; yòu; 'and', then the fractional part. Fraction Structure 2⁄3 三 sān 3 分 fēn parts 之 zhī of this 二 èr 2 三分之二 sān fēn zhī èr 3 parts {of this} 2 15⁄32 三 sān 3 十 shí 10 二 èr 2 分 fēn parts 之 zhī of this 十 shí 10 五 wǔ 5 三十二分之十五 sān shí èr fēn zhī shí wǔ 3 10 2 parts {of this} 10 5 1⁄3000 三 sān 3 千 qiān 1000 分 fēn parts 之 zhī of this 一 yī 1 三千分之一 sān qiān fēn zhī yī 3 1000 parts {of this} 1 3 5⁄6 三 sān 3 又 yòu and 六 liù 6 分 fēn parts 之 zhī of this 五 wǔ 5 三又六分之五 sān yòu liù fēn zhī wǔ 3 and 6 parts {of this} 5 Percentages are constructed similarly, using 百; bǎi; '100' as the denominator. (The number 100 is typically expressed as 一百; yībǎi; 'one hundred', like the English "one hundred". However, for percentages, 百 is used on its own.) Percentage Structure 25% 百 bǎi 100 分 fēn parts 之 zhī of this 二 èr 2 十 shí 10 五 wǔ 5 百分之二十五 bǎi fēn zhī èr shí wǔ 100 parts {of this} 2 10 5 110% 百 bǎi 100 分 fēn parts 之 zhī of this 一 yī 1 百 bǎi 100 一 yī 1 十 shí 10 百分之一百一十 bǎi fēn zhī yī bǎi yī shí 100 parts {of this} 1 100 1 10 Because percentages and other fractions are formulated the same, Chinese are more likely than not to express 10%, 20% etc. as "parts of 10" (or 1/10, 2/10, etc. i.e. 十分之一; shí fēnzhī yī, 十分之二; shí fēnzhī èr, etc.) rather than "parts of 100" (or 10/100, 20/100, etc. i.e. 百分之十; bǎi fēnzhī shí, 百分之二十; bǎi fēnzhī èrshí, etc.) In Taiwan, the most common formation of percentages in the spoken language is the number per hundred followed by the word 趴; pā, a contraction of the Japanese パーセント; pāsento, itself taken from the English "percent". Thus 25% is 二十五趴; èrshíwǔ pā.[nb 2] Decimal numbers are constructed by first writing the whole number part, then inserting a point (simplified Chinese: 点; traditional Chinese: 點; pinyin: diǎn), and finally the fractional part. The fractional part is expressed using only the numbers for 0 to 9, similarly to English. Decimal expression Structure 16.98 十 shí 10 六 liù 6 點 diǎn point 九 jiǔ 9 八 bā 8 十六點九八 shí liù diǎn jiǔ bā 10 6 point 9 8 12345.6789 一 yī 1 萬 wàn 10000 兩 liǎng 2 千 qiān 1000 三 sān 3 百 bǎi 100 四 sì 4 十 shí 10 五 wǔ 5 點 diǎn point 六 liù 6 七 qī 7 八 bā 8 九 jiǔ 9 一萬兩千三百四十五點六七八九 yī wàn liǎng qiān sān bǎi sì shí wǔ diǎn liù qī bā jiǔ 1 10000 2 1000 3 100 4 10 5 point 6 7 8 9 75.4025 七七 qī 7 十十 shí 10 五五 wǔ 5 點點 diǎn point 四四 sì 4 〇零 líng 0 二二 èr 2 五五 wǔ 5 七十五點四〇二五七十五點四零二五 qī shí wǔ diǎn sì líng èr wǔ 7 10 5 point 4 0 2 5 0.1 零 líng 0 點 diǎn point 一 yī 1 零點一 líng diǎn yī 0 point 1 半; bàn; 'half' functions as a number and therefore requires a measure word. For example: 半杯水; bàn bēi shuǐ; 'half a glass of water'. Ordinal numbers Ordinal numbers are formed by adding 第; dì ("sequence") before the number. Ordinal Structure 1st 第 dì sequence 一 yī 1 第一 dì yī sequence 1 2nd 第 dì sequence 二 èr 2 第二 dì èr sequence 2 82nd 第 dì sequence 八 bā 8 十 shí 10 二 èr 2 第八十二 dì bā shí èr sequence 8 10 2 The Heavenly Stems are a traditional Chinese ordinal system. Negative numbers Negative numbers are formed by adding fù (负; 負) before the number. Number Structure −1158 負 fù negative 一 yī 1 千 qiān 1000 一 yī 1 百 bǎi 100 五 wǔ 5 十 shí 10 八 bā 8 負一千一百五十八 fù yī qiān yī bǎi wǔ shí bā negative 1 1000 1 100 5 10 8 −3 5/6 負 fù negative 三 sān 3 又 yòu and 六 liù 6 分 fēn parts 之 zhī of this 五 wǔ 5 負三又六分之五 fù sān yòu liù fēn zhī wǔ negative 3 and 6 parts {of this} 5 −75.4025 負 fù negative 七 qī 7 十 shí 10 五 wǔ 5 點 diǎn point 四 sì 4 零 líng 0 二 èr 2 五 wǔ 5 負七十五點四零二五 fù qī shí wǔ diǎn sì líng èr wǔ negative 7 10 5 point 4 0 2 5 Usage Chinese grammar requires the use of classifiers (measure words) when a numeral is used together with a noun to express a quantity. For example, "three people" is expressed as 三个人; 三個人; sān ge rén , "three (ge particle) person", where 个/個 ge is a classifier. There exist many different classifiers, for use with different sets of nouns, although 个/個 is the most common, and may be used informally in place of other classifiers. Chinese uses cardinal numbers in certain situations in which English would use ordinals. For example, 三楼/三樓; sān lóu (literally "three story/storey") means "third floor" ("second floor" in British § Numbering). Likewise, 二十一世纪/二十一世紀; èrshí yī shìjì (literally "twenty-one century") is used for "21st century".[9] Numbers of years are commonly spoken as a sequence of digits, as in 二零零一; èr líng líng yī ("two zero zero one") for the year 2001.[10] Names of months and days (in the Western system) are also expressed using numbers: 一月; yīyuè ("one month") for January, etc.; and 星期一; xīngqīyī ("week one") for Monday, etc. There is only one exception: Sunday is 星期日; xīngqīrì, or informally 星期天; xīngqītiān, both literally "week day". When meaning "week", "星期" xīngqī and "禮拜; 礼拜" lǐbài are interchangeable. "禮拜天" lǐbàitiān or "禮拜日" lǐbàirì means "day of worship". Chinese Catholics call Sunday "主日" zhǔrì, "Lord's day".[11] Full dates are usually written in the format 2001年1月20日 for January 20, 2001 (using 年; nián "year", 月; yuè "month", and 日; rì "day") – all the numbers are read as cardinals, not ordinals, with no leading zeroes, and the year is read as a sequence of digits. For brevity the nián, yuè and rì may be dropped to give a date composed of just numbers. For example "6-4" in Chinese is "six-four", short for "month six, day four" i.e. June Fourth, a common Chinese shorthand for the 1989 Tiananmen Square protests (because of the violence that occurred on June 4). For another example 67, in Chinese is sixty seven, short for year nineteen sixty seven, a common Chinese shorthand for the Hong Kong 1967 leftist riots. Counting rod and Suzhou numerals In the same way that Roman numerals were standard in ancient and medieval Europe for mathematics and commerce, the Chinese formerly used the rod numerals, which is a positional system. The Suzhou numerals (simplified Chinese: 苏州花码; traditional Chinese: 蘇州花碼; pinyin: Sūzhōu huāmǎ) system is a variation of the Southern Song rod numerals. Nowadays, the huāmǎ system is only used for displaying prices in Chinese markets or on traditional handwritten invoices. Hand gestures There is a common method of using of one hand to signify the numbers one to ten. While the five digits on one hand can easily express the numbers one to five, six to ten have special signs that can be used in commerce or day-to-day communication. Historical use of numerals in China Most Chinese numerals of later periods were descendants of the Shang dynasty oracle numerals of the 14th century BC. The oracle bone script numerals were found on tortoise shell and animal bones. In early civilizations, the Shang were able to express any numbers, however large, with only nine symbols and a counting board though it was still not positional .[13] Some of the bronze script numerals such as 1, 2, 3, 4, 10, 11, 12, and 13 became part of the system of rod numerals. In this system, horizontal rod numbers are used for the tens, thousands, hundred thousands etc. It is written in Sunzi Suanjing that "one is vertical, ten is horizontal".[14] 七一八二四 7 1 8 2 4 The counting rod numerals system has place value and decimal numerals for computation, and was used widely by Chinese merchants, mathematicians and astronomers from the Han dynasty to the 16th century. In 690 AD, Empress Wǔ promulgated Zetian characters, one of which was "〇". The word is now used as a synonym for the number zero.[nb 3] Alexander Wylie, Christian missionary to China, in 1853 already refuted the notion that "the Chinese numbers were written in words at length", and stated that in ancient China, calculation was carried out by means of counting rods, and "the written character is evidently a rude presentation of these". After being introduced to the rod numerals, he said "Having thus obtained a simple but effective system of figures, we find the Chinese in actual use of a method of notation depending on the theory of local value [i.e. place-value], several centuries before such theory was understood in Europe, and while yet the science of numbers had scarcely dawned among the Arabs."[15] During the Ming and Qing dynasties (after Arabic numerals were introduced into China), some Chinese mathematicians used Chinese numeral characters as positional system digits. After the Qing period, both the Chinese numeral characters and the Suzhou numerals were replaced by Arabic numerals in mathematical writings. Cultural influences Traditional Chinese numeric characters are also used in Japan and Korea and were used in Vietnam before the 20th century. In vertical text (that is, read top to bottom), using characters for numbers is the norm, while in horizontal text, Arabic numerals are most common. Chinese numeric characters are also used in much the same formal or decorative fashion that Roman numerals are in Western cultures. Chinese numerals may appear together with Arabic numbers on the same sign or document. See also Wikimedia Commons has media related to Chinese numerals. • Chinese number gestures • Numbers in Chinese culture • Chinese units of measurement • Chinese classifier • Chinese grammar • Japanese numerals • Korean numerals • Vietnamese numerals • Celestial stem • List of numbers in Sinitic languages Notes 1. Variant Chinese character of 肆, with a 镸 radical next to a 四 character. Not all browsers may be able to display this character, which forms a part of the Unicode CJK Unified Ideographs Extension A group. 2. This usage can also be found in written sources, such as in the headline of this article (while the text uses "%") and throughout this article. 3. The code for the lowercase 〇 (IDEOGRAPHIC NUMBER ZERO) is U+3007, not to be confused with the O mark (CIRCLE). References 1. 大寫數字『 Archived 2011-07-22 at the Wayback Machine 2. Li, Suming (18 March 2016). Qiao, Meng (ed.). ""军语"里的那些秘密武警少将亲自为您揭开" [Secrets in the "Military Lingo", Reveled by PAP General]. People's Armed Police. Retrieved 2021-06-18. 3. 飛航管理程序 [Air Traffic Management Procedures] (14 ed.). 30 November 2015. 4. (in Chinese) 1981 Gazette of the State Council of the People's Republic of China Archived 2012-01-11 at the Wayback Machine, No. 365 Archived 2014-11-04 at the Wayback Machine, page 575, Table 7: SI prefixes 5. 中華民國統計資訊網（專業人士）. 中華民國統計資訊網 (in Chinese). Archived from the original on 5 August 2016. Retrieved 31 July 2016. 6. 中華民國統計資訊網（專業人士） (in Chinese). 中華民國統計資訊網. Archived from the original on 28 August 2016. Retrieved 31 July 2016. 7. "石化氣爆高市府代位求償訴訟中". 中央社即時新聞 CNA NEWS. 中央社即時新聞 CNA NEWS. Archived from the original on 1 August 2016. Retrieved 31 July 2016. 8. "陳子豪雙響砲兄弟連2天轟猿動紫趴". 中央社即時新聞 CNA NEWS. 中央社即時新聞 CNA NEWS. Archived from the original on 31 July 2016. Retrieved 31 July 2016. 9. Yip, Po-Ching; Rimmington, Don, Chinese: A Comprehensive Grammar, Routledge, 2004, p. 12. 10. Yip, Po-Ching; Rimmington, Don, Chinese: A Comprehensive Grammar, Routledge, 2004, p. 13. 