Who is considered the founder of algebra?
Islamic Scholarship &amp; Learning in Central Asia
Algebra was also pioneered by Persian Scientist Muhammad ibn Mūsā al-Khwārizmī during this time in his landmark text, Kitab al-Jabr wa-l-Muqabala, from which the term algebra is derived. He is thus considered to be the father of algebra. The terms algorism and algorithm are also derived from the name of al-Khwarizmi, who was responsible for introducing the Arabic numerals and Hindu-Arabic numeral system beyond the Indian Subcontinent.
History of algebra
It is sometimes alleged that the Greeks had no algebra, but this is inaccurate. By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them, and with this new form of algebra they were able to find solutions to equations by using a process that they invented, known as "the application of areas". "The application of areas" is only a part of geometric algebra and it is thoroughly covered in Euclid's "Elements".
Abstract algebraic logic
The archetypal association of this kind, one fundamental to the historical origins of algebraic logic and lying at the heart of all subsequently developed subtheories, is the association between the class of Boolean algebras and classical propositional calculus. This association was discovered by George Boole in the 1850s, and refined by others, especially Ernst Schröder in the 1890s. This work culminated in Lindenbaum–Tarski algebras, devised by Alfred Tarski and his student Adolf Lindenbaum in the 1930s. Later, Tarski and his American students (whose ranks include Don Pigozzi) went on to discover cylindric algebra, which algebraizes all of classical first-order logic, and revived relation algebra, whose models include all well-known axiomatic set theories.
Jean-Robert Argand
Jean-Robert Argand (July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a bookstore in Paris, he published the idea of geometrical interpretation of complex numbers known as the Argand diagram and is known for the first rigorous proof of the Fundamental Theorem of Algebra.
Dirac algebra
In mathematical physics, the Dirac algebra is the Clifford algebra Cℓ(C), which may be thought of as Cℓ(C). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation with the Dirac gamma matrices, which represent the generators of the algebra.
Algebrator
Algebrator (also called Softmath) is a computer algebra system (CAS), which was developed in the late 1990s by Neven Jurkovic of Softmath, San Antonio, Texas. This is a CAS specifically geared towards algebra education. Beside the computation results, it shows step by step the solution process and context sensitive explanations.
Algebra
The roots of algebra can be traced to the ancient Babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the "Rhind Mathematical Papyrus", Euclid's "Elements", and "The Nine Chapters on the Mathematical Art". The geometric work of the Greeks, typified in the "Elements", provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, although this would not be realized until mathematics developed in medieval Islam.
Emmy Noether
Noether's work continues to be relevant for the development of theoretical physics and mathematics and she is consistently ranked as one of the greatest mathematicians of the twentieth century. In his obituary, fellow algebraist BL van der Waerden says that her mathematical originality was "absolute beyond comparison", and Hermann Weyl said that Noether "changed the face of algebra by her work". During her lifetime and even until today, Noether has been characterized as the greatest woman mathematician in recorded history by mathematicians such as Pavel Alexandrov, Hermann Weyl, and Jean Dieudonné.
Boolean algebra (structure)
The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, "The Mathematical Analysis of Logic", published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, "The Laws of Thought", published in 1854. Boole's formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons and Charles Sanders Peirce. The first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 "Vorlesungen" of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whitehead's 1898 "Universal Algebra". Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 "Lattice Theory". In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.
Emmy Noether
In the century from 1832 to Noether's death in 1935, the field of mathematics – specifically algebra – underwent a profound revolution, whose reverberations are still being felt. Mathematicians of previous centuries had worked on practical methods for solving specific types of equations, e.g., cubic, quartic, and quintic equations, as well as on the related problem of constructing regular polygons using compass and straightedge. Beginning with Carl Friedrich Gauss's 1832 proof that prime numbers such as five can be factored in Gaussian integers, Évariste Galois's introduction of permutation groups in 1832 (although, because of his death, his papers were published only in 1846, by Liouville), William Rowan Hamilton's discovery of quaternions in 1843, and Arthur Cayley's more modern definition of groups in 1854, research turned to determining the properties of ever-more-abstract systems defined by ever-more-universal rules. Noether's most important contributions to mathematics were to the development of this new field, abstract algebra.
History of mathematics
Around the same time, Eratosthenes of Cyrene (c. 276–194 BC) devised the Sieve of Eratosthenes for finding prime numbers. The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline. Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers. Hipparchus of Nicaea (c. 190–120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle. Heron of Alexandria (c. 10–70 AD) is credited with Heron's formula for finding the area of a scalene triangle and with being the first to recognize the possibility of negative numbers possessing square roots. Menelaus of Alexandria (c. 100 AD) pioneered spherical trigonometry through Menelaus' theorem. The most complete and influential trigonometric work of antiquity is the "Almagest" of Ptolemy (c. AD 90–168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years. Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.
Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics. During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis, which is also known as "Diophantine analysis". The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the "Arithmetica", a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations. The "Arithmetica" had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the "Arithmetica" (that of dividing a square into two squares). Diophantus also made significant advances in notation, the "Arithmetica" being the first instance of algebraic symbolism and syncopation.
