What mathematical developments/discoveries caused imaginary numbers to gain acceptance at the time (18th century) they did?

Question

In a Wiki article on imaginary numbers it was asserted that "the use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855)."

What motivated Euler's and Gauss's contributions to the theory of imaginary numbers? For instance, I know that Euler produced the formula that later led to DeMoivre's theorem, but don't quite understand why. And their lives barely overlapped, so why did no one "in between" pick up the "baton" from Euler to Gauss?

(Ironically, Rene Descartes, who derided imaginary numbers, founded the "Cartesian" (2x2) co-ordinate system, which parallels the plane on which imaginary numbers are also graphed. This may have been a case of an "accidental" contribution.)

Answer 1

The first serious use of complex numbers is in finding the roots of quadratic, cubic, and quartic polynomials. Cardano, in his Ars Magna (1545), first showed that quadratic equations could have (formally) complex roots, although he didn't call them that; he said they were "as subtle as [they are] useless". In Bombelli's algebra text (1572), he developed the rules of complex arithmetic, and showed that Cardano's formula for the cubic could lead to real solutions even though intermediate results were imaginary. By the way, I've been told on multiple occassions that the notation $i= \sqrt{-1}$ was only developed to guard against the common mistake of 'proving' $$(\sqrt{-1})^2=\sqrt{(-1)^2}=\sqrt{1}=1$$

A key insight that was achieved in the early 18th century is the deep connection between complex numbers and geometry. It was observed that $i$ can be used to simplify many trigonometric identities, and in 1748 Euler discovered his famous and beautiful formula $$e^{it}=\cos t+i \sin t$$ (The derivation was rather different from the one usually presented in today's textbooks; see this entry in the series How Euler Did It.)

The conception of a complex number as a point in the plane is another discovery worth noting. This construction was already used by Wessel in 1799, and was independently rediscovered by Argand, but it really gained popularity when Gauss published his treatise on complex numbers. This book also established much of the modern notation and terminology that is used in complex analysis.

Answer 2

Just to complement Danu's answer. Some people used complex numbers since the 16th century, however, WIDE acceptance came later (at the end of the 18th century) when several people (Argand, Vessel, Gauss) discovered the geometric interpretation.

This was apparently a crucial step. Still, they were not universally recognized. They say that even Chebyshev never used them.

Another event which could be significant: in the early 19th century, physicists started to use them (Fresnel).

Answer 3

Apart from the necessity in the calculation of roots of cubic polynomials, there is another, more fundamental role complex numbers play in polynomial equations, which was only beginning to be appreciated in the 17th century. This role is expressed through the fundamental theorem of algebra, which says that any nonconstant polynomial equation has at least one root, if we allow complex numbers to be roots. That is, if $a_0,a_1,\ldots,a_n$ are real numbers such that at least one of $a_1,a_2,\ldots,a_n$ is nonzero, then the equation \begin{equation}\label{e:polynomial-x-0} p(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0 , \end{equation} has a solution, provided $x$ may have complex values. If $a_1=a_2=\ldots=a_n=0$, then the equation $p(x)=0$ becomes $a_0=0$, which does not have any (complex) solution when $a_0\neq0$. So the condition that at least one of $a_1,a_2,\ldots,a_n$ is nonzero (i.e., $p(x)$ is nonconstant) is simply to rule out this trivial case. The fundamental theorem of algebra is miraculous because complex numbers are designed to solve any quadratic equation, and it is a priori conceivable that we need to introduce a new kind of "number" every time we increase the degree of a polynomial equation. The first formulation of the fundamental theorem of algebra was given by Albert Girard (1595-1632) in 1629, although he did not attempt a proof. Indeed, rigorous proofs of this theorem did not appear until the early 19th century, which incidentally marks the beginning of an era when the existence and usefulness of complex numbers were widely accepted.

Any doubts on the existence and importance of complex numbers were completely disposed of after the development of complex analysis, which is also known as function theory. The initial motivation for studying functions of a complex variable was to use them to compute (or simplify) real definite integrals, and the pioneering works in this direction were done by Euler and Joseph-Louis Lagrange (1736-1813) around 1760-1780. Their research was taken up later in the 1810's by Augustin Louis Cauchy (1789-1857), who realized by 1821 that complex functions have a rich theory of their own. Gauss reached the same understanding as early as 1811, and played a major role in popularizing complex numbers, but he did not directly contribute to the development of complex analysis. Thus roughly between 1820-1850, Cauchy singlehandedly developed all the basic results of complex analysis, perhaps with the exception of Laurent series, which first appeared in a paper submitted by Pierre Alphonse Laurent (1813-1854) in 1843.