Who introduced the concept of complex numbers?
Complex number
The complex numbers give rise to the fundamental theorem of algebra: every non-constant polynomial equation with complex coefficients has a complex solution. This property is true of the complex numbers, but not the reals. The 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations.
Complex analysis
Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has become very popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis is in string theory which studies conformal invariants in quantum field theory.
Complex number
William Rowan Hamilton introduced the approach to define the set of complex numbers as the set of of real numbers, in which the following rules for addition and multiplication are imposed:
Complex number
The "complex conjugate" of the complex number is given by . It is denoted by either formula_23 or . This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.
Number
This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When René Descartes coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation
seemed capriciously inconsistent with the algebraic identity
which is valid for positive real numbers "a" and "b", and was also used in complex number calculations with one of "a", "b" positive and the other negative. The incorrect use of this identity, and the related identity
in the case when both "a" and "b" are negative even bedeviled Euler. This difficulty eventually led him to the convention of using the special symbol "i" in place of formula_19 to guard against this mistake.
Complex-base system
In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955) or complex number (proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965).
Complex number
Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function is related to the distribution of prime numbers.
Struve function
introduced by . The complex number α is the order of the Struve function, and is often an integer. The modified Struve functions are equal to .
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.
Number
The 18th century saw the work of Abraham de Moivre and Leonhard Euler. De Moivre's formula (1730) states:
while Euler's formula of complex analysis (1748) gave us:
Number
Gauss studied complex numbers of the form , where "a" and "b" are integral, or rational (and "i" is one of the two roots of ). His student, Gotthold Eisenstein, studied the type , where "ω" is a complex root of Other such classes (called cyclotomic fields) of complex numbers derive from the roots of unity for higher values of "k". This generalization is largely due to Ernst Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893.
Split-complex number
In 1935 J.C. Vignaux and A. Durañona y Vedia developed the split-complex geometric algebra and function theory in four articles in "Contribución a las Ciencias Físicas y Matemáticas", National University of La Plata, República Argentina (in Spanish). These expository and pedagogical essays presented the subject for broad appreciation.
Complex number
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren, Français and his brother, Bellavitis.
Integral
The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.
While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits. Integration was first rigorously formalized, using limits, by Riemann. Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real analysis). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system.
Number theory
Algebraic number theory may be said to start with the study of reciprocity and cyclotomy, but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory is Dirichlet's theorem on arithmetic progressions (1837), whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable. The first use of analytic ideas in number theory actually
goes back to Euler (1730s), who used formal power series and non-rigorous (or implicit) limiting arguments. The use of "complex" analysis in number theory comes later: the work of Bernhard Riemann (1859) on the zeta function is the canonical starting point; Jacobi's four-square theorem (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory (modular forms).
Complex number
The set of all complex numbers is denoted by formula_6 (upright bold) or formula_7 (blackboard bold).
Split-complex number
The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines. William Kingdon Clifford used split-complex numbers to represent sums of spins. Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions. He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from the circle group. Extending the analogy, functions of a motor variable contrast to functions of an ordinary complex variable.
A Course of Pure Mathematics
III COMPLEX NUMBERS
Gaussian integer
The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity (1832). The theorem of quadratic reciprocity (which he had first succeeded in proving in 1796) relates the solvability of the congruence to that of . Similarly, cubic reciprocity relates the solvability of to that of , and biquadratic (or quartic) reciprocity is a relation between and . Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about "whole complex numbers" (i.e. the Gaussian integers) than they are as statements about ordinary whole numbers (i.e. the integers).
Complex number
Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.
Ehrhart polynomial
This series can be expressed as a rational function. Specifically, Ehrhart proved (1962) that there exist complex numbers , , such that the Ehrhart series of is
Complex number
Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law
must hold for any three elements , and of a field. The set of real numbers does form a field. A polynomial with real coefficients is an expression of the form
where the are real numbers. The usual addition and multiplication of polynomials endows the set of all such polynomials with a ring structure. This ring is called the polynomial ring over the real numbers.