11. Days of the Week in Chinese: Three Different Words for 'Week' http://www.cjvlang.com/Dow/dowchin.html Archived 2016-03-06 at the Wayback Machine 12. The Shorter Science & Civilisation in China Vol 2, An abridgement by Colin Ronan of Joseph Needham's original text, Table 20, p. 6, Cambridge University Press ISBN 0-521-23582-0 13. The Shorter Science & Civilisation in China Vol 2, An abridgement by Colin Ronan of Joseph Needham's original text, p5, Cambridge University Press ISBN 0-521-23582-0 14. Chinese Wikisource Archived 2012-02-22 at the Wayback Machine 孫子算經: 先識其位，一從十橫，百立千僵，千十相望，萬百相當。 15. 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Toshikazu Sunada Toshikazu Sunada (砂田利一, Sunada Toshikazu, born September 7, 1948) is a Japanese mathematician and author of many books and essays on mathematics and mathematical sciences. He is professor emeritus of both Meiji University and Tohoku University. He is also distinguished professor of emeritus at Meiji in recognition of achievement over the course of an academic career. Before he joined Meiji University in 2003, he was professor of mathematics at Nagoya University (1988–1991), at the University of Tokyo (1991–1993), and at Tohoku University (1993–2003). Sunada was involved in the creation of the School of Interdisciplinary Mathematical Sciences at Meiji University and is its first dean (2013–2017). Since 2019, he is President of Mathematics Education Society of Japan. Toshikazu Sunada Born1948 (age 74–75) Tokyo, Japan Alma materTokyo Institute of Technology Awards • 1987 Iyanaga Award of Mathematical Society of Japan • 2013 Publication Prize of Mathematical Society of Japan • 2017 Hiroshi Fujiwara Prize for Mathematical Sciences • 2018 Prize for Science and Technology (the Commendation for Science and Technology by the Minister of Education, Culture, Sports, Science and Technology) • 2019 the 1st Kodaira Kunihiko Prize Scientific career FieldsMathematics (Spectral geometry and discrete geometric analysis) InstitutionsNagoya University Tokyo University Tohoku University Meiji University Main work Sunada's work covers complex analytic geometry, spectral geometry, dynamical systems, probability, graph theory, discrete geometric analysis, and mathematical crystallography. Among his numerous contributions, the most famous one is a general construction of isospectral manifolds (1985), which is based on his geometric model of number theory, and is considered to be a breakthrough in the problem proposed by Mark Kac in "Can one hear the shape of a drum?" (see Hearing the shape of a drum). Sunada's idea was taken up by Carolyn S. Gordon, David Webb, and Scott A. Wolpert when they constructed a counterexample for Kac's problem. For this work, Sunada was awarded the Iyanaga Prize of the Mathematical Society of Japan (MSJ) in 1987. He was also awarded Publication Prize of MSJ in 2013, the Hiroshi Fujiwara Prize for Mathematical Sciences in 2017, the Prize for Science and Technology (the Commendation for Science and Technology by the Minister of Education, Culture, Sports, Science and Technology) in 2018, and the 1st Kodaira Kunihiko Prize in 2019. In a joint work with Atsushi Katsuda, Sunada also established a geometric analogue of Dirichlet's theorem on arithmetic progressions in the context of dynamical systems (1988). One can see, in this work as well as the one above, how the concepts and ideas in totally different fields (geometry, dynamical systems, and number theory) are put together to formulate problems and to produce new results. His study of discrete geometric analysis includes a graph-theoretic interpretation of Ihara zeta functions, a discrete analogue of periodic magnetic Schrödinger operators as well as the large time asymptotic behaviors of random walk on crystal lattices. The study of random walk led him to the discovery of a "mathematical twin" of the diamond crystal out of an infinite universe of hypothetical crystals (2005). He named it the K4 crystal due to its mathematical relevance (see the linked article). What was noticed by him is that the K4 crystal has the "strong isotropy property", meaning that for any two vertices x and y of the crystal net, and for any ordering of the edges adjacent to x and any ordering of the edges adjacent to y, there is a net-preserving congruence taking x to y and each x-edge to the similarly ordered y-edge. This property is shared only by the diamond crystal (the strong isotropy should not be confused with the edge-transitivity or the notion of symmetric graph; for instance, the primitive cubic lattice is a symmetric graph, but not strongly isotropic). The K4 crystal and the diamond crystal as networks in space are examples of “standard realizations”, the notion introduced by Sunada and Motoko Kotani as a graph-theoretic version of Albanese maps (Abel-Jacobi maps) in algebraic geometry. For his work, see also Isospectral, Reinhardt domain, Ihara zeta function, Ramanujan graph, quantum ergodicity, quantum walk. Selected publications by Sunada • T. Sunada, Holomorphic equivalence problem for bounded Reinhardt domains, Mathematische Annalen 235 (1978), 111–128 • T. Sunada, Rigidity of certain harmonic mappings, Inventiones Mathematicae 51 (1979), 297–307 • J. Noguchi and T. Sunada, Finiteness of the family of rational and meromorphic mappings into algebraic varieties, American Journal of Mathematics 104 (1982), 887–900 • T. Sunada, Riemannian coverings and isospectral manifolds, Annals of Mathematics 121 (1985), 169–186 • T. Sunada, L-functions and some applications, Lecture Notes in Mathematics 1201 (1986), Springer-Verlag, 266–284 • A. Katsuda and T. Sunada, Homology and closed geodesics in a compact Riemann surface, American Journal of Mathematics 110(1988), 145–156 • T. Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, Topology 28 (1989), 125–132 • A. Katsuda and T. Sunada, Closed orbits in homology classes, Publications Mathématiques de l'IHÉS 71 (1990), 5–32 • M. Nishio and T. Sunada, Trace formulae in spectral geometry, Proc. ICM-90 Kyoto, Springer-Verlag, Tokyo, (1991), 577–585 • T. Sunada, Quantum ergodicity, Trend in Mathematics, Birkhauser Verlag, Basel, 1997, 175–196 • M. Kotani and T. Sunada, Albanese maps and an off diagonal long time asymptotic for the heat kernel, Communications in Mathematical Physics 209 (2000), 633–670 • M. Kotani and T. Sunada, Spectral geometry of crystal lattices, Contemporary Mathematics 338 (2003), 271–305 • T. Sunada, Crystals that nature might miss creating, Notices of the American Mathematical Society 55 (2008), 208–215 • T. Sunada, Discrete geometric analysis, Proceedings of Symposia in Pure Mathematics (ed. by P. Exner, J. P. Keating, P. Kuchment, T. Sunada, A. Teplyaev), 77 (2008), 51–86 • K. Shiga and T. Sunada, A Mathematical Gift, III, American Mathematical Society • T. Sunada, Lecture on topological crystallography, Japan Journal of Mathematics 7 (2012), 1–39 • T. Sunada, Topological Crystallography, With a View Towards Discrete Geometric Analysis, Springer, 2013, ISBN 978-4-431-54176-9 (print) ISBN 978-4-431-54177-6 (online) • T. Sunada, Generalized Riemann sums, in From Riemann to Differential Geometry and Relativity, Editors: Lizhen Ji, Athanase Papadopoulos, Sumio Yamada, Springer (2017), 457–479 • T. Sunada, Topics on mathematical crystallography, Proceedings of the symposium Groups, graphs and random walks, London Mathematical Society Lecture Note Series 436, Cambridge University Press, 2017, 473–513 • T. Sunada, From Euclid to Riemann and beyond, in Geometry in History, Editors: S. G. Dani, Athanase Papadopoulos, Springer (2019), 213–304 References • Atsushi Katsuda and Polly Wee Sy,, An overview of Sunada's work • Meiji U. Homepage (Mathematics Department) • David Bradley, , Diamond's chiral chemical cousin • M. Itoh et al., New metallic carbon crystal, Phys. Rev. Lett. 102, 055703 (2009) • Diamond twin, Meiji U. 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Shisanji Hokari Dr. Shisanji Hokari (穂刈四三二, Hokari Shisanji, 28 March 1908 – 2 January 2004) was a Japanese mathematician. He was admitted to the American Mathematical Society in 1966.[1] He was a professor emeritus of Tokyo Metropolitan University and the president of Josai University. Year Age Milestone 1926 18 Enrolled to Tokyo University of Science. 1928 20 Graduated from Tokyo University of Science. 1931 23 Enrolled to Hokkaido University. 1934 26 Graduated from Hokkaido University. 1939 31 Lecturer at Hokkaido University. 1940 32 Received a doctorate. Assistant Professor at Hokkaido University. 1949 41 Professor at Tokyo Metropolitan University. 1971 63 Professor at Josai University. Dean of Faculty of Science. 1971 63 Professor Emeritus at Tokyo Metropolitan University. 1977 69 Honorary Member of Japan Society of Mathematical Education. 1978 70 President of Josai University. 1980 72 President Emeritus of Josai University. 1982 74 Professor Emeritus at Josai University. 1987 79 Received 3rd Class Order of the Rising Sun. References 1. Green, John W.; Sherman, Seymour. The annual meeting in Chicago. Bull. Amer. Math. Soc. 72 (1966), no. 3, p. 476 http://projecteuclid.org/euclid.bams/1183527951. Authority control International • ISNI • VIAF National • United States • Japan Academics • CiNii • zbMATH
1 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer.[1] It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. ← 0 1 2 → −1 0 1 2 3 4 5 6 7 8 9 → • List of numbers • Integers ← 0 10 20 30 40 50 60 70 80 90 → Cardinalone Ordinal1st (first) Numeral systemunary Factorization∅ Divisors1 Greek numeralΑ´ Roman numeralI, i Greek prefixmono-/haplo- Latin prefixuni- Binary12 Ternary13 Senary16 Octal18 Duodecimal112 Hexadecimal116 Greek numeralα' Arabic, Kurdish, Persian, Sindhi, Urdu١ Assamese & Bengali১ Chinese numeral一/弌/壹 Devanāgarī१ Ge'ez፩ GeorgianႠ/ⴀ/ა(Ani) Hebrewא Japanese numeral一/壱 Kannada೧ Khmer១ Malayalam൧ Meitei꯱ Thai๑ Tamil௧ Telugu೧ Counting rod𝍠 The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is the smallest possible difference between two distinct natural numbers. The unique mathematical properties of the number have led to its unique uses in other fields, ranging from science to sports. It commonly denotes the first, leading, or top thing in a group. As a word One is most commonly a determiner used with singular countable nouns, as in one day at a time.[2] One is also a pronoun used to refer to an unspecified person or to people in general as in one should take care of oneself.[3] Finally, one is a noun when it refers to the number one as in one plus one is two and when it is used as a pro form, as in the green one is nice or those ones look good. Etymology One comes from the English word an,[4] which comes from the Proto-Germanic root ainaz.[4] The Proto-Germanic root ainaz comes from the Proto-Indo-European root oi-no-.[4] Compare the Proto-Germanic root ainaz to Old Frisian an, Gothic ains, Danish en, Dutch een, German eins and Old Norse einn. Compare the Proto-Indo-European root *oi-no- (which means "one, single")[4] to Greek oinos (which means "ace" on dice),[4] Latin unus (one),[4] Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin and Breton un (one).[4] As a number One, sometimes referred to as unity,[5][1] is the first non-zero natural number. It is thus the integer after zero. Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, its own square and square root, its own cube and cube root, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but is instead considered a unit (meaning of ring theory). As a digit Main article: History of the Hindu–Arabic numeral system The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmic script of ancient India, where it was a simple vertical line. It was transmitted to Europe via the Maghreb and Andalusia during the Middle Ages, through scholarly works written in Arabic. In some countries, the serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph used for seven in other countries. In styles in which the digit 1 is written with a long upstroke, the digit 7 is often written with a horizontal stroke through the vertical line, to disambiguate them. Styles that do not use the long upstroke on digit 1 usually do not use the horizontal stroke through the vertical of the digit 7 either. While the shape of the character for the digit 1 has an ascender in most modern typefaces, in typefaces with text figures, the glyph usually is of x-height, as, for example, in . Many older typewriters lack a separate key for 1, using the lowercase letter l or uppercase I instead. It is possible to find cases when the uppercase J is used, though it may be for decorative purposes. In some typefaces, different glyphs are used for I and 1, but the numeral 1 resembles a small caps version of I, with parallel serifs at top and bottom, with the capital I being full-height. Mathematics Definitions Mathematically, 1 is: • in arithmetic (algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers; • more generally, in algebra, the multiplicative identity (also called unity), usually of a group or a ring. Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the successor of 0. In Principia Mathematica, it is defined as the set of all singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the set {0}. In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e (from the German Einheit, "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields. By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). The term unit matrix is sometimes used to mean a matrix composed entirely of 1s. By definition, 1 is the probability of an event that is absolutely or almost certain to occur. In category theory, 1 is sometimes used to denote the terminal object of a category. In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899. Properties Tallying is often referred to as "base 1", since only one mark – the tally itself – is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a positional notation. Since the base 1 exponential function (1x) always equals 1, its inverse does not exist (which would be called the logarithm base 1 if it did exist). In many mathematical and engineering problems, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions, too, are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application. Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must be equal to 1. There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999.... 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few. 1 is also the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences. The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all. 1 is the most common leading digit in many sets of data, a consequence of Benford's law. 1 is the only known Tamagawa number for a simply connected algebraic group over a number field. The generating function that has all coefficients equal to 1 is a geometric series, given by ${\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\ldots $ The zeroth metallic mean is 1, with the golden section equal to the continued fraction [1;1,1,...], and the infinitely nested square root $\scriptstyle {\sqrt {1+{\sqrt {{\text{ }}1+\cdots {\text{ }}}}}}.$ The series of unit fractions that most rapidly converge to 1 are the reciprocals of Sylvester's sequence, which generate the infinite Egyptian fraction $1={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{43}}+\cdots $ Primality Main article: Prime number § Primality of one 1 is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) like −1 and, in the Gaussian integers, i and −i. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units. For example, 4 = 22, but if units are included, is also equal to, say, (−1)6 × 123 × 22, among infinitely many similar "factorizations". 1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1). As such, some mathematicians considered it a prime number as late as the middle of the 20th century, but mathematical consensus has generally and since then universally been to exclude it for a variety of reasons (such as complicating the fundamental theorem of arithmetic and other theorems related to prime numbers). 1 is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers. Table of basic calculations Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000 1 × x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000 Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 ÷ x 1 0.5 0.3 0.25 0.2 0.16 0.142857 0.125 0.1 0.1 0.09 0.083 0.076923 0.0714285 0.06 x ÷ 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1x 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 In technology • The resin identification code used in recycling to identify polyethylene terephthalate.[6] • The ITU country code for the North American Numbering Plan area, which includes the United States, Canada, and parts of the Caribbean. • A binary code is a sequence of 1 and 0 that is used in computers for representing any kind of data. • In many physical devices, 1 represents the value for "on", which means that electricity is flowing.[7][8] • The numerical value of true in many programming languages. • 1 is the ASCII code of "Start of Header". In science • Dimensionless quantities are also known as quantities of dimension one. • 1 is the atomic number of hydrogen. • +1 is the electric charge of positrons and protons. • Group 1 of the periodic table consists of the alkali metals. • Period 1 of the periodic table consists of just two elements, hydrogen and helium. • The dwarf planet Ceres has the minor-planet designation 1 Ceres because it was the first asteroid to be discovered. • The Roman numeral I often stands for the first-discovered satellite of a planet or minor planet (such as Neptune I, a.k.a. Triton). For some earlier discoveries, the Roman numerals originally reflected the increasing distance from the primary instead. In philosophy In the philosophy of Plotinus (and that of other neoplatonists), The One is the ultimate reality and source of all existence.[9] Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers ("De Allegoriis Legum," ii.12 [i.66]). The Neopythagorean philosopher Nicomachus of Gerasa affirmed that one is not a number, but the source of number. He also believed the number two is the embodiment of the origin of otherness. His number theory was recovered by Boethius in his Latin translation of Nicomachus's treatise Introduction to Arithmetic.[10] In sports In many professional sports, the number 1 is assigned to the player who is first or leading in some respect, or otherwise important; the number is printed on his or her sports uniform or equipment. This is the pitcher in baseball, the goalkeeper in association football (soccer), the starting fullback in most of rugby league, the starting loosehead prop in rugby union and the previous year's world champion in Formula One. 1 may be the lowest possible player number, like in the American–Canadian National Hockey League (NHL) since the 1990s or in American football. In other fields • Number One is Royal Navy informal usage for the chief executive officer of a ship, the captain's deputy responsible for discipline and all normal operation of a ship and its crew. • 1 is the value of an ace in many playing card games, such as cribbage. • List of highways numbered 1 • List of public transport routes numbered 1 • 1 is often used to denote the Gregorian calendar month of January. • 1 CE, the first year of the Common Era • 01, the former dialling code for Greater London (now 020) • For Pythagorean numerology (a pseudoscience), the number 1 is the number that means beginning, new beginnings, new cycles, it is a unique and absolute number. • PRS One, a German paraglider design • In some countries, a street address of "1" is considered prestigious and developers will attempt to obtain such an address for a building, to the point of lobbying for a street or portion of a street to be renamed, even if this makes the address less useful for wayfinding. The construction of a new street to serve the development may also provide the possibility of a "1" address. An example of such an address is the Apple Campus, located at 1 Infinite Loop, Cupertino, California. See also • −1 • +1 (disambiguation) • List of mathematical constants • One (word) • Root of unity • List of highways numbered 1 References Wikimedia Commons has media related to: 1 (number) (category) 1. Weisstein, Eric W. "1". mathworld.wolfram.com. Archived from the original on 2020-07-26. Retrieved 2020-08-10. 2. Huddleston, Rodney D.; Pullum, Geoffrey K.; Reynolds, Brett (2022). A student's introduction to English grammar (2nd ed.). Cambridge, United Kingdom: Cambridge University Press. p. 117. ISBN 978-1-316-51464-1. OCLC 1255524478. 3. Huddleston, Rodney D.; Pullum, Geoffrey K.; Reynolds, Brett (2022). A student's introduction to English grammar (2nd ed.). Cambridge, United Kingdom: Cambridge University Press. p. 140. ISBN 978-1-316-51464-1. OCLC 1255524478. 4. "Online Etymology Dictionary". etymonline.com. Douglas Harper. Archived from the original on 2013-12-30. Retrieved 2013-12-30. 5. Skoog, Douglas. Principles of Instrumental Analysis. Brooks/Cole, 2007, p. 758. 6. "Plastic Packaging Resins" (PDF). American Chemistry Council. Archived from the original (PDF) on 2011-07-21. 7. Woodford, Chris (2006), Digital Technology, Evans Brothers, p. 9, ISBN 978-0-237-52725-9, retrieved 2016-03-24 8. Godbole, Achyut S. (1 September 2002), Data Comms & Networks, Tata McGraw-Hill Education, p. 34, ISBN 978-1-259-08223-8 9. Olson, Roger (2017). The Essentials of Christian Thought: Seeing Reality through the Biblical Story. Zondervan Academic. ISBN 9780310521563.{{cite book}}: CS1 maint: location missing publisher (link) 10. British Society for the History of Science (July 1, 1977). "From Abacus to Algorism: Theory and Practice in Medieval Arithmetic". The British Journal for the History of Science. Cambridge University PRess. 