Among the last great Greek mathematicians is Pappus of Alexandria (4th century AD). He is known for his hexagon theorem and centroid theorem, as well as the Pappus configuration and Pappus graph. His "Collection" is a major source of knowledge on Greek mathematics as most of it has survived. Pappus is considered the last major innovator in Greek mathematics, with subsequent work consisting mostly of commentaries on earlier work.
Frobenius theorem (real division algebras)
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following:
These algebras have real dimension , and , respectively. Of these three algebras, and are commutative, but is not.
History of algebra
Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century.
Algebra
Algebra (from Arabic ""al-jabr"", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.
History of algebra
The 12th century saw a flood of translations from Arabic into Latin and by the 13th century, European mathematics was beginning to rival the mathematics of other lands. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra.
Beniamino Segre
Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry.
Emmy Noether
Noether also was responsible for a number of other advances in the field of algebra. With Emil Artin, Richard Brauer, and Helmut Hasse, she founded the theory of central simple algebras.
Generality of algebra
In the history of mathematics, the generality of algebra was a phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange, particularly in manipulating infinite series. According to Koetsier, the generality of algebra principle assumed, roughly, that the algebraic rules that hold for a certain class of expressions can be extended to hold more generally on a larger class of objects, even if the rules are no longer obviously valid. As a consequence, 18th century mathematicians believed that they could derive meaningful results by applying the usual rules of algebra and calculus that hold for finite expansions even when manipulating infinite expansions. In works such as "Cours d'Analyse", Cauchy rejected the use of "generality of algebra" methods and sought a more rigorous foundation for mathematical analysis.
Algebra
The word "algebra" comes from the Arabic ("" lit. "the reunion of broken parts") from the title of the book "Ilm al-jabr wa'l-muḳābala" by the Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the fifteenth century, from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century.
History of algebra
Diophantus was a Hellenistic mathematician who lived c. 250 CE, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written "Arithmetica", a treatise that was originally thirteen books but of which only the first six have survived. "Arithmetica" has very little in common with traditional Greek mathematics since it is divorced from geometric methods, and it is different from Babylonian mathematics in that Diophantus is concerned primarily with exact solutions, both determinate and indeterminate, instead of simple approximations.
Federigo Enriques
Abramo Giulio Umberto Federigo Enriques (5 January 1871 – 14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry.
Number
Also in 1799, Gauss provided the first generally accepted proof of the fundamental theorem of algebra, showing that every polynomial over the complex numbers has a full set of solutions in that realm. The general acceptance of the theory of complex numbers is due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known.
François Viète
In 1841, the French mathematician, Michel Chasles was one of the first to reevaluate his role in the development of modern algebra.
Equation
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: "Cartesius") revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane, centered on a particular point called the origin, may be described as the set of all points whose coordinates "x" and "y" satisfy the equation .
History of algebra
The geometric work of the Greeks, typified in Euclid's "Elements", provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations.
Leibniz algebra
In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module "L" over a commutative ring "R" with a bilinear product [ _ , _ ] satisfying the Leibniz identity
History of geometry
Al-Mahani (born 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Al-Karaji (born 953) completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today.
Ockham algebra
In mathematics, an Ockham algebra is a bounded distributive lattice with a dual endomorphism. They were introduced by , and were named after William of Ockham by . Ockham algebras form a variety.
History of algebra
In addition to the three stages of expressing algebraic ideas, some authors recognized four conceptual stages in the development of algebra that occurred alongside the changes in expression. These four stages were as follows:The origins of algebra can be traced to the ancient Babylonians, who developed a positional number system that greatly aided them in solving their rhetorical algebraic equations. The Babylonians were not interested in exact solutions but approximations, and so they would commonly use linear interpolation to approximate intermediate values. One of the most famous tablets is the Plimpton 322 tablet, created around 1900–1600 BC, which gives a table of Pythagorean triples and represents some of the most advanced mathematics prior to Greek mathematics.
History of algebra
Mathematical historians generally agree that the use of "x" in algebra was introduced by René Descartes and was first published in his treatise "La Géométrie" (1637). In that work, he used letters from the beginning of the alphabet ("a", "b", "c"...) for known quantities, and letters from the end of the alphabet ("z", "y", "x"...) for unknowns. It has been suggested that he later settled on "x" (in place of "z") for the first unknown because of its relatively greater abundance in the French and Latin typographical fonts of the time.
History of algebra
'Abd al-Hamīd ibn Turk authored a manuscript entitled "Logical Necessities in Mixed Equations", which is very similar to al-Khwarzimi's "Al-Jabr" and was published at around the same time as, or even possibly earlier than, "Al-Jabr". The manuscript gives exactly the same geometric demonstration as is found in "Al-Jabr", and in one case the same example as found in "Al-Jabr", and even goes beyond "Al-Jabr" by giving a geometric proof that if the discriminant is negative then the quadratic equation has no solution. The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.
Algebra
By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them. Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called "Arithmetica". These texts deal with solving algebraic equations, and have led, in number theory to the modern notion of Diophantine equation.
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra.
