Who introduced the concept of complex numbers?
5826#4
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 number
14554#3
Although Greek mathematician and engineer Heron of Alexandria is noted as the first to have conceived these numbers, Rafael Bombelli first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time imaginary numbers, as well as negative numbers, were poorly understood and regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his "La Géométrie", where the term "imaginary" was used and meant to be derogatory. The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818).
Imaginary number
21690#72
In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points. This eventually led to the concept of the extended complex plane.
Number
19629934#0
The Oedipus complex (also spelled Œdipus complex) is a concept of psychoanalytic theory. Sigmund Freud introduced the concept in his "Interpretation of Dreams" (1899) and coined the expression in his "A Special Type of Choice of Object made by Men" (1910). The positive Oedipus complex refers to a child's unconscious sexual desire for the opposite-sex parent and hatred for the same-sex parent. The negative Oedipus complex refers to a child's unconscious sexual desire for the same-sex parent and hatred for the opposite-sex parent. Freud considered that the child's identification with the same-sex parent is the successful outcome of the complex and that unsuccessful outcome of the complex might lead to neurosis, pedophilia, and homosexuality.
Oedipus complex
5826#129
A further source of confusion was that the equation formula_135 seemed to be capriciously inconsistent with the algebraic identity formula_136, which is valid for non-negative real numbers and , and which was also used in complex number calculations with one of , positive and the other negative. The incorrect use of this identity (and the related identity formula_137) in the case when both and are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol in place of to guard against this mistake. Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
Complex number
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Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a real number; its point lies on the horizontal axis of the complex plane. Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number.
Complex number
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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.
Split-complex number
450541#40
Zero-knowledge proofs were first conceived in 1985 by Shafi Goldwasser, Silvio Micali, and Charles Rackoff in their paper "The Knowledge Complexity of Interactive Proof-Systems". This paper introduced the IP hierarchy of interactive proof systems ("see interactive proof system") and conceived the concept of "knowledge complexity", a measurement of the amount of knowledge about the proof transferred from the prover to the verifier. They also gave the first zero-knowledge proof for a concrete problem, that of deciding quadratic nonresidues mod "m" (this more or less means that there isn't any number "x" where formula_86 is "equivalent" to some given number). Together with a paper by László Babai and Shlomo Moran, this landmark paper invented interactive proof systems, for which all five authors won the first Gödel Prize in 1993.
Zero-knowledge proof
7787730#7
Muses also envisioned a mathematical number concept, Musean hypernumbers, that includes hypercomplex number algebras such as complex numbers and split-complex numbers as primitive types. He assigned them levels based on certain arithmetical properties they may possess. While many open questions remain, in particular about defining relations of these levels, Muses pictured a wide range of applicability for this concept. Some of these are based on properties of magic squares, and even related to religious belief. He believed that these hypernumbers were central to issues of consciousness.
Charles Musès
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The common terms used in the theory are chiefly due to the founders. Argand called formula_140 the "direction factor", and formula_141 the "modulus"; Cauchy (1828) called formula_140 the "reduced form" (l'expression réduite) and apparently introduced the term "argument"; Gauss used for formula_143, introduced the term "complex number" for , and called the "norm". The expression "direction coefficient", often used for formula_140, is due to Hankel (1867), and "absolute value," for "modulus," is due to Weierstrass.
Complex number