10 (2): Abstract. doi:10.1017/S0007087400015375. S2CID 145065082. Archived from the original on May 16, 2021. Retrieved May 16, 2021. External links Wikiquote has quotations related to 1 (number). • The Number 1 • The Positive Integer 1 • Prime curiosities: 1 Integers 0s • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 100s • 100 • 101 • 102 • 103 • 104 • 105 • 106 • 107 • 108 • 109 • 110 • 111 • 112 • 113 • 114 • 115 • 116 • 117 • 118 • 119 • 120 • 121 • 122 • 123 • 124 • 125 • 126 • 127 • 128 • 129 • 130 • 131 • 132 • 133 • 134 • 135 • 136 • 137 • 138 • 139 • 140 • 141 • 142 • 143 • 144 • 145 • 146 • 147 • 148 • 149 • 150 • 151 • 152 • 153 • 154 • 155 • 156 • 157 • 158 • 159 • 160 • 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Percent sign The percent sign % (sometimes per cent sign in British English) is the symbol used to indicate a percentage, a number or ratio as a fraction of 100. Related signs include the permille (per thousand) sign ‰ and the permyriad (per ten thousand) sign ‱ (also known as a basis point), which indicate that a number is divided by one thousand or ten thousand, respectively. Higher proportions use parts-per notation. "%" redirects here. For the mod sign as used in modular arithmetic, see Modular arithmetic. % Percent sign In UnicodeU+0025 % PERCENT SIGN (&percnt;) Different from Different fromU+2052 ⁒ COMMERCIAL MINUS SIGN U+00F7 ÷ DIVISION SIGN Related See alsoU+2030 ‰ PER MILLE SIGN U+2031 ‱ PER TEN THOUSAND SIGN (Basis point) Correct style Form and spacing English style guides prescribe writing the percent sign following the number without any space between (e.g. 50%).[sources 1] However, the International System of Units and ISO 31-0 standard prescribe a space between the number and percent sign,[8][9][10] in line with the general practice of using a non-breaking space between a numerical value and its corresponding unit of measurement. Other languages have other rules for spacing in front of the percent sign: • In Czech and in Slovak, the percent sign is spaced with a non-breaking space if the number is used as a noun.[11] In Czech, no space is inserted if the number is used as an adjective (e.g. “a 50% increase”),[12] whereas Slovak uses a non-breaking space in this case as well.[13] • In Finnish, the percent sign is always spaced, and a case suffix can be attached to it using the colon (e.g. 50 %:n kasvu 'an increase of 50%').[14] • In French, the percent sign must be spaced with a non-breaking space.[15][16] • According to the Real Academia Española, in Spanish, the percent sign should be spaced now, despite the fact that it is not the linguistic norm.[17] Despite that, in North American Spanish (Mexico and the US), several style guides and institutions either recommend the percent sign be written following the number without any space between or do so in their own publications in accordance with common usage in that region.[18][19] • In Russian, the percent sign is rarely spaced, contrary to the guidelines of the GOST 8.417-2002 state standard. • In Chinese, the percent sign is almost never spaced, probably because Chinese does not use spaces to separate characters or words at all. • According to the Swedish Language Council, the percent sign should be preceded by a space in Swedish, as all other units. • In German, the space is prescribed by the regulatory body in the national standard DIN 5008. • In Turkish and some other Turkic languages, the percent sign precedes rather than follows the number, without an intervening space. • In Persian texts, the percent sign may either precede or follow the number, in either case without a space. • In Arabic, the percent sign follows the number; as Arabic is written from right to left, this means that the percent sign is to the left of the number, usually without a space. • In Hebrew, the percent sign is written to the right of the number, just as in English, without an intervening space. This is because numbers in Hebrew (which otherwise is written from right to left) are written from left to right, as in English. • In Dutch, the official rule (NBN Z 01-002) is to place a space between the number and the sign (e.g. "een stijging van 50 %"), but most of the time, the space is missing (e.g. "een stijging van 50%").[20] Usage in text It is often recommended that the percent sign only be used in tables and other places with space restrictions. In running text, it should be spelled out as percent or per cent (often in newspapers). For example, not "Sales increased by 24% over 2006" but "Sales increased by 24 percent over 2006".[21][22][23] Evolution Prior to 1425, there is no known evidence of a special symbol being used for percentage. The Italian term per cento, "for a hundred", was used as well as several different abbreviations (e.g. "per 100", "p 100", "p cento", etc.). Examples of this can be seen in the 1339 arithmetic text (author unknown) depicted below.[24] The letter p with its descender crossed by a horizontal or diagonal strike (ꝑ in Unicode) conventionally stood for per, por, par, or pur in Medieval and Renaissance palaeography.[25] At some point, a scribe used the abbreviation "pc" with a tiny loop or circle (depicting the ending -o used in Italian ordinals, as in primo, secondo, etc.; it is analogous to the English "-th" as in "25th"). This appears in some additional pages of a 1425 text which were probably added around 1435.[26] This is shown below (source, Rara Arithmetica p. 440). The "pc" with a loop eventually evolved into a horizontal fraction sign by 1650 (see below for an example in a 1684 text[27]) and thereafter lost the "per".[28] In 1925, D. E. Smith wrote, "The solidus form () is modern."[29] Usage Unicode The Unicode code points are: • U+0025 % PERCENT SIGN (HTML %, &percnt;[30]), • U+FF05 ％ FULLWIDTH PERCENT SIGN see fullwidth forms • U+FE6A ﹪ SMALL PERCENT SIGN see Small Form Variants • U+066A ٪ ARABIC PERCENT SIGN, which has the circles replaced by square dots set on edge, the shape of the digit 0 in Eastern Arabic numerals. ASCII The ASCII code for the percent character is 37, or 0x25 in hexadecimal. In computers Names for the percent sign include percent sign (in ITU-T), mod, grapes (in hacker jargon) , and the humorous double-oh-seven (in INTERCAL). In computing, the percent character is also used for the modulo operation in programming languages that derive their syntax from the C programming language, which in turn acquired this usage from the earlier B.[31] In the textual representation of URIs, a % immediately followed by a 2-digit hexadecimal number denotes an octet specifying (part of) a character that might otherwise not be allowed in URIs (see percent-encoding). In SQL, the percent sign is a wildcard character in "LIKE" expressions, for example SELECT * FROM table WHERE fullname LIKE 'Lisa %' will fetch all records whose names start with "Lisa ". In TeX (and therefore also in LaTeX) and PostScript, and in GNU Octave and MATLAB,[32] a % denotes a line comment. In BASIC, Visual Basic, ASP, and VBA a trailing % after a variable name marks it as an integer. In ASP, the percent sign can be used to indicate the start and end of the ASP code <%...... %> In Perl % is the sigil for hashes. In many programming languages' string formatting operations (performed by functions such as printf and scanf), the percent sign denotes parts of the template string that will be replaced with arguments. (See printf format string.) In Python and Ruby the percent sign is also used as the string formatting operator.[33][34][35] In the command processors COMMAND.COM (DOS) and CMD.EXE (OS/2 and Windows), %1, %2,... stand for the first, second,... parameters of a batch file. %0 stands for the specification of the batch file itself as typed on the command line. The % sign is also used similarly in the FOR command. %VAR1% represents the value of an environment variable named VAR1. Thus: set PATH=c:\;%PATH% sets a new value for PATH, that being the old value preceded by "c:\;". Because these uses give the percent sign special meaning, the sequence %% (two percent signs) is used to represent a literal percent sign, so that: set PATH=c:\;%%PATH%% would set PATH to the literal value "c:\;%PATH%". In the C Shell, % is part of the default command prompt. In linguistics In linguistics, the percent sign is prepended to an example string to show that it is judged well-formed by some speakers and ill-formed by others. This may be due to differences in dialect or even individual idiolects. This is similar to the asterisk to mark ill-formed or reconstructed strings, the question mark to mark strings where well-formedness is unclear, and the number sign to mark strings that are syntactically well-formed but semantically nonsensical. See also • Metacharacter • U+2030 ‰ PER MILLE SIGN, • U+2031 ‱ PER TEN THOUSAND SIGN (also known as basis point) Reference notes 1. [1][2][3][4][5][6][7] Notes 1. "Guardian and Observer style guide: P". The Guardian. 30 April 2021. Archived from the original on 28 December 2019. Retrieved 16 March 2023. 2. "The Chicago Manual of Style". University of Chicago Press. 2003. Archived from the original on 5 January 2009. Retrieved 5 January 2007. 3. Publication Manual of the American Psychological Association. 1994. Washington, DC: American Psychological Association, p. 114. 4. Merriam-Webster's Manual for Writers and Editors. 1998. Springfield, MA: Merriam-Webster, p. 128. 5. Jenkins, Jana et al. 2011. The IBM Style Guide: Conventions for Writers and Editors. Boston, MA: Pearson Education, p. 162. 6. Covey, Stephen R. FranklinCovey Style Guide: For Business and Technical Communication. Salt Lake City, UT: FranklinCovey, p. 287. 7. Dodd, Janet S. 1997. The ACS Style Guide: A Manual for Authors and Editors. Washington, DC: American Chemical Society, p. 264. 8. "SI Brochure". International Bureau of Weights and Measures. 2006. Archived from the original on 21 March 2019. Retrieved 5 May 2016. 9. "The International System of Units" (PDF). International Bureau of Weights and Measures. 2006. Archived (PDF) from the original on 13 March 2020. Retrieved 6 August 2007. 10. "Quantities and units – Part 0: General principles". International Organization for Standardization. 22 December 1999. Archived from the original on 29 May 2007. Retrieved 5 January 2007. 11. "Internetová jazyková příručka". Ústav pro jazyk český Akademie věd ČR. 2014. Archived from the original on 14 February 2015. Retrieved 24 November 2014. 12. "Jazyková poradna ÚJČ AV ČR: FAQ". Ústav pro jazyk český Akademie věd ČR. 2002. Archived from the original on 19 April 2002. Retrieved 16 March 2009. 13. "Jazyková poradňa". Petit Press, a.s. 2013. Archived from the original on 21 February 2009. Retrieved 26 October 2019. 14. "Kielikello 2/2006". kotus.fi. Kotimaisten kielten keskus. 2006. Archived from the original on 1 May 2015. Retrieved 30 June 2015. 15. Guide des principales règles typographiques (PDF). Université Joseph-Fourier. Archived (PDF) from the original on 3 March 2016. Retrieved 8 June 2022. 16. André, Jacques. Petites leçons de typographie (PDF). Rennes: Institut de recherche en informatique et systèmes aléatoires. p. 34. Archived (PDF) from the original on 20 January 2011. Retrieved 8 June 2022. 17. "El % se escribe separado de la cifra a la que acompaña". fundeu.es. Fundeu. 2012. Archived from the original on 24 November 2021. Retrieved 24 November 2021. 18. "Normas particulares de estilo". colmex.mx. Colegio de México. 