List of important publications in mathematics
This book by Tang dynasty mathematician Wang Xiaotong Contains the world's earliest third order equation.Contained rules for manipulating both negative and positive numbers, rules for dealing the number zero, a method for computing square roots, and general methods of solving linear and some quadratic equations, solution to Pell's equation.The first book on the systematic algebraic solutions of linear and quadratic equations by the Persian scholar Muhammad ibn Mūsā al-Khwārizmī. The book is considered to be the foundation of modern algebra and Islamic mathematics. The word "algebra" itself is derived from the "al-Jabr" in the title of the book.
Aloor, Kerala
A prominent Kerala mathematician-astronomer Sangamagrama Madhava (Mādhava)(AD1350 –1425) who founded the Kerala school of astronomy and mathematics was from Irinjalappilly, Aloor. Sangamagrama (Saṅgamagrāma) in medieval Kerala believed to be Irinjalakuda Brahminical grama which includes Aloor . Madhava of Sangamagrama is described as "the greatest mathematician-astronomer of medieval India" or as "the founder of mathematical analysis". He was the first to have developed infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".His discoveries opened the doors to what has today come to be known as Mathematical Analysis. One of the greatest mathematician-astronomers of the Middle Ages, Mādhavan made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry and algebra in his works Mahajyānayana prakāra ("Methods for the great sines") and Venuaroham.
His Contributions
History of algebra
In the early 15th century, Jamshīd al-Kāshī developed an early form of Newton's method to numerically solve the equation formula_37 to find roots of formula_38. Al-Kāshī also developed decimal fractions and claimed to have discovered it himself. However, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.
History of algebra
Bhāskara II (1114 – c. 1185) was the leading mathematician of the 12th century. In Algebra, he gave the general solution of Pell's equation. He is the author of "Lilavati" and "Vija-Ganita", which contain problems dealing with determinate and indeterminate linear and quadratic equations, and Pythagorean triples and he fails to distinguish between exact and approximate statements. Many of the problems in "Lilavati" and "Vija-Ganita" are derived from other Hindu sources, and so Bhaskara is at his best in dealing with indeterminate analysis.
Algebra
A mathematician who does research in algebra is called an algebraist.
Universal algebra
In Alfred North Whitehead's book "A Treatise on Universal Algebra," published in 1898, the term "universal algebra" had essentially the same meaning that it has today. Whitehead credits William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself.
Vector space
Vector spaces stem from affine geometry via the introduction of coordinates in the plane or three-dimensional space. Around 1636, Descartes and Fermat founded analytic geometry by equating solutions to an equation of two variables with points on a plane curve. In 1804, to achieve geometric solutions without using coordinates, Bolzano introduced certain operations on points, lines and planes, which are predecessors of vectors. His work was then used in the conception of barycentric coordinates by Möbius in 1827. In 1828 C. V. Mourey suggested the existence of an algebra surpassing not only ordinary algebra but also two-dimensional algebra created by him searching a geometrical interpretation of complex numbers.
Diophantus
Diophantus' work has had a large influence in history. Editions of Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the 17th and 18th centuries. Diophantus and his works have also influenced Arab mathematics and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. As far as we know Diophantus did not affect the lands of the Orient much and how much he affected India is a matter of debate.
Abraham Adrian Albert
Abraham Adrian Albert (November 9, 1905 – June 6, 1972) was an American mathematician. In 1939, he received the American Mathematical Society's Cole Prize in Algebra for his work on Riemann matrices. He is best known for his work on the Albert–Brauer–Hasse–Noether theorem on finite-dimensional division algebras over number fields and as the developer of Albert algebras, which are also known as exceptional Jordan algebras.
Algebra
Earlier traditions discussed above had a direct influence on the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c. 780–850). He later wrote "The Compendious Book on Calculation by Completion and Balancing", which established algebra as a mathematical discipline that is independent of geometry and arithmetic.
Pythagoras
Isaac Newton firmly believed in the Pythagorean teaching of the mathematical harmony and order of the universe. Though Newton was notorious for rarely giving others credit for their discoveries, he attributed the discovery of the Law of Universal Gravitation to Pythagoras. Albert Einstein believed that a scientist may also be "a Platonist or a Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research." The English philosopher Alfred North Whitehead argued that "In a sense, Plato and Pythagoras stand nearer to modern physical science than does Aristotle. The two former were mathematicians, whereas Aristotle was the son of a doctor". By this measure, Whitehead declared that Einstein and other modern scientists like him are "following the pure Pythagorean tradition."
Abbasid Caliphate
Algebra was significantly developed by Persian scientist Muhammad ibn Mūsā al-Khwārizmī during this time in his landmark text, "Kitab al-Jabr wa-l-Muqabala", from which the term "algebra" is derived. He is thus considered to be the father of algebra by some, although the Greek mathematician Diophantus has also been given this title. The terms algorism and algorithm are derived from the name of al-Khwarizmi, who was also responsible for introducing the Arabic numerals and Hindu-Arabic numeral system beyond the Indian subcontinent.
Arab scientist Ibn al-Haytham (Alhazen) developed an early scientific method in his "Book of Optics" (1021). The most important development of the scientific method was the use of experiments to distinguish between competing scientific theories set within a generally empirical orientation, which began among Muslim scientists. Ibn al-Haytham's empirical proof of the intromission theory of light (that is, that light rays entered the eyes rather than being emitted by them) was particularly important. Alhazen was significant in the history of scientific method, particularly in his approach to experimentation, and has been referred to as the "world's first true scientist".
