Content deleted Content added
No edit summary |
No edit summary |
||
Line 25:
quantities was coined by [[Rene Descartes|René Descartes]] in
the [[17th century]]
and was meant to be derogatory. The [[18th century]] saw the labors of [[De Moivre]] and [[Euler]]. To De Moivre is due (1730) the well-known formula which bears his name, <math>(\cos \theta + i \sin \theta)^{n} = \cos n \theta + i \sin n \theta</math>, and to Euler (1748) the formula <math>\cos \theta + i \sin \theta = e ^{\theta i}</math>. The existence of complex numbers was
not completely accepted until the
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.
formally correct definition using pairs of real numbers was given in▼
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 <math>\pm\sqrt{-1}</math> should represent a unit line, and its negative, perpendicular to the real axis. [[Buée]]'s paper was not published until 1806, in which year [[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 [[Cauchy]] and [[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 $\cos \phi + i \sin \phi$ the ``direction
factor'', and $r = \sqrt{a^2+b^2}$ the ``modulus''; Cauchy (1828)
called $\cos \phi + i \sin \phi$ the ``reduced form'' (l'expression
r\'eduite); Gauss used $i$ for $\sqrt{-1}$, introduced the term
``complex number'' for $a+bi$, and called $a^2+b^2$ the ``norm.'' The
expression ``direction coefficient'', often used for $\cos \phi + i
\sin \phi$, 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), [[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 <math>a + bi</math>, in which <math>i</math> is the root of <math>x^2 + 1 = 0</math>. Thus [[Eisenstein]] has studied the type <math>a + bj</math>, <math>j</math> being a complex root of <math>x^3 - 1 = 0</math>. Similarly, complex types have been derived from <math>x^k - 1 = 0</math> (<math>k</math> prime). This generalization is largely due to [[Kummer]], to whom is also due the theory of [[Ideal number]]s, which has recently been simplified by [[Klein]] (1893) from the point of view of geometry. A further complex theory is due to [[Galois]], the basis being the imaginary roots of an irreducible congruence, <math>F(x) \equiv 0</math> (mod <math>p</math>, a prime). The late writers (from 1884) on the general theory include [[Weierstrass]], [[Schwarz]], [[Dedekind]], [[Hölder]], [[Berloty]], [[Poincaré]], [[Study]], and [[Macfarlane]].
▲The formally correct definition using pairs of real numbers was given in the [[19th century]].
==Definition==
| |||