2020. Archived from the original on 10 May 2022. Retrieved 5 April 2022. 19. "¿En un texto, es correcto usar el signo de porcentaje o tiene que escribirse por ciento?". academia.org.mx. Academia Mexicana de la Lengua. 2020. Archived from the original on 6 February 2023. Retrieved 5 April 2022. 20. "procentteken (spatie)". www.vlaanderen.be (in Dutch). Archived from the original on 27 November 2021. Retrieved 27 November 2021. 21. American Economic Review: Style Guide Archived 2007-12-25 at the Wayback Machine 22. "UNC Pharmacy style guide". Archived from the original on 12 June 2007. Retrieved 16 October 2007. 23. "University of Colorado style guide". Archived from the original on 2 November 2007. Retrieved 16 October 2007. 24. Smith 1898, p. 437 25. Letter p. Archived 18 April 2009 at the Wayback Machine / Cappelli, Adriano: Lexicon Abbreviaturarum Archived 8 May 2015 at the Wayback Machine. 2. verb. Aufl. Leipzig 1928. Wörterbuch der Abkürzungen: P. pages 256–257 26. Smith 1898, pp. 439-440 27. Smith 1898, p. 441 28. Smith 1898, p. 440 29. Smith 1925, Vol. 2, p. 250 in Dover reprint of 1958, ISBN 0-486-20430-8 30. HTML5 is the only version of HTML that has a named entity for the percent sign, see https://www.w3.org/TR/html4/sgml/entities.html Archived 1 April 2018 at the Wayback Machine ("The following sections present the complete lists of character entity references.") and https://www.w3.org/TR/2014/CR-html5-20140731/syntax.html#named-character-references Archived 5 August 2017 at the Wayback Machine ("percnt;"). 31. Thompson, Ken (1996). "Users' Reference to B". Archived from the original on 6 July 2006. 32. "2.7.1 Single Line Comments". GNU. Archived from the original on 20 July 2018. Retrieved 20 July 2018. 33. "Python 2 – String Formatting Operations". Archived from the original on 4 November 2015. Retrieved 28 October 2015. 34. "Python 3 – printf-style String Formatting". Archived from the original on 14 June 2020. Retrieved 28 October 2015. 35. "Ruby – String#%". Archived from the original on 8 October 2011. Retrieved 28 October 2015. References • Smith, D. E. (1898), Rara Arithmetica: a catalogue of the arithmetics written before MDCI, with description of those in the library of George Arthur Plimpton of New York, Boston: Ginn • Smith, D. E. (1925), History of Mathematics, Boston: Ginn Common punctuation marks and other typographical marks or symbols • space • , comma • : colon • ; semicolon • ‐ hyphen • ’ ' apostrophe • ′ ″ ‴ prime • . full stop • & ampersand • @ at sign • ^ caret • / slash • \ backslash • … ellipsis • * asterisk • ⁂ asterism • * * * dinkus • - hyphen-minus • ‒ – — dash • = ⸗ double hyphen • ? question mark • ! exclamation mark • ‽ interrobang • ¡ ¿ inverted ! and ? • ⸮ irony punctuation • # number sign • № numero sign • º ª ordinal indicator • % percent sign • ‰ per mille • ‱ basis point • ° degree symbol • ⌀ diameter sign • + − plus and minus signs • × multiplication sign • ÷ division sign • ~ tilde • ± plus–minus sign • ∓ minus-plus sign • _ underscore • ⁀ tie • \| ¦ ‖ vertical bar • • bullet • · interpunct • © copyright symbol • © copyleft • ℗ sound recording copyright • ® registered trademark • SM service mark symbol • TM trademark symbol • ‘ ’ “ ” ' ' " " quotation mark • ‹ › « » guillemet • ( ) [ ] { } ⟨ ⟩ bracket • ” 〃 ditto mark • † ‡ dagger • ❧ hedera/floral heart • ☞ manicule • ◊ lozenge • ¶ ⸿ pilcrow (paragraph mark) • ※ reference mark • § section mark • Version of this table as a sortable list • Currency symbols • Diacritics (accents) • Logic symbols • Math symbols • Whitespace • Chinese punctuation • Hebrew punctuation • Japanese punctuation • Korean punctuation
Material conditional The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol $\rightarrow $ is interpreted as material implication, a formula $P\rightarrow Q$ is true unless $P$ is true and $Q$ is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum. Not to be confused with Material inference or Material implication (rule of inference). Material conditional IMPLY Definition$x\rightarrow y$ Truth table$(1011)$ Logic gate Normal forms Disjunctive${\overline {x}}+y$ Conjunctive${\overline {x}}+y$ Zhegalkin polynomial$1\oplus x\oplus xy$ Post's lattices 0-preservingno 1-preservingyes Monotoneno Affineno Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language. Notation In logic and related fields, the material conditional is customarily notated with an infix operator $\to $.[1] The material conditional is also notated using the infixes $\supset $ and $\Rightarrow $. In the prefixed Polish notation, conditionals are notated as $Cpq$. In a conditional formula $p\to q$, the subformula $p$ is referred to as the antecedent and $q$ is termed the consequent of the conditional. Conditional statements may be nested such that the antecedent or the consequent may themselves be conditional statements, as in the formula $(p\to q)\to (r\to s)$. History In Arithmetices Principia: Nova Methodo Exposita (1889), Peano expressed the proposition “If $A$ then $B$” as $A$ Ɔ $B$ with the symbol Ɔ, which is the opposite of C.[2] He also expressed the proposition $A\supset B$ as $A$ Ɔ $B$.[lower-alpha 1][3][4] Hilbert expressed the proposition “If A then B” as $A\to B$ in 1918.[1] Russell followed Peano in his Principia Mathematica (1910–1913), in which he expressed the proposition “If A then B” as $A\supset B$. Following Russell, Gentzen expressed the proposition “If A then B” as $A\supset B$. Heyting expressed the proposition “If A then B” as $A\supset B$ at first but later came to express it as $A\to B$ with a right-pointing arrow. Bourbaki expressed the proposition “If A then B” as $A\Rightarrow B$ in 1954.[5] Definitions Semantics From a semantic perspective, material implication is the binary truth functional operator which returns "true" unless its first argument is true and its second argument is false. This semantics can be shown graphically in a truth table such as the one below. Truth table The truth table of p → q: $p$$q$$p$ → $q$ TrueTrueTrue TrueFalseFalse FalseTrueTrue FalseFalseTrue The 3rd and 4th logical cases of this truth table, where the antecedent p is false and p → q is true, are called vacuous truths. Deductive definition Material implication can also be characterized deductively in terms of the following rules of inference. 1. Modus ponens 2. Conditional proof 3. Classical contraposition 4. Classical reductio ad absurdum Unlike the semantic definition, this approach to logical connectives permits the examination of structurally identical propositional forms in various logical systems, where somewhat different properties may be demonstrated. For example, in intuitionistic logic, which rejects proofs by contraposition as valid rules of inference, (p → q) ⇒ ¬p ∨ q is not a propositional theorem, but the material conditional is used to define negation. Formal properties When disjunction, conjunction and negation are classical, material implication validates the following equivalences: • Contraposition: $P\to Q\equiv \neg Q\to \neg P$ • Import-Export: $P\to (Q\to R)\equiv (P\land Q)\to R$ • Negated conditionals: $\neg (P\to Q)\equiv P\land \neg Q$ • Or-and-if: $P\to Q\equiv \neg P\lor Q$ • Commutativity of antecedents: ${\big (}P\to (Q\to R){\big )}\equiv {\big (}Q\to (P\to R){\big )}$ • Distributivity: ${\big (}R\to (P\to Q){\big )}\equiv {\big (}(R\to P)\to (R\to Q){\big )}$ Similarly, on classical interpretations of the other connectives, material implication validates the following entailments: • Antecedent strengthening: $P\to Q\models (P\land R)\to Q$ • Vacuous conditional: $\neg P\models P\to Q$ • Transitivity: $(P\to Q)\land (Q\to R)\models P\to R$ • Simplification of disjunctive antecedents: $(P\lor Q)\to R\models (P\to R)\land (Q\to R)$ Tautologies involving material implication include: • Reflexivity: $\models P\to P$ • Totality: $\models (P\to Q)\lor (Q\to P)$ • Conditional excluded middle: $\models (P\to Q)\lor (P\to \neg Q)$ Discrepancies with natural language Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris is in France". These classic problems have been called the paradoxes of material implication.[6] In addition to the paradoxes, a variety of other arguments have been given against a material implication analysis. For instance, counterfactual conditionals would all be vacuously true on such an account.[7] In the mid-20th century, a number of researchers including H. P. Grice and Frank Jackson proposed that pragmatic principles could explain the discrepancies between natural language conditionals and the material conditional. On their accounts, conditionals denote material implication but end up conveying additional information when they interact with conversational norms such as Grice's maxims.[6][8] Recent work in formal semantics and philosophy of language has generally eschewed material implication as an analysis for natural-language conditionals.[8] In particular, such work has often rejected the assumption that natural-language conditionals are truth functional in the sense that the truth value of "If P, then Q" is determined solely by the truth values of P and Q.[6] Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as modal logic, relevance logic, probability theory, and causal models.[8][6][9] Similar discrepancies have been observed by psychologists studying conditional reasoning. For instance, the notorious Wason selection task study, less than 10% of participants reasoned according to the material conditional. Some researchers have interpreted this result as a failure of the participants to confirm to normative laws of reasoning, while others interpret the participants as reasoning normatively according to nonclassical laws.[10][11][12] See also • Boolean domain • Boolean function • Boolean logic • Conditional quantifier • Implicational propositional calculus • Laws of Form • Logical graph • Logical equivalence • Material implication (rule of inference) • Peirce's law • Propositional calculus • Sole sufficient operator Conditionals • Counterfactual conditional • Indicative conditional • Corresponding conditional • Strict conditional Notes 1. Note that the horseshoe symbol Ɔ has been flipped to become a subset symbol ⊂. References 1. Hilbert, D. (1918). Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.). 2. Jean van Heijenoort, ed. (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press. pp. 84–87. ISBN 0-674-32449-8. 3. Michael Nahas (25 Apr 2022). "English Translation of "Arithmetices Principia, Nova Methodo Exposita"" (PDF). GitHub. p. VI. Retrieved 2022-08-10. 4. Mauro ALLEGRANZA (2015-02-13). "elementary set theory - Is there any connection between the symbol $\supset $ when it means implication and its meaning as superset?". Mathematics Stack Exchange. Stack Exchange Inc. Answer. Retrieved 2022-08-10. 5. Bourbaki, N. (1954). Théorie des ensembles. Paris: Hermann & Cie, Éditeurs. p. 14. 6. Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.). 