Apollonius of Perga
Heath goes on to use the term geometrical algebra for the methods of the entire golden age. The term is “not inappropriately” called that, he says. Today the term has been resurrected for use in other senses (see under geometric algebra). Heath was using it as it had been defined by Henry Burchard Fine in 1890 or before. Fine applies it to La Géométrie of René Descartes, the first full-blown work of analytic geometry. Establishing as a precondition that “two algebras are formally identical whose fundamental operations are formally the same,” Fine says that Descartes’ work “is not ... mere numerical algebra, but what may for want of a better name be called the algebra of line segments. Its symbolism is the same as that of numerical algebra; ...”
Garrett Birkhoff
During the 1930s, Birkhoff, along with his Harvard colleagues Marshall Stone and Saunders Mac Lane, substantially advanced American teaching and research in abstract algebra. In 1941 he and Mac Lane published "A Survey of Modern Algebra", the second undergraduate textbook in English on the subject (Cyrus Colton MacDuffee's "An Introduction to Abstract Algebra" was published in 1940). Mac Lane and Birkhoff's "Algebra" (1967) is a more advanced text on abstract algebra. A number of papers he wrote in the 1930s, culminating in his monograph, "Lattice Theory" (1940; the third edition remains in print), turned lattice theory into a major branch of abstract algebra. His 1935 paper, "On the Structure of Abstract Algebras" founded a new branch of mathematics, universal algebra. Birkhoff's approach to this development of universal algebra and lattice theory acknowledged prior ideas of Charles Sanders Peirce, Ernst Schröder, and Alfred North Whitehead; in fact, Whitehead had written an 1898 monograph entitled "Universal Algebra". Further, in 1935, Birkoff showed that any equivalence between expressions that holds for all possible forms of operator must have a finite proof using certain underlying rules about equality. However, as soon as one introduces actual axioms that constrain the operators this is no longer true—and in general it can be undecidable whether or not a particular equivalence holds.
History of algebra
Omar Khayyám (c. 1050 – 1123) wrote a book on Algebra that went beyond "Al-Jabr" to include equations of the third degree. Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic solutions were impossible. His method of solving cubic equations by using intersecting conics had been used by Menaechmus, Archimedes, and Ibn al-Haytham (Alhazen), but Omar Khayyám generalized the method to cover all cubic equations with positive roots. He only considered positive roots and he did not go past the third degree. He also saw a strong relationship between Geometry and Algebra.
History of algebra
The Hellenistic mathematician Diophantus has traditionally been known as "the father of algebra" but debate now exists as to whether or not Al-Khwarizmi deserves this title instead. Those who support Diophantus point to the fact that the algebra found in "Al-Jabr" is more elementary than the algebra found in "Arithmetica" and that "Arithmetica" is syncopated while "Al-Jabr" is fully rhetorical.
Algebra
The Hellenistic mathematicians Hero of Alexandria and Diophantus as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though Diophantus' "Arithmetica" and Brahmagupta's "Brāhmasphuṭasiddhānta" are on a higher level. For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book "Brahmasphutasiddhanta". Later, Persian and Arabic mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special "ad hoc" methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations.
Augustus De Morgan
George Peacock's theory of algebra was much improved by D. F. Gregory, a younger member of the Cambridge School, who laid stress not on the permanence of equivalent forms, but on the permanence of certain formal laws. This new theory of algebra as the science of symbols and of their laws of combination was carried to its logical issue by De Morgan; and his doctrine on the subject is still followed by English algebraists in general. Thus George Chrystal founds his "Textbook of Algebra" on De Morgan's theory; although an attentive reader may remark that he practically abandons it when he takes up the subject of infinite series. De Morgan's theory is stated in his volume on "Trigonometry and Double Algebra", where in Book II, Chapter II, headed "On symbolic algebra", he writes:
Euclid
Euclid (; – "Eukleídēs", ; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His "Elements" is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the "Elements", Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and rigor.
Henk J. M. Bos
Bos has contributed to the study of the mathematical works of the seventeenth-century philosopher René Descartes, including Descartes’ contribution to the development of algebra and geometry.
History of algebra
There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.
Albert Girard
Albert Girard () (1595 in Saint-Mihiel, France − 8 December 1632 in Leiden) was a French-born mathematician. He studied at the University of Leiden. He "had early thoughts on the fundamental theorem of algebra" and gave the inductive definition for the Fibonacci numbers.
He was the first to use the abbreviations 'sin', 'cos' and 'tan' for the trigonometric functions in a treatise. Girard was the first to state, in 1625, that each prime of the form 1 mod 4 is the sum of two squares. (See Fermat's theorem on sums of two squares.) It was said that he was quiet-natured and, unlike most mathematicians, did not keep a journal for his personal life.
Emmy Noether
Amalie Emmy Noether (; 23 March 1882 – 14 April 1935) was a German mathematician who made important contributions to abstract algebra and theoretical physics. She invariably used the name "Emmy Noether" in her life and publications. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed the theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.