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz (sometimes spelled Leibnitz) (; or ; ; – 14 November 1716) was a prominent German polymath and philosopher in the history of mathematics and the history of philosophy. His most notable accomplishment was conceiving the ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments. Mathematical works have always favored Leibniz's notation as the conventional expression of calculus, while Newton's notation became unused. It was only in the 20th century that Leibniz's law of continuity and transcendental law of homogeneity found mathematical implementation (by means of non-standard analysis). He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of all digital computers.
Ordinal number
Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and to classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.
Infinite-valued logic
Isaac Newton and Gottfried Wilhelm Leibniz used both infinities and infinitesimals to develop the differential and integral calculus in the late 17th century. Richard Dedekind, who defined real numbers in terms of certain sets of rational numbers in the 19th century, also developed an axiom of continuity stating that a single correct value exists at the limit of any trial and error approximation. Felix Hausdorff demonstrated the logical possibility of an absolutely continuous ordering of words comprising bivalent values, each word having absolutely infinite length, in 1938. However, the definition of a random real number, meaning a real number that has no finite description whatsoever, remains somewhat in the realm of paradox.
Georg Cantor
The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper, "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"). This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements). Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable. His proof differs from diagonal argument that he gave in 1891. Cantor's article also contains a new method of constructing transcendental numbers. Transcendental numbers were first constructed by Joseph Liouville in 1844.
Hypercomplex number
In mathematics, a hypercomplex number is a traditional term for an element of a unital algebra over the field of real numbers. 
The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.
Maurice René Fréchet
Maurice Fréchet (; 2 September 1878 – 4 June 1973) was a French mathematician. He made major contributions to the topology of point sets and introduced the entire concept of metric spaces. He also made several important contributions to the field of statistics and probability, as well as calculus. His dissertation opened the entire field of functionals on metric spaces and introduced the notion of compactness. Independently of Riesz, he discovered the representation theorem in the space of Lebesgue square integrable functions.
Rafael Bombelli
Perhaps more importantly than his work with algebra, however, the book also includes Bombelli's monumental contributions to complex number theory. Before he writes about complex numbers, he points out that they occur in solutions of equations of the form formula_1 given that formula_2 which is another way of stating that the discriminant of the cubic is negative. The solution of this kind of equation requires taking the cube root of the sum of one number and the square root of some negative number.
Mathematical logic
Georg Cantor developed the fundamental concepts of infinite set theory. His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities (Cantor 1874). Over the next twenty years, Cantor developed a theory of transfinite numbers in a series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the diagonal argument, and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset. Cantor believed that every set could be well-ordered, but was unable to produce a proof for this result, leaving it as an open problem in 1895 (Katz 1998, p. 807).
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors.
Binary number
In November 1937, George Stibitz, then working at Bell Labs, completed a relay-based computer he dubbed the "Model K" (for "Kitchen", where he had assembled it), which calculated using binary addition. Bell Labs authorized a full research program in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed 8 January 1940, was able to calculate complex numbers. In a demonstration to the American Mathematical Society conference at Dartmouth College on 11 September 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a teletype. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were John von Neumann, John Mauchly and Norbert Wiener, who wrote about it in his memoirs.
Complex number
In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.
Complex number
The set of complex numbers is defined as the quotient ring . This extension field contains two square roots of , namely (the cosets of) and , respectively. (The cosets of) and form a basis of as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs of real numbers. The quotient ring is a field, because is irreducible over , so the ideal it generates is maximal.
Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and had sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy. He is known for his numerous contributions to calculus, and along with his brother Johann, was one of the founders of the calculus of variations. He also discovered the fundamental mathematical constant e. However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work "Ars Conjectandi".
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.
Kurt Hensel
Hensel is well known for his introduction of "p"-adic numbers. First described by him in 1897, they became increasingly important in number theory and other fields during the twentieth century.