7. Starr, Will (2019). "Counterfactuals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy. 8. Gillies, Thony (2017). "Conditionals" (PDF). In Hale, B.; Wright, C.; Miller, A. (eds.). A Companion to the Philosophy of Language. Wiley Blackwell. pp. 401–436. doi:10.1002/9781118972090.ch17. ISBN 9781118972090. 9. von Fintel, Kai (2011). "Conditionals" (PDF). In von Heusinger, Klaus; Maienborn, Claudia; Portner, Paul (eds.). Semantics: An international handbook of meaning. de Gruyter Mouton. pp. 1515–1538. doi:10.1515/9783110255072.1515. hdl:1721.1/95781. ISBN 978-3-11-018523-2. 10. Oaksford, M.; Chater, N. (1994). "A rational analysis of the selection task as optimal data selection". Psychological Review. 101 (4): 608–631. CiteSeerX 10.1.1.174.4085. doi:10.1037/0033-295X.101.4.608. 11. Stenning, K.; van Lambalgen, M. (2004). "A little logic goes a long way: basing experiment on semantic theory in the cognitive science of conditional reasoning". Cognitive Science. 28 (4): 481–530. CiteSeerX 10.1.1.13.1854. doi:10.1016/j.cogsci.2004.02.002. 12. von Sydow, M. (2006). Towards a Flexible Bayesian and Deontic Logic of Testing Descriptive and Prescriptive Rules. Göttingen: Göttingen University Press. doi:10.53846/goediss-161. S2CID 246924881. Further reading • Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003. • Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell. • Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA. • Stalnaker, Robert, "Indicative Conditionals", Philosophia, 5 (1975): 269–286. External links • Media related to Material conditional at Wikimedia Commons • Edgington, Dorothy. "Conditionals". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Common logical connectives • Tautology/True $\top $ • Alternative denial (NAND gate) $\uparrow $ • Converse implication $\leftarrow $ • Implication (IMPLY gate) $\rightarrow $ • Disjunction (OR gate) $\lor $ • Negation (NOT gate) $\neg $ • Exclusive or (XOR gate) $\not \leftrightarrow $ • Biconditional (XNOR gate) $\leftrightarrow $ • Statement (Digital buffer) • Joint denial (NOR gate) $\downarrow $ • Nonimplication (NIMPLY gate) $\nrightarrow $ • Converse nonimplication $\nleftarrow $ • Conjunction (AND gate) $\land $ • Contradiction/False $\bot $ Philosophy portal Common logical symbols ∧ or & and ∨ or ¬ or ~ not → implies ⊃ implies, superset ↔ or ≡ iff \| nand ∀ universal quantification ∃ existential quantification ⊤ true, tautology ⊥ false, contradiction ⊢ entails, proves ⊨ entails, therefore ∴ therefore ∵ because Philosophy portal Mathematics portal Mathematical logic General • Axiom • list • Cardinality • First-order logic • Formal proof • Formal semantics • Foundations of mathematics • Information theory • Lemma • Logical consequence • Model • Theorem • Theory • Type theory Theorems (list) & Paradoxes • Gödel's completeness and incompleteness theorems • Tarski's undefinability • Banach–Tarski paradox • Cantor's theorem, paradox and diagonal argument • Compactness • Halting problem • Lindström's • Löwenheim–Skolem • Russell's paradox Logics Traditional • Classical logic • Logical truth • Tautology • Proposition • Inference • Logical equivalence • Consistency • Equiconsistency • Argument • Soundness • Validity • Syllogism • Square of opposition • Venn diagram Propositional • Boolean algebra • Boolean functions • Logical connectives • Propositional calculus • Propositional formula • Truth tables • Many-valued logic • 3 • Finite • ∞ Predicate • First-order • list • Second-order • Monadic • Higher-order • Free • Quantifiers • Predicate • Monadic predicate calculus Set theory • Set • Hereditary • Class • (Ur-)Element • Ordinal number • Extensionality • Forcing • Relation • Equivalence • Partition • Set operations: • Intersection • Union • Complement • Cartesian product • Power set • Identities Types of Sets • Countable • Uncountable • Empty • Inhabited • Singleton • Finite • Infinite • Transitive • Ultrafilter • Recursive • Fuzzy • Universal • Universe • Constructible • Grothendieck • Von Neumann Maps & Cardinality • Function/Map • Domain • Codomain • Image • In/Sur/Bi-jection • Schröder–Bernstein theorem • Isomorphism • Gödel numbering • Enumeration • Large cardinal • Inaccessible • Aleph number • Operation • Binary Set theories • Zermelo–Fraenkel • Axiom of choice • Continuum hypothesis • General • Kripke–Platek • Morse–Kelley • Naive • New Foundations • Tarski–Grothendieck • Von Neumann–Bernays–Gödel • Ackermann • Constructive Formal systems (list), Language & Syntax • Alphabet • Arity • Automata • Axiom schema • Expression • Ground • Extension • by definition • Conservative • Relation • Formation rule • Grammar • Formula • Atomic • Closed • Ground • Open • Free/bound variable • Language • Metalanguage • Logical connective • ¬ • ∨ • ∧ • → • ↔ • = • Predicate • Functional • Variable • Propositional variable • Proof • Quantifier • ∃ • ! • ∀ • rank • Sentence • Atomic • Spectrum • Signature • String • Substitution • Symbol • Function • Logical/Constant • Non-logical • Variable • Term • Theory • list Example axiomatic systems (list) • of arithmetic: • Peano • second-order • elementary function • primitive recursive • Robinson • Skolem • of the real numbers • Tarski's axiomatization • of Boolean algebras • canonical • minimal axioms • of geometry: • Euclidean: • Elements • Hilbert's • Tarski's • non-Euclidean • Principia Mathematica Proof theory • Formal proof • Natural deduction • Logical consequence • Rule of inference • Sequent calculus • Theorem • Systems • Axiomatic • Deductive • Hilbert • list • Complete theory • Independence (from ZFC) • Proof of impossibility • Ordinal analysis • Reverse mathematics • Self-verifying theories Model theory • Interpretation • Function • of models • Model • Equivalence • Finite • Saturated • Spectrum • Submodel • Non-standard model • of arithmetic • Diagram • Elementary • Categorical theory • Model complete theory • Satisfiability • Semantics of logic • Strength • Theories of truth • Semantic • Tarski's • Kripke's • T-schema • Transfer principle • Truth predicate • Truth value • Type • Ultraproduct • Validity Computability theory • Church encoding • Church–Turing thesis • Computably enumerable • Computable function • Computable set • Decision problem • Decidable • Undecidable • P • NP • P versus NP problem • Kolmogorov complexity • Lambda calculus • Primitive recursive function • Recursion • Recursive set • Turing machine • Type theory Related • Abstract logic • Category theory • Concrete/Abstract Category • Category of sets • History of logic • History of mathematical logic • timeline • Logicism • Mathematical object • Philosophy of mathematics • Supertask Mathematics portal
Power set In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself.[1] In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.[2] The powerset of S is variously denoted as P(S), 𝒫(S), P(S), $\mathbb {P} (S)$, $\wp (S)$, or 2S. The notation 2S, meaning the set of all functions from S to a given set of two elements (e.g., {0, 1}), is used because the powerset of S can be identified with, equivalent to, or bijective to the set of all the functions from S to the given two elements set.[1] Power set The elements of the power set of {x, y, z} ordered with respect to inclusion. TypeSet operation FieldSet theory StatementThe power set is the set that contains all subsets of a given set. Symbolic statement$x\in P(S)\iff x\subseteq S$ Any subset of P(S) is called a family of sets over S. Example If S is the set {x, y, z}, then all the subsets of S are • {} (also denoted $\varnothing $ or $\emptyset $, the empty set or the null set) • {x} • {y} • {z} • {x, y} • {x, z} • {y, z} • {x, y, z} and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.[3] Properties If S is a finite set with the cardinality \|S\| = n (i.e., the number of all elements in the set S is n), then the number of all the subsets of S is \|P(S)\| = 2n. This fact as well as the reason of the notation 2S denoting the power set P(S) are demonstrated in the below. An indicator function or a characteristic function of a subset A of a set S with the cardinality \|S\| = n is a function from S to the two elements set {0, 1}, denoted as IA: S → {0, 1}, and it indicates whether an element of S belongs to A or not; If x in S belongs to A, then IA(x) = 1, and 0 otherwise. Each subset A of S is identified by or equivalent to the indicator function IA, and {0,1}S as the set of all the functions from S to {0,1} consists of all the indicator functions of all the subsets of S. In other words, {0,1}S is equivalent or bijective to the power set P(S). Since each element in S corresponds to either 0 or 1 under any function in {0,1}S, the number of all the functions in {0,1}S is 2n. Since the number 2 can be defined as {0,1} (see, for example, von Neumann ordinals), the P(S) is also denoted as 2S. Obviously \|2S\| = 2\|S\| holds. Generally speaking, XY is the set of all functions from Y to X and \|XY\| = \|X\|\|Y\|. Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum). The power set of a set S, together with the operations of union, intersection and complement, can be viewed as the prototypical example of a Boolean algebra. In fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. For infinite Boolean algebras, this is no longer true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra (see Stone's representation theorem). The power set of a set S forms an abelian group when it is considered with the operation of symmetric difference (with the empty set as the identity element and each set being its own inverse), and a commutative monoid when considered with the operation of intersection. It can hence be shown, by proving the distributive laws, that the power set considered together with both of these operations forms a Boolean ring. Representing subsets as functions In set theory, XY is the notation representing the set of all functions from Y to X. As "2" can be defined as {0,1} (see, for example, von Neumann ordinals), 2S (i.e., {0,1}S) is the set of all functions from S to {0,1}. As shown above, 2S and the power set of S, P(S), are considered identical set-theoretically. This equivalence can be applied to the example above, in which S = {x, y, z}, to get the isomorphism with the binary representations of numbers from 0 to 2n − 1, with n being the number of elements in the set S or \|S\| = n. First, the enumerated set { (x, 1), (y, 2), (z, 3) } is defined in which the number in each ordered pair represents the position of the paired element of S in a sequence of binary digits such as {x, y} = 011(2); x of S is located at the first from the right of this sequence and y is at the second from the right, and 1 in the sequence means the element of S corresponding to the position of it in the sequence exists in the subset of S for the sequence while 0 means it does not. For the whole power set of S, we get: Subset Sequence of binary digits Binary interpretation Decimal equivalent { }0, 0, 0000(2)0(10) { x }0, 0, 1001(2)1(10) { y }0, 1, 0010(2)2(10) { x, y }0, 1, 1011(2)3(10) { z }1, 0, 0100(2)4(10) { x, z }1, 0, 