History of algebra
Euclid (Greek: ) was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BCE). Neither the year nor place of his birth have been established, nor the circumstances of his death.
Culture of Greece
The period of Classical Greece (from 800BC until the rise of Macedon, a Greek state in the north) is that most often associated with Greek advances in science. Thales of Miletus is regarded by many as the father of science; he was the first of the ancient philosophers to seek to explain the physical world in terms of natural rather than supernatural causes. Pythagoras was a mathematician often described as the "father of numbers"; it is believed that he had the pioneering insight into the numerical ratios that determine the musical scale, and the Pythagorean theorem is commonly attributed to him. Diophantus of Alexandria, in turn, was the "father of algebra". Many parts of modern geometry are based on the work of Euclid, while Eratosthenes was one of the first scientific geographers, calculating the circumference of the earth and conceiving the first maps based on scientific principles.
Rota–Baxter algebra
In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map "R" which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gian-Carlo Rota, Pierre Cartier, and Frederic V. Atkinson, among others. Baxter’s derivation of this identity that later bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory.
History of mathematics
Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others. His Platonic Academy, in Athens, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus, came. Plato also discussed the foundations of mathematics, clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions. The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.
History of science
In mathematics, the mathematician Muhammad ibn Musa al-Khwarizmi (c. 780–850) gave his name to the concept of the algorithm, while the term algebra is derived from "al-jabr", the beginning of the title of one of his publications. What is now known as Arabic numerals originally came from India, but Muslim mathematicians made several key refinements to the number system, such as the introduction of decimal point notation.
House of Wisdom
Besides their translations of earlier works and their commentaries on them, scholars at the Bayt al-Hikma produced important original research. For example, the noted mathematician al-Khwarizmi worked in al-Ma'mun's House of Wisdom and is famous for his contributions to the development of algebra. Muhammad ibn Musa al-Khwarizmi was born around 780 and died around 850. He was known as a mathematician and an astronomer in the House of Wisdom, and is also known for his book "Kitab al-Jabr" in which he develops a number of algorithms. The application of the word "algebra" to mathematics and the etymology of the word "algorithm" can be traced back to al-Khwarizmi — the actual concept of an algorithm dates back before the time of Euclid. The term 'algorithm' is derived from a Latin version of al-Khwarizmi's name which is 'Algorithmus'. He was instrumental in introducing the Arabs to Hindu numerals and algebra so he is known as "The Father of Algebra". Additionally, al-Khwarizmi wrote about the astrolabe, sundial, and elaborated on Ptolemy's geometric model. Al-Khwarizmi is also known as the first geographer of Islam with his famous "Picture of the Earth" treatise. In "Picture of the Earth", he arranged the coordinates of hundreds of cities in the world at that time and gave instructions for drawing a new map of the world. George Sarton, one the most famous historians of science known for his book, "Introduction to the History of Science", called the period between 800 and 850 AD "The Time of al-Khwarizmi". Besides that, this mathematician is responsible for the introduction of the Hindu decimal system to the Arab world, and through it to Europe. Al-Kindi also made important breakthroughs in cryptanalysis. In al-Khwarizmi's early career, he proposed ideas towards the Hindu astronomical tables known as Sindhind. As a result, Caliph Al-Ma'mun sought al-Khwarizmi out to work on the science of equations.
Plato
Plato (; "Plátōn", in Classical Attic; 428/427 or 424/423 – 348/347 BC) was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world. He is widely considered the pivotal figure in the development of Western philosophy. Unlike nearly all of his philosophical contemporaries, Plato's entire work is believed to have survived intact for over 2,400 years.
Philbert Maurice d'Ocagne
Philbert Maurice d'Ocagne (25 March 1862 – 23 September 1938) was a French engineer and mathematician. He founded the field of nomography, the graphic computation of algebraic equations, on charts which he called a nomogram.
Calabi–Yau algebra
In algebra, a Calabi–Yau algebra was introduced by Victor Ginzburg to transport the geometry of a Calabi–Yau manifold to noncommutative algebraic geometry.
Muhammad ibn Musa al-Khwarizmi
Al-Khwarizmi's popularizing treatise on algebra ("The Compendious Book on Calculation by Completion and Balancing", c. 813–833 CE) presented the first systematic solution of linear and quadratic equations. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because he was the first to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation), he has been described as the father or founder of algebra. The term "algebra" itself comes from the title of his book (specifically the word "al-jabr" meaning "completion" or "rejoining"). His name gave rise to the terms "algorism" and "algorithm". His name is also the origin of (Spanish) "guarismo" and of (Portuguese) "algarismo", both meaning digit.
Boolean algebra
Boolean algebra was introduced by George Boole in his first book "The Mathematical Analysis of Logic" (1847), and set forth more fully in his "An Investigation of the Laws of Thought" (1854).
According to Huntington, the term "Boolean algebra" was first suggested by Sheffer in 1913,
although Charles Sanders Peirce in 1880 gave the title "A Boolian Algebra with One Constant" to the first chapter of his "The Simplest Mathematics".
Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages. It is also used in set theory and statistics.
