The impetus to study complex numbers proper first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). 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. As an example, Tartaglia's cubic formula gives the solution to the equation x3 − x = 0 as
The three cube roots of −1, two of which are complex
\frac{1}{\sqrt{3}}\left((\sqrt{-1})^{1/3}+\frac{1}{(\sqrt{-1})^{1/3}}\right),
At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z3 = i has solutions –i, {\scriptstyle\frac{\sqrt{3}}{2}}+{\scriptstyle\fra c{1}{2}}i and {\scriptstyle\frac{-\sqrt{3}}{2}}+{\scriptstyle\frac{1}{2}}i. Substituting these in turn for {\scriptstyle\sqrt{-1}^{1/3}} in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x3 – x = 0. Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, as later mathematicians showed rigorously, the use of complex numbers is unavoidable. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.
The term "imaginary" for these quantities was coined by René Descartes in 1637, although he was at pains to stress their imaginary nature[15]
[...] quelquefois seulement imaginaires c’est-à-dire que l’on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu’il n’y a quelquefois aucune quantité qui corresponde à celle qu’on imagine.
([...] sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.)
A further source of confusion was that the equation sqrt(-1)^2 = sqrt(-1)*sqrt(-1) = -1 seemed to be capriciously inconsistent with the algebraic identity sqrt(ab) = sqrt(a)sqrt(b), which is valid for non-negative real numbers a and b, and which 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 1/sqrt(a) = sqrt(1/a)) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of sqrt(-1) to guard against this mistake[citation needed]. 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.
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:
(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta. \,
In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:
\cos \theta + i\sin \theta = e ^{i\theta } \,
by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.
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.
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.
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. The English mathematician G. H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.[16] Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.
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 réduite) and apparently introduced the term argument; Gauss used i for \sqrt{-1}, introduced the term complex number for a + bi, and called a2 + b2 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.
Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.
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