1101(2)5(10) { y, z }1, 1, 0110(2)6(10) { x, y, z }1, 1, 1111(2)7(10) Such a injective mapping from P(S) to integers is arbitrary, so this representation of all the subsets of S is not unique, but the sort order of the enumerated set does not change its cardinality. (E.g., { (y, 1), (z, 2), (x, 3) } can be used to construct another injective mapping from P(S) to the integers without changing the number of one-to-one correspondences.) However, such finite binary representation is only possible if S can be enumerated. (In this example, x, y, and z are enumerated with 1, 2, and 3 respectively as the position of binary digit sequences.) The enumeration is possible even if S has an infinite cardinality (i.e., the number of elements in S is infinite), such as the set of integers or rationals, but not possible for example if S is the set of real numbers, in which case we cannot enumerate all irrational numbers. Relation to binomial theorem The binomial theorem is closely related to the power set. A k–elements combination from some set is another name for a k–elements subset, so the number of combinations, denoted as C(n, k) (also called binomial coefficient) is a number of subsets with k elements in a set with n elements; in other words it's the number of sets with k elements which are elements of the power set of a set with n elements. For example, the power set of a set with three elements, has: • C(3, 0) = 1 subset with 0 elements (the empty subset), • C(3, 1) = 3 subsets with 1 element (the singleton subsets), • C(3, 2) = 3 subsets with 2 elements (the complements of the singleton subsets), • C(3, 3) = 1 subset with 3 elements (the original set itself). Using this relationship, we can compute $ \left\|2^{S}\right\|$ using the formula: $\left\|2^{S}\right\|=\sum _{k=0}^{\|S\|}{\binom {\|S\|}{k}}$ Therefore, one can deduce the following identity, assuming $ \|S\|=n$: $\left\|2^{S}\right\|=2^{n}=\sum _{k=0}^{n}{\binom {n}{k}}$ Recursive definition If $S$ is a finite set, then a recursive definition of $P(S)$ proceeds as follows: • If $S=\{\}$, then $P(S)=\{\,\{\}\,\}$. • Otherwise, let $e\in S$ and $T=S\setminus \{e\}$; then $P(S)=P(T)\cup \{t\cup \{e\}:t\in P(T)\}$. In words: • The power set of the empty set is a singleton whose only element is the empty set. • For a non-empty set $S$, let $e$ be any element of the set and $T$ its relative complement; then the power set of $S$ is a union of a power set of $T$ and a power set of $T$ whose each element is expanded with the $e$ element. Subsets of limited cardinality The set of subsets of S of cardinality less than or equal to κ is sometimes denoted by Pκ(S) or [S]κ, and the set of subsets with cardinality strictly less than κ is sometimes denoted P< κ(S) or [S]<κ. Similarly, the set of non-empty subsets of S might be denoted by P≥ 1(S) or P+(S). Power object A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective, the idea of the power set of X as the set of subsets of X generalizes naturally to the subalgebras of an algebraic structure or algebra. The power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an algebraic lattice, and every algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard, subalgebras behave analogously to subsets. However, there are two important properties of subsets that do not carry over to subalgebras in general. First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class, although they can always be organized as a lattice. Secondly, whereas the subsets of a set are in bijection with the functions from that set to the set {0,1} = 2, there is no guarantee that a class of algebras contains an algebra that can play the role of 2 in this way. Certain classes of algebras enjoy both of these properties. The first property is more common, the case of having both is relatively rare. One class that does have both is that of multigraphs. Given two multigraphs G and H, a homomorphism h: G → H consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set HG of homomorphisms from G to H can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set. Furthermore, the subgraphs of a multigraph G are in bijection with the graph homomorphisms from G to the multigraph Ω definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of G as the multigraph ΩG, called the power object of G. What is special about a multigraph as an algebra is that its operations are unary. A multigraph has two sorts of elements forming a set V of vertices and E of edges, and has two unary operations s,t: E → V giving the source (start) and target (end) vertices of each edge. An algebra all of whose operations are unary is called a presheaf. Every class of presheaves contains a presheaf Ω that plays the role for subalgebras that 2 plays for subsets. Such a class is a special case of the more general notion of elementary topos as a category that is closed (and moreover cartesian closed) and has an object Ω, called a subobject classifier. Although the term "power object" is sometimes used synonymously with exponential object YX, in topos theory Y is required to be Ω. Functors and quantifiers In category theory and the theory of elementary topoi, the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifier is the left adjoint.[4] See also • Cantor's theorem • Family of sets • Field of sets • Combination References 1. Weisstein, Eric W. "Power Set". mathworld.wolfram.com. Retrieved 2020-09-05. 2. Devlin 1979, p. 50 3. Puntambekar 2007, pp. 1–2 4. Saunders Mac Lane, Ieke Moerdijk, (1992) Sheaves in Geometry and Logic Springer-Verlag. ISBN 0-387-97710-4 See page 58 Bibliography • Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer-Verlag. ISBN 0-387-90441-7. Zbl 0407.04003. • Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. Zbl 0087.04403. • Puntambekar, A. A. (2007). Theory Of Automata And Formal Languages. Technical Publications. ISBN 978-81-8431-193-8. External links Look up power set in Wiktionary, the free dictionary. • Power set at PlanetMath. • Power set at the nLab • Power object at the nLab • Power set Algorithm in C++ Mathematical logic General • Axiom • list • Cardinality • First-order logic • Formal proof • Formal semantics • Foundations of mathematics • Information theory • Lemma • Logical consequence • Model • Theorem • Theory • Type theory Theorems (list) & Paradoxes • Gödel's completeness and incompleteness theorems • Tarski's undefinability • Banach–Tarski paradox • Cantor's theorem, paradox and diagonal argument • Compactness • Halting problem • Lindström's • Löwenheim–Skolem • Russell's paradox Logics Traditional • Classical logic • Logical truth • Tautology • Proposition • Inference • Logical equivalence • Consistency • Equiconsistency • Argument • Soundness • Validity • Syllogism • Square of opposition • Venn diagram Propositional • Boolean algebra • Boolean functions • Logical connectives • Propositional calculus • Propositional formula • Truth tables • Many-valued logic • 3 • Finite • ∞ Predicate • First-order • list • Second-order • Monadic • Higher-order • Free • Quantifiers • Predicate • Monadic predicate calculus Set theory • Set • Hereditary • Class • (Ur-)Element • Ordinal number • Extensionality • Forcing • Relation • Equivalence • Partition • Set operations: • Intersection • Union • Complement • Cartesian product • Power set • Identities Types of Sets • Countable • Uncountable • Empty • Inhabited • Singleton • Finite • Infinite • Transitive • Ultrafilter • Recursive • Fuzzy • Universal • Universe • Constructible • Grothendieck • Von Neumann Maps & Cardinality • Function/Map • Domain • Codomain • Image • In/Sur/Bi-jection • Schröder–Bernstein theorem • Isomorphism • Gödel numbering • Enumeration • Large cardinal • Inaccessible • Aleph number • Operation • Binary Set theories • Zermelo–Fraenkel • Axiom of choice • Continuum hypothesis • General • Kripke–Platek • Morse–Kelley • Naive • New Foundations • Tarski–Grothendieck • Von Neumann–Bernays–Gödel • Ackermann • Constructive Formal systems (list), Language & Syntax • Alphabet • Arity • Automata • Axiom schema • Expression • Ground • Extension • by definition • Conservative • Relation • Formation rule • Grammar • Formula • Atomic • Closed • Ground • Open • Free/bound variable • Language • Metalanguage • Logical connective • ¬ • ∨ • ∧ • → • ↔ • = • Predicate • Functional • Variable • Propositional variable • Proof • Quantifier • ∃ • ! • ∀ • rank • Sentence • Atomic • Spectrum • Signature • String • Substitution • Symbol • Function • Logical/Constant • Non-logical • Variable • Term • Theory • list Example axiomatic systems (list) • of arithmetic: • Peano • second-order • elementary function • primitive recursive • Robinson • Skolem • of the real numbers • Tarski's axiomatization • of Boolean algebras • canonical • minimal axioms • of geometry: • Euclidean: • Elements • Hilbert's • Tarski's • non-Euclidean • Principia Mathematica Proof theory • Formal proof • Natural deduction • Logical consequence • Rule of inference • Sequent calculus • Theorem • Systems • Axiomatic • Deductive • Hilbert • list • Complete theory • Independence (from ZFC) • Proof of impossibility • Ordinal analysis • Reverse mathematics • Self-verifying theories Model theory • Interpretation • Function • of models • Model • Equivalence • Finite • Saturated • Spectrum • Submodel • Non-standard model • of arithmetic • Diagram • Elementary • Categorical theory • Model complete theory • Satisfiability • Semantics of logic • Strength • Theories of truth • Semantic • Tarski's • Kripke's • T-schema • Transfer principle • Truth predicate • Truth value • Type • Ultraproduct • Validity Computability theory • Church encoding • Church–Turing thesis • Computably enumerable • Computable function • Computable set • Decision problem • Decidable • Undecidable • P • NP • P versus NP problem • Kolmogorov complexity • Lambda calculus • Primitive recursive function • Recursion • Recursive set • Turing machine • Type theory Related • Abstract logic • Category theory • Concrete/Abstract Category • Category of sets • History of logic • History of mathematical logic • timeline • Logicism • Mathematical object • Philosophy of mathematics • Supertask Mathematics portal Set theory Overview • Set (mathematics) Axioms • Adjunction • Choice • countable • dependent • global • Constructibility (V=L) • Determinacy • Extensionality • Infinity • Limitation of size • Pairing • Power set • Regularity • Union • Martin's axiom • Axiom schema • replacement • specification Operations • Cartesian product • Complement (i.e. set difference) • De Morgan's laws • Disjoint union • Identities • Intersection • Power set • Symmetric difference • Union • Concepts • Methods • Almost • Cardinality • Cardinal number (large) • Class • Constructible universe • Continuum hypothesis • Diagonal argument • Element • ordered pair • tuple • Family • Forcing • One-to-one correspondence • Ordinal number • Set-builder notation • Transfinite induction • Venn diagram Set types • Amorphous • Countable • Empty • Finite (hereditarily) • Filter • base • subbase • Ultrafilter • Fuzzy • Infinite (Dedekind-infinite) • Recursive • Singleton • Subset · Superset • Transitive • Uncountable • Universal Theories • Alternative • Axiomatic • Naive • Cantor's theorem • Zermelo • General • Principia Mathematica • New Foundations • Zermelo–Fraenkel • von Neumann–Bernays–Gödel • Morse–Kelley • Kripke–Platek • Tarski–Grothendieck • Paradoxes • Problems • Russell's paradox • Suslin's problem • Burali-Forti paradox Set theorists • Paul Bernays • Georg Cantor • Paul Cohen • Richard Dedekind • Abraham Fraenkel • Kurt Gödel • Thomas Jech • John von Neumann • Willard Quine • Bertrand Russell • Thoralf Skolem • Ernst Zermelo