Paradoxes of set theory
In 1905, the Hungarian mathematician Julius König published a paradox based on the fact that there are only countably many finite definitions. If we imagine the real numbers as a well-ordered set, those real numbers which can be finitely defined form a subset. Hence in this well-order there should be a first real number that is not finitely definable. This is paradoxical, because this real number has just been finitely defined by the last sentence. This leads to a contradiction in naive set theory.
Henri Laborit
Laborit was one of the pioneers of complexity theory and self-organization in France and the initiator of "complex thought" in his meetings with the "Groupe des Dix". "Complex thought" was later popularized by Edgar Morin.
Bicomplex number
In 1892 Corrado Segre introduced bicomplex numbers in "Mathematische Annalen", which form an algebra isomorphic to the tessarines.
Hypercomplex number
The cataloguing project began in 1872 when Benjamin Peirce first published his "Linear Associative Algebra", and was carried forward by his son Charles Sanders Peirce. Most significantly, they identified the nilpotent and the idempotent elements as useful hypercomplex numbers for classifications. The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals ℝ, the complexes ℂ, the quaternions ℍ, and the octonions 𝕆, and the Frobenius theorem says the only real associative division algebras are ℝ, ℂ, and ℍ. In 1958 J. Frank Adams published a further generalization in terms of Hopf invariants on "H"-spaces which still limits the dimension to 1, 2, 4, or 8.
Georg Cantor
Between 1879 and 1884, Cantor published a series of six articles in "Mathematische Annalen" that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole. Cantor also introduced the Cantor set during this period.
Hermann Grassmann
One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was Hermann Hankel, whose 1867 "Theorie der complexen Zahlensysteme"
Laurent series
In mathematics, the Laurent series of a complex function "f"("z") is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where
a Taylor series expansion cannot be applied. The Laurent series was named
after and first published by Pierre Alphonse Laurent in 1843.
Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.
Intelligent design
In 1986, Charles B. Thaxton, a physical chemist and creationist, used the term "specified complexity" from information theory when claiming that messages transmitted by DNA in the cell were specified by intelligence, and must have originated with an intelligent agent.
The intelligent design concept of "specified complexity" was developed in the 1990s by mathematician, philosopher, and theologian William A. Dembski. Dembski states that when something exhibits specified complexity (i.e., is both complex and "specified", simultaneously), one can infer that it was produced by an intelligent cause (i.e., that it was designed) rather than being the result of natural processes. He provides the following examples: "A single letter of the alphabet is specified without being complex. A long sentence of random letters is complex without being specified. A Shakespearean sonnet is both complex and specified." He states that details of living things can be similarly characterized, especially the "patterns" of molecular sequences in functional biological molecules such as DNA.
Dembski defines complex specified information (CSI) as anything with a less than 1 in 10 chance of occurring by (natural) chance. Critics say that this renders the argument a tautology: complex specified information cannot occur naturally because Dembski has defined it thus, so the real question becomes whether or not CSI actually exists in nature.
Bicomplex number
The subject of multiple imaginary units was examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 in Philosophical Magazine, William Rowan Hamilton communicated a system multiplying according to the quaternion group. In 1848 Thomas Kirkman reported on his correspondence with Arthur Cayley regarding equations on the units determining a system of hypercomplex numbers.
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.
Binary code
The modern binary number system, the basis for binary code, was invented by Gottfried Leibniz in 1689 and appears in his article "Explication de l'Arithmétique Binaire". The full title is translated into English as the "Explanation of the binary arithmetic", which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of Fu Xi." (1703). Leibniz's system uses 0 and 1, like the modern binary numeral system. Leibniz encountered the "I Ching" through French Jesuit Joachim Bouvet and noted with fascination how its hexagrams correspond to the binary numbers from 0 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired. Leibniz saw the hexagrams as an affirmation of the universality of his own religious belief.