Journal of Pure and Applied Algebra
The Journal of Pure and Applied Algebra is a monthly peer-reviewed scientific journal covering that part of algebra likely to be of general mathematical interest: algebraic results with immediate applications, and the development of algebraic theories of sufficiently general relevance to allow for future applications. Its founding editors-in-chief were P. Freyd (University of Pennsylvania) and A. Heller (City University of New York). The current managing editors are Eric Friedlander (University of Southern California), Chuck Weibel (Rutgers University), and Srikanth Iyengar (University of Utah).
Hossein Zakeri
Zakeri was named the father of commutative algebra of Iran in 2012, for his efforts and contributions in Commutative Algebra to Iranian mathematical community. The 10th seminar of "Commutative Algebra and Related Topics" of Institute for Research in Fundamental Sciences (December 2013), and the 24th Iranian Algebra Seminar (November 2014), were held in honour of him, with many of his former students and outstanding members of universities participating in it.
History of algebra
The word "algebra" is derived from the Arabic word الجبر "al-jabr", and this comes from the treatise written in the year 830 by the medieval Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī, whose Arabic title, "Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala", can be translated as "The Compendious Book on Calculation by Completion and Balancing". The treatise provided for the systematic solution of linear and quadratic equations. According to one history, "[i]t is not certain just what the terms "al-jabr" and "muqabalah" mean, but the usual interpretation is similar to that implied in the previous translation. The word 'al-jabr' presumably meant something like 'restoration' or 'completion' and seems to refer to the transposition of subtracted terms to the other side of an equation; the word 'muqabalah' is said to refer to 'reduction' or 'balancing'—that is, the cancellation of like terms on opposite sides of the equation. Arabic influence in Spain long after the time of al-Khwarizmi is found in "Don Quixote", where the word 'algebrista' is used for a bone-setter, that is, a 'restorer'." The term is used by al-Khwarizmi to describe the operations that he introduced, "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.
History of algebra
In "Arithmetica", Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations; thus he used what is now known as "syncopated" algebra. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. So, for example, what we would write as
Diophantus would have written this as
where the symbols represent the following:
Note that the coefficients come after the variables and that addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's syncopated equation into a modern symbolic equation would be the following:
Commutation theorem
The theory of Hilbert algebras was introduced by Godement (under the name "unitary algebras"), Segal and Dixmier to formalize the classical method of defining the trace for trace class operators starting from Hilbert–Schmidt operators. Applications in the representation theory of groups naturally lead to examples of Hilbert algebras. Every von Neumann algebra endowed with a semifinite trace has a canonical "completed" or "full" Hilbert algebra associated with it; and conversely a completed Hilbert algebra of exactly this form can be canonically associated with every Hilbert algebra. The theory of Hilbert algebras can be used to deduce the commutation theorems of Murray and von Neumann; equally well the main results on Hilbert algebras can also be deduced directly from the commutation theorems for traces. The theory of Hilbert algebras was generalised by Takesaki as a tool for proving commutation theorems for semifinite weights in Tomita–Takesaki theory; they can be dispensed with when dealing with states.
Benjamin Fillon
Benjamin Fillon (15 March 1819 – 23 May 1881) was a French numismatist and archaeologist. Much of his lifetime's work was devoted to researching the French mathematician, Franciscus Vieta, a key figure in developing new algebra.
Boolean algebra
Boole's algebra predated the modern developments in abstract algebra and mathematical logic; it is however seen as connected to the origins of both fields. In an abstract setting, Boolean algebra was perfected in the late 19th century by Jevons, Schröder, Huntington, and others until it reached the modern conception of an (abstract) mathematical structure. For example, the empirical observation that one can manipulate expressions in the algebra of sets by translating them into expressions in Boole's algebra is explained in modern terms by saying that the algebra of sets is "a" Boolean algebra (note the indefinite article). In fact, M. H. Stone proved in 1936 that every Boolean algebra is isomorphic to a field of sets.
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. 
The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.
Fundamental theorem of algebra
A first attempt at proving the theorem was made by d'Alembert in 1746, but his proof was incomplete. Among other problems, it assumed implicitly a theorem (now known as Puiseux's theorem) which would not be proved until more than a century later, and furthermore the proof assumed the fundamental theorem of algebra. Other attempts were made by Euler (1749), de Foncenex (1759), Lagrange (1772), and Laplace (1795). These last four attempts assumed implicitly Girard's assertion; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was "a" + "bi" for some real numbers "a" and "b". In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial "p"("z").
History of algebra
"Ssy-yüan yü-chien"《四元玉鑒》, or "Precious Mirror of the Four Elements", was written by Chu Shih-chieh in 1303 and it marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. The "Ssy-yüan yü-chien" deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of "fan fa", today called Horner's method, to solve these equations.
Von Neumann algebra
Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.
Lindenbaum–Tarski algebra
The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski. 
It was first introduced by Tarski in 1935
as a device to establish correspondence between classical propositional calculus and Boolean algebras. 
The Lindenbaum–Tarski algebra is considered the origin of the modern algebraic logic.