Rhombus In plane Euclidean geometry, a rhombus (PL: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after the French sweet[1] – also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Rhombus A rhombus in two different orientations Typequadrilateral, trapezoid, parallelogram, kite Edges and vertices4 Schläfli symbol{ } + { } {2α} Coxeter–Dynkin diagrams Symmetry groupDihedral (D2), [2], (*22), order 4 Area$K={\frac {p\cdot q}{2}}$ (half the product of the diagonals) Propertiesconvex, isotoxal Dual polygonrectangle Every rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. A rhombus with right angles is a square.[2] Etymology The word "rhombus" comes from Ancient Greek: ῥόμβος, romanized: rhombos, meaning something that spins,[3] which derives from the verb ῥέμβω, romanized: rhémbō, meaning "to turn round and round."[4] The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for a bicone, two right circular cones sharing a common base.[5] The surface we refer to as rhombus today is a cross section of the bicone on a plane through the apexes of the two cones. Characterizations A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:[6][7] • a parallelogram in which a diagonal bisects an interior angle • a parallelogram in which at least two consecutive sides are equal in length • a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram) • a quadrilateral with four sides of equal length (by definition) • a quadrilateral in which the diagonals are perpendicular and bisect each other • a quadrilateral in which each diagonal bisects two opposite interior angles • a quadrilateral ABCD possessing a point P in its plane such that the four triangles ABP, BCP, CDP, and DAP are all congruent[8] • a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point[9] Basic properties Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties: • Opposite angles of a rhombus have equal measure. • The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral. • Its diagonals bisect opposite angles. The first property implies that every rhombus is a parallelogram. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel; adjacent angles are supplementary; the two diagonals bisect one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the sum of the squares of the diagonals (the parallelogram law). Thus denoting the common side as a and the diagonals as p and q, in every rhombus $\displaystyle 4a^{2}=p^{2}+q^{2}.$ Not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals (the second property) is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite. Every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral.[10] That is, it has an inscribed circle that is tangent to all four sides. Diagonals The length of the diagonals p = AC and q = BD can be expressed in terms of the rhombus side a and one vertex angle α as $p=a{\sqrt {2+2\cos {\alpha }}}$ and $q=a{\sqrt {2-2\cos {\alpha }}}.$ These formulas are a direct consequence of the law of cosines. Inradius The inradius (the radius of a circle inscribed in the rhombus), denoted by r, can be expressed in terms of the diagonals p and q as[10] $r={\frac {p\cdot q}{2{\sqrt {p^{2}+q^{2}}}}},$ or in terms of the side length a and any vertex angle α or β as $r={\frac {a\sin \alpha }{2}}={\frac {a\sin \beta }{2}}.$ Area As for all parallelograms, the area K of a rhombus is the product of its base and its height (h). The base is simply any side length a: $K=a\cdot h.$ The area can also be expressed as the base squared times the sine of any angle: $K=a^{2}\cdot \sin \alpha =a^{2}\cdot \sin \beta ,$ or in terms of the height and a vertex angle: $K={\frac {h^{2}}{\sin \alpha }},$ or as half the product of the diagonals p, q: $K={\frac {p\cdot q}{2}},$ or as the semiperimeter times the radius of the circle inscribed in the rhombus (inradius): $K=2a\cdot r.$ Another way, in common with parallelograms, is to consider two adjacent sides as vectors, forming a bivector, so the area is the magnitude of the bivector (the magnitude of the vector product of the two vectors), which is the determinant of the two vectors' Cartesian coordinates: K = x1y2 – x2y1.[11] Dual properties The dual polygon of a rhombus is a rectangle:[12] • A rhombus has all sides equal, while a rectangle has all angles equal. • A rhombus has opposite angles equal, while a rectangle has opposite sides equal. • A rhombus has an inscribed circle, while a rectangle has a circumcircle. • A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides. • The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. • The figure formed by joining the midpoints of the sides of a rhombus is a rectangle, and vice versa. Cartesian equation The sides of a rhombus centered at the origin, with diagonals each falling on an axis, consist of all points (x, y) satisfying $\left\|{\frac {x}{a}}\right\|\!+\left\|{\frac {y}{b}}\right\|\!=1.$ The vertices are at $(\pm a,0)$ and $(0,\pm b).$ This is a special case of the superellipse, with exponent 1. Other properties • One of the five 2D lattice types is the rhombic lattice, also called centered rectangular lattice. • Identical rhombi can tile the 2D plane in three different ways, including, for the 60° rhombus, the rhombille tiling. As topological square tilings As 30-60 degree rhombille tiling • Three-dimensional analogues of a rhombus include the bipyramid and the bicone as a surface of revolution. As the faces of a polyhedron Convex polyhedra with rhombi include the infinite set of rhombic zonohedrons, which can be seen as projective envelopes of hypercubes. • A rhombohedron (also called a rhombic hexahedron) is a three-dimensional figure like a cuboid (also called a rectangular parallelepiped), except that its 3 pairs of parallel faces are up to 3 types of rhombi instead of rectangles. • The rhombic dodecahedron is a convex polyhedron with 12 congruent rhombi as its faces. • The rhombic triacontahedron is a convex polyhedron with 30 golden rhombi (rhombi whose diagonals are in the golden ratio) as its faces. • The great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron with 30 intersecting rhombic faces. • The rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. • The rhombic enneacontahedron is a polyhedron composed of 90 rhombic faces, with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim ones. • The rhombic icosahedron is a polyhedron composed of 20 rhombic faces, of which three, four, or five meet at each vertex. It has 10 faces on the polar axis with 10 faces following the equator. Example polyhedra with all rhombic faces Isohedral Isohedral golden rhombi 2-isohedral 3-isohedral Trigonal trapezohedron Rhombic dodecahedron Rhombic triacontahedron Rhombic icosahedron Rhombic enneacontahedron Rhombohedron See also • Merkel-Raute • Rhombus of Michaelis, in human anatomy • Rhomboid, either a parallelepiped or a parallelogram that is neither a rhombus nor a rectangle • Rhombic antenna • Rhombic Chess • Flag of the Department of North Santander of Colombia, containing four stars in the shape of a rhombus • Superellipse (includes a rhombus with rounded corners) References 1. Alsina, Claudi; Nelsen, Roger B. (31 December 2015). A Mathematical Space Odyssey: Solid Geometry in the 21st Century. ISBN 9781614442165. 2. Note: Euclid's original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition. See, e.g., De Villiers, Michael (February 1994). "The role and function of a hierarchical classification of quadrilaterals". For the Learning of Mathematics. 14 (1): 11–18. JSTOR 40248098. 3. ῥόμβος Archived 2013-11-08 at the Wayback Machine, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus 4. ρέμβω Archived 2013-11-08 at the Wayback Machine, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus 5. "The Origin of Rhombus". Archived from the original on 2015-04-02. Retrieved 2005-01-25. 6. Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition Archived 2020-02-26 at the Wayback Machine", Information Age Publishing, 2008, pp. 55-56. 7. Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry Archived 2019-09-01 at the Wayback Machine, Mathematical Association of America, 2010, p. 53. 8. Paris Pamfilos (2016), "A Characterization of the Rhombus", Forum Geometricorum 16, pp. 331–336, Archived 2016-10-23 at the Wayback Machine 9. "IMOmath, "26-th Brazilian Mathematical Olympiad 2004"" (PDF). Archived (PDF) from the original on 2016-10-18. Retrieved 2020-01-06. 10. Weisstein, Eric W. "Rhombus". MathWorld. 11. WildLinAlg episode 4 Archived 2017-02-05 at the Wayback Machine, Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube 12. de Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", Mathematical Gazette 95, March 2011, 102-107. External links Look up rhombus in Wiktionary, the free dictionary. Wikimedia Commons has media related to Rhombi. • Parallelogram and Rhombus - Animated course (Construction, Circumference, Area) • Rhombus definition, Math Open Reference with interactive applet. • Rhombus area, Math Open Reference - shows three different ways to compute the area of a rhombus, with interactive applet Polygons (List) Triangles • Acute • Equilateral • Ideal • Isosceles • Kepler • Obtuse • Right Quadrilaterals • Antiparallelogram • Bicentric • Crossed • Cyclic • Equidiagonal • Ex-tangential • Harmonic • Isosceles trapezoid • Kite • Orthodiagonal • Parallelogram • Rectangle • Right kite • Right trapezoid • Rhombus • Square • Tangential • Tangential trapezoid • Trapezoid By number of sides 1–10 sides • Monogon (1) • Digon (2) • Triangle (3) • Quadrilateral (4) • Pentagon (5) • Hexagon (6) • Heptagon (7) • Octagon (8) • Nonagon (Enneagon, 9) • Decagon (10) 11–20 sides • Hendecagon (11) • Dodecagon (12) • Tridecagon (13) • Tetradecagon (14) • Pentadecagon (15) • Hexadecagon (16) • Heptadecagon (17) • Octadecagon (18) • Icosagon (20) >20 sides • Icositrigon (23) • Icositetragon (24) • Triacontagon (30) • 257-gon • Chiliagon (1000) • Myriagon (10,000) • 65537-gon • Megagon (1,000,000) • Apeirogon (∞) Star polygons • Pentagram • Hexagram • Heptagram • Octagram • Enneagram • Decagram • Hendecagram • Dodecagram Classes • Concave • Convex • Cyclic • Equiangular • Equilateral • Infinite skew • Isogonal • Isotoxal • Magic • Pseudotriangle • Rectilinear • Regular • Reinhardt • Simple • Skew • Star-shaped • Tangential • Weakly simple Authority control: National • Germany

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