Complex number
This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate , for which the relation is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation induces the equalities and which hold for all integers these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in , again of the form with real coefficients
Catalan number
The Catalan sequence was described in the 18th century by Leonhard Euler, who was interested in the number of different ways of dividing a polygon into triangles. The sequence is named after Eugène Charles Catalan, who discovered the connection to parenthesized expressions during his exploration of the Towers of Hanoi puzzle. The counting trick for Dyck words was found by D. André in 1887.
Gottfried Wilhelm Leibniz
Complex thoughts would be represented by combining characters for simpler thoughts. Leibniz saw that the uniqueness of prime factorization suggests a central role for prime numbers in the universal characteristic, a striking anticipation of Gödel numbering. Granted, there is no intuitive or mnemonic way to number any set of elementary concepts using the prime numbers.
Leonhard Euler
Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. For any real number  (taken to be radians), Euler's formula states that the complex exponential function satisfies
Norm (mathematics)
The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane R. This identification of the complex number as a vector in the Euclidean plane, makes the quantity formula_6 (as first suggested by Euler) the Euclidean norm associated with the complex number.
Complex plane
While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers and is given by formula_5; then for a complex number its absolute value || coincides with its Euclidean norm, and its argument with the angle turning from 1 to .
Quater-imaginary base
The quater-imaginary numeral system was first proposed by Donald Knuth in 1960. It is a non-standard positional numeral system which uses the imaginary number 2"i" as its base. It is able to (almost) uniquely represent every complex number using only the digits 0, 1, 2, and 3. (Numbers less than zero, which are ordinarily represented with a minus sign, are representable as digit strings in quater-imaginary; for example, the number −1 is represented as "103" in quater-imaginary notation.)
Bicomplex number
Bicomplex numbers form an algebra over C of dimension two, and since C is of dimension two over R, the bicomplex numbers are an algebra over R of dimension four. In fact the real algebra is older than the complex one; it was labelled "tessarines" in 1848 while the complex algebra was not introduced until 1892.
Complex number
The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called casus irreducibilis). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545, though his understanding was rudimentary.
C-number
The term Number C (or C number) is an old nomenclature used by Paul Dirac which refers to real and complex numbers. It is used to distinguish from operators (q-numbers or quantum numbers) in quantum mechanics.
Bicomplex number
As commutative hypercomplex numbers, the tessarine algebra has been advocated by Clyde M. Davenport (1978, 1991, 2008) (exchange "j" and −"k" in his multiplication table). In particular Davenport notes the utility of the isomorphic correspondence between the bicomplex numbers and the direct sum of a pair of complex planes. Tessarines have also been applied in digital signal processing.
Complex polygon
A complex number may be represented in the form formula_1, where formula_2 and formula_3 are real numbers, and formula_4 is the square root of formula_5. Multiples of formula_4 such as formula_7 are called "imaginary numbers". A complex number lies in a complex plane having one real and one imaginary dimension, which may be represented as an Argand diagram. So a single complex dimension comprises two spatial dimensions, but of different kinds - one real and the other imaginary.
Complex number
Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate , by the ideal generated by the polynomial (see below).
Real number
The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was published by Georg Cantor in 1871. In 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. See Cantor's first uncountability proof.
Complex number
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers.
Ezra T. Newman
In 1973 he advocated the use of complex numbers in relativity, and consideration of complex spacetime.
Hossein Zakeri
Hossein Zakeri (Persian حسین ذاکری), Prof. Dr. (born 27 December 1942) is an Iranian mathematician. He, along with Prof. R. Y. Sharp, are the founders of generalized fractions, a branch in theory of commutative algebra which expands the concept of fractions in commutative rings by introducing the modules of generalized fractions. This topic later found applications in local cohomology, in the monomial conjecture, and other branches of commutative algebra.