Mathematics in medieval Islam
The study of algebra, the name of which is derived from the Arabic word meaning completion or "reunion of broken parts", flourished during the Islamic golden age. Muhammad ibn Musa al-Khwarizmi, a scholar in the House of Wisdom in Baghdad, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book "The Compendious Book on Calculation by Completion and Balancing", Al-Khwarizmi deals with ways to solve for the positive roots of first and second degree (linear and quadratic) polynomial equations. He also introduces the method of reduction, and unlike Diophantus, gives general solutions for the equations he deals with.
François Viète
Above all, Vieta was the first mathematician who introduced notations for the problem (and not just for the unknowns). As a result, his algebra was no longer limited to the statement of rules, but relied on an efficient computer algebra, in which the operations act on the letters and the results can be obtained at the end of the calculations by a simple replacement. This approach, which is the heart of contemporary algebraic method, was a fundamental step in the development of mathematics. With this, Vieta marked the end of medieval algebra (from Al-Khwarizmi to Stevin) and opened the modern period.
Copernican Revolution
Mohammed ibn Musa al-Khowarizimi, developed algebra and his work would help develop European algebra. Arabic algebra conversion expanded Copernicus understanding to mathematics.
Jean le Rond d'Alembert
In France, the fundamental theorem of algebra is known as the d'Alembert/Gauss theorem, as an error in d'Alembert's proof was caught by Gauss.
Italian school of algebraic geometry
According to Guerraggio & Nastasi (page 9, 2005) Luigi Cremona is "considered the founder of the Italian school of algebraic geometry". Later they explain that in Turin the collaboration of Enrico D'Ovidio and Corrado Segre "would bring, either by their own efforts or those of their students, Italian algebraic geometry to full maturity". A one-time student of Segre, H.F. Baker wrote (1926, page 269), [Corrado Segre] "may probably be said to be the father of that wonderful Italian school which has achieved so much in the birational theory of algebraical loci." On this topic, Brigaglia & Ciliberto (2004) say "Segre had headed and maintained the school of geometry that Luigi Cremona had established in 1860." Reference to the Mathematics Genealogy Project shows that, in terms of "Italian doctorates", the real productivity of the school began with Guido Castelnuovo and Federigo Enriques. In the USA Oscar Zariski inspired many Ph.D.s.
Journal of Algebra
Journal of Algebra (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. "Journal of Algebra" was founded by Graham Higman, who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor-in-chief.
Geometric algebra
Also developed are the concept of vector manifold and geometric integration theory (which generalizes Cartan's differential forms).Although the connection of geometry with algebra dates as far back at least to Euclid's "Elements" in the third century B.C. (see Greek geometric algebra), GA in the sense used in this article was not developed until 1844, when it was used in a "systematic way" to describe the geometrical properties and "transformations" of a space. In that year, Hermann Grassmann introduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to the propositional calculus) that encoded all of the geometrical information of a space. Grassmann's algebraic system could be applied to a number of different kinds of spaces, the chief among them being Euclidean space, affine space, and projective space. Following Grassmann, in 1878 William Kingdon Clifford examined Grassmann's algebraic system alongside the quaternions of William Rowan Hamilton in . From his point of view, the quaternions described certain "transformations" (which he called "rotors"), whereas Grassmann's algebra described certain "properties" (or "Strecken" such as length, area, and volume). His contribution was to define a new product — the "geometric product" – on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently, Rudolf Lipschitz in 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations in formula_102 dimensions. Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra.
Linear algebra
There is a strong relationship between linear algebra and geometry, which started with the introduction by René Descartes, in 1637, of Cartesian coordinates. In this new (at that time) geometry, now called Cartesian geometry, points are represented by Cartesian coordinates, which are sequences of three real numbers (in the case of the usual three-dimensional space). The basic objects of geometry, which are lines and planes are represented by linear equations. Thus, computing intersections of lines and planes amounts solving systems of linear equations. This was one of the main motivations for developing linear algebra.
René Descartes
One of Descartes' most enduring legacies was his development of Cartesian or analytic geometry, which uses algebra to describe geometry. He "invented the convention of representing unknowns in equations by "x", "y", and "z", and knowns by "a", "b", and "c"". He also "pioneered the standard notation" that uses superscripts to show the powers or exponents; for example, the 2 used in x to indicate x squared. He was first to assign a fundamental place for algebra in our system of knowledge, using it as a method to automate or mechanize reasoning, particularly about abstract, unknown quantities. European mathematicians had previously viewed geometry as a more fundamental form of mathematics, serving as the foundation of algebra. Algebraic rules were given geometric proofs by mathematicians such as Pacioli, Cardan, Tartaglia and Ferrari. Equations of degree higher than the third were regarded as unreal, because a three-dimensional form, such as a cube, occupied the largest dimension of reality. Descartes professed that the abstract quantity "a" could represent length as well as an area. This was in opposition to the teachings of mathematicians, such as Vieta, who argued that it could represent only area. Although Descartes did not pursue the subject, he preceded Gottfried Wilhelm Leibniz in envisioning a more general science of algebra or "universal mathematics," as a precursor to symbolic logic, that could encompass logical principles and methods symbolically, and mechanize general reasoning.
René Descartes
Descartes' "Meditations on First Philosophy" (1641) continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system (see below) was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry, used in the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution.
