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The sum and product of two complex numbers are:
Complex numbers were first introduced in connection with explicit formulas for the roots of cubic polynomials. In mathematics, the term "complex" when used as an adjective means that the field of complex numbers is the underlying number field considered. For example complex matrix, complex polynomial and complex Lie algebra.
The earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century15th century 16th century 17th century more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. Events Beginning of the " Little Ice Age" a cooling period that resulted in lower crop yi closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see Niccolo Fontana TartagliaNiccolo Fontana Tartaglia ( 1499 or 1500 December 13, 1557) was a mathematician, an engineer (designing fortifications), surveyor (of topography, seeking the best means of defense or offense) and bookkeeper from the then Republic of Venice (now Italy)., Gerolamo CardanoGerolamo Cardano or Jerome Cardan ( September 24, 1501 September 21 1576) was a celebrated Renaissance mathematician, physician, astrologer, and gambler. He was born in Pavia, Italy, the illegitimate child of a mathematically gifted lawyer who was a frien). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René DescartesRene Descartes ( IPA: rne. dekt) ( March 31, 1596 February 11, 1650), also known as Cartesius worked as a philosopher and mathematician. While most notable for his groundbreaking work in philosophy, he has achieved wide fame as the inventor of the Cartesi in the 17th century16th century 17th century 18th century more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601- 1700. During this period, the power of England and the United Provinces increased; while that of and was meant to be derogatory. (See imaginary numberIn mathematics, an imaginary number (or purely imaginary number is a complex number whose square is negative or zero. The term was coined by Rene Descartes in 1637 in his La Geometrie and was meant to be derogatory: obviously, such numbers were thought no for a discussion of the "reality" of complex numbers.) The 18th century17th century 18th century 19th century more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701- 1800; however, historians will sometimes specifically refer to the 18th Century as 1715- 89, saw the labors of Abraham de MoivreAbraham de Moivre ( May 26, 1667 November 27, 1754), was a French mathematician famous for de Moivre's formula which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was elected a Fellow of the and Leonhard Euler. To De Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula:
and to Euler (1748) Euler's formula of complex analysis:
The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, 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' De Algebra tractatus.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that should represent a unit line, and its negative, perpendicular to the real axis. Buée 's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Mention should also be made of an excellent little treatise by Mourey (1828), in which the foundations for the theory of directional numbers are scientifically laid. The general acceptance of the theory is not a little 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.
The common terms used in the theory are chiefly due to the founders. Argand called the direction factor, and the modulus; Cauchy (1828) called the reduced form (l'expression réduite); Gauss used for , introduced the term complex number for , and called the norm.
The expression direction coefficient, often used for , is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.
Following Cauchy and Gauss have come a number of contributors of high rank, of whom the following may be especially mentioned: Kummer (1844), Leopold Kronecker (1845), Scheffler (1845, 1851, 1880), Bellavitis (1835, 1852), Peacock (1845), and De Morgan (1849). Möbius must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers.
Other types have been studied, besides the familiar , in which is the root of . Thus Ferdinand Eisenstein has studied the type , being a complex root of . Similarly, complex types have been derived from ( prime). This generalization is largely due to Kummer, to whom is also due the theory of ideal numbers, which has recently been simplified by Felix Klein (1893) from the point of view of geometry. A further complex theory is due to Evariste Galois, the basis being the imaginary roots of an irreducible congruence,
The late writers (from 1884) on the general theory include Weierstrass, Schwarz , Richard Dedekind, Otto Hölder, Berloty , Henri Poincaré, Eduard Study, and Macfarlane.
The formally correct definition using pairs of real numbers was given in the 19th century.