Ideal number
It is widely believed that Kummer was led to his "ideal complex numbers" by his interest in Fermat's Last Theorem; there is even a story often told that Kummer, like Lamé, believed he had proven Fermat's Last Theorem until Lejeune Dirichlet told him his argument relied on unique factorization; but the story was first told by Kurt Hensel in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources. Harold Edwards says the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken" (Edwards 1977, p. 79). Kummer's use of the letter λ to represent a prime number, α to denote a λth root of unity, and his study of the factorization of prime number formula_12 into "complex numbers composed of formula_13th roots of unity" all derive directly from a paper of Jacobi which is concerned with higher reciprocity laws. Kummer's 1844 memoir was in honor of the jubilee celebration of the University of Königsberg and was meant as a tribute to Jacobi. Although Kummer had studied Fermat's Last Theorem in the 1830s and was probably aware that his theory would have implications for its study, it is more likely that the subject of Jacobi's (and Gauss's) interest, higher reciprocity laws, held more importance for him. Kummer referred to his own partial proof of Fermat's Last Theorem for regular primes as "a curiosity of number theory rather than a major item" and to the higher reciprocity law (which he stated as a conjecture) as "the principal subject and the pinnacle of contemporary number theory." On the other hand, this latter pronouncement was made when Kummer was still excited about the success of his work on reciprocity and when his work on Fermat's Last Theorem was running out of steam, so it may perhaps be taken with some skepticism.
Ring theory
The various hypercomplex numbers were identified with matrix rings by Joseph Wedderburn (1908) and Emil Artin (1928). Wedderburn's structure theorems were formulated for finite-dimensional algebras over a field while Artin generalized them to Artinian rings.
Complex number
In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name, de Moivre's formula:
Complex number
Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.
Complex number
Based on the concept of real numbers, a complex number is a number of the form , where and are real numbers and is an indeterminate satisfying . For example, is a complex number.
Complex number
The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
Complex number
The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's "De Algebra tractatus".
Irreducible complexity
In 1974 the young Earth creationist Henry M. Morris introduced a similar concept in his book "Scientific Creationism", in which he wrote; "This issue can actually be attacked quantitatively, using simple principles of mathematical probability. The problem is simply whether a complex system, in which many components function unitedly together, and in which each component is uniquely necessary to the efficient functioning of the whole, could ever arise by random processes."
Complex-base system
Base , using digits and , was proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965. The rounding region of an integer – i.e., a set formula_56 of complex (non-integer) numbers that share the integer part of their representation in this system – has in the complex plane a fractal shape: the twindragon (see figure). This set formula_56 is, by definition, all points that can be written as formula_58 with formula_59. formula_56 can be decomposed into 16 pieces congruent to formula_61. Notice that if formula_56 is rotated counterclockwise by 135°, we obtain two adjacent sets congruent to formula_63, because formula_64. The rectangle formula_65 in the center intersects the coordinate axes counterclockwise at the following points: formula_66, formula_67, and formula_68, and formula_69. Thus, formula_56 contains all complex numbers with absolute value ≤.
Italophilia
The University of Bologna, the first modern university, was a leading center of mathematical studies in the 16th century, and it was there that Niccolò Fontana Tartaglia developed the method of solving cubic equations, an achievement previously considered impossible. Extending Tartaglia’s work, Lodovico Ferrari devised a similar method to solve quartic equations, and both solutions were published by Gerolamo Cardano. The solutions to the cubic and quartic equations used, for the first time, what are now known as complex numbers, although it was another Bologna mathematician, Rafael Bombelli, who first explained what complex numbers are, and how they could be used. Complex numbers have since found practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. Later, Cardano published the first systematic treatment of probability.
John Horton Conway
He invented a new system of numbers, the surreal numbers, which are closely related to certain games and have been the subject of a mathematical novel by Donald Knuth. He also invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of "ONAG".