Philosophy of mathematics
Gottlob Frege was the founder of logicism. In his seminal "Die Grundgesetze der Arithmetik" ("Basic Laws of Arithmetic") he built up arithmetic from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts "F" and "G", the extension of "F" equals the extension of "G" if and only if for all objects "a", "Fa" equals "Ga"), a principle that he took to be acceptable as part of logic.
Frege's construction was flawed. Russell discovered that Basic Law V is inconsistent (this is Russell's paradox). Frege abandoned his logicist program soon after this, but it was continued by Russell and Whitehead. They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex form (for example, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop so much of mathematics, such as an "axiom of reducibility". Even Russell said that this axiom did not really belong to logic.
History of algebra
Those who support Al-Khwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, and was the first to teach algebra in an elementary form and for its own sake, whereas Diophantus was primarily concerned with the theory of numbers. Al-Khwarizmi also introduced the fundamental concept of "reduction" and "balancing" (which he originally used the term "al-jabr" to refer to), referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. Other supporters of Al-Khwarizmi point to his algebra no longer being concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." They also point to his treatment of an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."
History of algebra
Aryabhata (476–550) was an Indian mathematician who authored "Aryabhatiya". In it he gave the rules,
and
Geometric algebra
The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics.
History of mathematics
In the 9th century, the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote several important books on the Hindu–Arabic numerals and on methods for solving equations. His book "On the Calculation with Hindu Numerals", written about 825, along with the work of Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. The word "algorithm" is derived from the Latinization of his name, Algoritmi, and the word "algebra" from the title of one of his works, "Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala" ("The Compendious Book on Calculation by Completion and Balancing"). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, and he was the first to teach algebra in an elementary form and for its own sake. He also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as "al-jabr". His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."
Pythagoras
During the Middle Ages, Pythagoras was revered as the founder of mathematics and music, two of the Seven Liberal Arts. He appears in numerous medieval depictions, in illuminated manuscripts and in the relief sculptures on the portal of the Cathedral of Chartres. The "Timaeus" was the only dialogue of Plato to survive in Latin translation in western Europe, which led William of Conches (c. 1080-1160) to declare that Plato was Pythagorean. In the 1430s, the Camaldolese friar Ambrose Traversari translated Diogenes Laërtius's "Lives and Opinions of Eminent Philosophers" from Greek into Latin and, in the 1460s, the philosopher Marsilio Ficino translated Porphyry and Iamblichus's "Lives of Pythagoras" into Latin as well, thereby allowing them to be read and studied by western scholars. In 1494, the Greek Neopythagorean scholar Constantine Lascaris published "The Golden Verses of Pythagoras", translated into Latin, with a printed edition of his "Grammatica", thereby bringing them to a widespread audience. In 1499, he published the first Renaissance biography of Pythagoras in his work "Vitae illustrium philosophorum siculorum et calabrorum", issued in Messina.
History of algebra
Brahmagupta (fl. 628) was an Indian mathematician who authored "Brahma Sphuta Siddhanta". In his work Brahmagupta solves the general quadratic equation for both positive and negative roots. In indeterminate analysis Brahmagupta gives the Pythagorean triads formula_19, formula_20, formula_21, but this is a modified form of an old Babylonian rule that Brahmagupta may have been familiar with. He was the first to give a general solution to the linear Diophantine equation ax + by = c, where a, b, and c are integers. Unlike Diophantus who only gave one solution to an indeterminate equation, Brahmagupta gave "all" integer solutions; but that Brahmagupta used some of the same examples as Diophantus has led some historians to consider the possibility of a Greek influence on Brahmagupta's work, or at least a common Babylonian source.
Geometric Algebra
Geometric Algebra is a book written by Emil Artin and published by Interscience Publishers, New York, in 1957. It was republished in 1988 in the Wiley Classics series ().
R. H. Bruck
Richard Hubert Bruck (December 26, 1914 – 1991) was an American mathematician best known for his work in the field of algebra, especially in its relation to projective geometry and combinatorics.
Jacques Dixmier
Often there is made the erroneous claim that Dixmier originated the name "von Neumann algebra" for the operator algebras introduced by John von Neumann, but Dixmier said in an interview that the name originated from a proposal by Jean Dieudonné.
Dixmier was an invited speaker at the International Congress of Mathematicians in 1966 at Moscow with talk "Espace dual d'une algèbre, ou d'un groupe localement compact" and again in 1978 at Helsinki with talk "Algèbres enveloppantes".
History of algebra
Iamblichus in "Introductio arithmatica" tells us that Thymaridas (c. 400 BCE – c. 350 BCE) worked with simultaneous linear equations. In particular, he created the then famous rule that was known as the "bloom of Thymaridas" or as the "flower of Thymaridas", which states that:
If the sum of "n" quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/ (n - 2) of the difference between the sums of these pairs and the first given sum.
Hodge algebra
In mathematics, a Hodge algebra or algebra with straightening law is a commutative algebra that is a free module over some ring "R", together with a given basis similar to the basis of standard monomials of the coordinate ring of a Grassmannian. Hodge algebras were introduced by , who named them after W. V. D. Hodge.