Jean-Robert Argand
Argand moved to Paris in 1806 with his family and, when managing a bookshop there, privately published his "Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques" (Essay on a method of representing imaginary quantities). In 1813, it was republished in the French journal "Annales de Mathématiques". The Essay discussed a method of graphing complex numbers via analytical geometry. It proposed the interpretation of the value as a rotation of 90 degrees in the Argand plane. In this essay he was also the first to propose the idea of modulus to indicate the magnitude of vectors and complex numbers, as well as the notation for vectors formula_1. The topic of complex numbers was also being studied by other mathematicians, notably Carl Friedrich Gauss and Caspar Wessel. Wessel's 1799 paper on a similar graphing technique did not attract attention.
Number
Numbers should be distinguished from numerals, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior Arabic numeral system around the late 14th century, and the Arabic numeral system remains the most common system for representing numbers in the world today. The key to the effectiveness of the system was the symbol for zero, which was developed by ancient Indian mathematicians around 500 AD.
Mathematical diagram
The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates — the magnitude or "modulus" of the product is the product of the two absolute values, or moduli, and the angle or "argument" of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
Analytic number theory
Johann Peter Gustav Lejeune Dirichlet is credited with the creation of analytic number theory, a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. In 1837 he published Dirichlet's theorem on arithmetic progressions, using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced the Dirichlet characters and L-functions. In 1841 he generalized his arithmetic progressions theorem from integers to the ring of Gaussian integers formula_3.
Number
Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose historically from trying to find closed formulas for the roots of cubic and quadratic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a square root of −1, denoted by "i", a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form
where "a" and "b" are real numbers. Because of this, complex numbers correspond to points on the complex plane, a vector space of two real dimensions. In the expression , the real number "a" is called the real part and "b" is called the imaginary part. If the real part of a complex number is 0, then the number is called an imaginary number or is referred to as "purely imaginary"; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or formula_15.
Caspar Wessel
Caspar Wessel (June 8, 1745, Vestby – March 25, 1818, Copenhagen) was a Danish–Norwegian mathematician and cartographer. In 1799, Wessel was the first person to describe the geometrical interpretation of complex numbers as points in the complex plane. He was the younger brother of poet and playwright Johan Herman Wessel.
Rolf Nevanlinna
Rolf Herman Nevanlinna (né Neovius; 22 October 1895 – 28 May 1980) was a Finnish mathematician who made significant contributions to complex analysis.
Contributions of Leonhard Euler to mathematics
Euler made important contributions to complex analysis. He introduced the scientific notation. He discovered what is now known as Euler's formula, that for any real number formula_3, the complex exponential function satisfies
Minggatu
He was the first person in China who calculated infinite series and obtained more than 10 formulae. In the 1730s, he first established and used what was later to be known as Catalan numbers. The Jesuit missionaries' influence can be seen by many traces of European mathematics in his works, including the use of Euclidean notions of continuous proportions, series addition, subtraction, multiplication and division, series reversion, and the binomial theorem. Minggatu's work is remarkable in that expansions in series, trigonometric and logarithmic were apprehended algebraically and inductively without the aid of differential and integral calculus.
Joseph Konhauser
Joseph D. E. Konhauser (1924 – February 1992) was an American mathematician who introduced Konhauser polynomials. He also organized many mathematical problems competitions. The annual Konhauser Problemfest is named after him.
Quaternions and spatial rotation
The complex numbers can be defined by introducing an abstract symbol which satisfies the usual rules of algebra and additionally the rule . This is sufficient to reproduce all of the rules of complex number arithmetic: for example:
Charles Méray
Hugues Charles Robert Méray (12 November 1835, Chalon-sur-Saône, Saône-et-Loire - 2 February 1911, Dijon) was a French mathematician. He is noted as the first to publish an arithmetical theory of irrational numbers. His work did not have much of a role in the history of mathematics because France, at that time, was less interested in such matters than Germany.
Euler's formula
The view of complex numbers as points in the complex plane was described about 50 years later by Caspar Wessel.This formula can be interpreted as saying that the function is a unit complex number, i.e., it traces out the unit circle in the complex plane as ranges through the real numbers. Here is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counterclockwise and in radians.
Gerolamo Cardano
Cardano was the first mathematician to make systematic use of negative numbers. He published with attribution the solution of Scipione del Ferro to the cubic equation and the solution of his student Lodovico Ferrari to the quartic equation in his 1545 book "Ars Magna". The solution to one particular case of the cubic equation formula_1 (in modern notation), had been communicated to him in 1539 by Niccolò Fontana Tartaglia (who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long dispute) in the form of a poem, but Ferro's solution predated Fontana's. In his exposition, he acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties, described for the first time by his Italian contemporary Rafael Bombelli. In "Opus novum de proportionibus" he introduced the binomial coefficients and the binomial theorem.
Highly composite number
A highly composite number, also known as an anti-prime, is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it.
Complex number
A complex number can thus be identified with an ordered pair of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called complex plane or Argand diagram, named after Jean-Robert Argand. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere.
Number
The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's "De Algebra tractatus".
Complex number
The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his "Stereometrica" he considers, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term formula_128 in his calculations, although negative quantities were not conceived of in Hellenistic mathematics and Heron merely replaced it by its positive (formula_129).
Mathematical diagram
A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram
The complex plane is sometimes called the "Argand plane" because it is used in "Argand diagrams". These are named after Jean-Robert Argand (1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.
Complex number
The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.
Split-complex number
In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in "Bulletin de l’Academie Polanaise des Sciences" (see link in References). 
He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra. D. H. Lehmer reviewed the article in Mathematical Reviews and observed that this second system was isomorphic to the "hyperbolic complex" numbers, the subject of this article.
Augustin-Louis Cauchy
In 1831, while in Turin, Cauchy submitted two papers to the Academy of Sciences of Turin. In the first he proposed the formula now known as Cauchy's integral formula,
where "f"("z") is analytic on "C" and within the region bounded by the contour "C" and the complex number "a" is somewhere in this region. The contour integral is taken counter-clockwise. Clearly, the integrand has a simple pole at "z" = "a". In the second paper he presented the residue theorem,
where the sum is over all the "n" poles of "f"("z") on and within the contour "C". These results of Cauchy's still form the core of complex function theory as it is taught today to physicists and electrical engineers. For quite some time, contemporaries of Cauchy ignored his theory, believing it to be too complicated. Only in the 1840s the theory started to get response, with Pierre Alphonse Laurent being the first mathematician, besides Cauchy, making a substantial contribution (his Laurent series published in 1843).
History of large numbers
However, since the 19th century, mathematicians have studied transfinite numbers, numbers which are not only greater than any finite number, but also, from the viewpoint of set theory, larger than the traditional concept of infinity. Of these transfinite numbers, perhaps the most extraordinary, and arguably, if they exist, "largest", are the large cardinals. The concept of transfinite numbers, however, was first considered by Indian Jaina mathematicians as far back as 400 BC.
Kolmogorov complexity
The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray Solomonoff, who published it in 1960, describing it in "A Preliminary Report on a General Theory of Inductive Inference" as part of his invention of algorithmic probability. He gave a more complete description in his 1964 publications, "A Formal Theory of Inductive Inference," Part 1 and Part 2 in "Information and Control".
Ars Magna (Gerolamo Cardano)
"Ars Magna" also contains the first occurrence of complex numbers (chapter XXXVII). The problem mentioned by Cardano which leads to square roots of negative numbers is: find two numbers whose sum is equal to 10 and whose product is equal to 40. The answer is 5 + √−15 and 5 − √−15. Cardano called this "sophistic," because he saw no physical meaning to it, but boldly wrote "nevertheless we will operate" and formally calculated that their product does indeed equal 40. Cardano then says that this answer is “as subtle as it is useless”.