Although the Fundamental Theorem of Algebra was still not proved in the 18th century, and complex numbers were not fully understood, the square root of minus one was being used more and more.

Analysis, especially calculus and the theory of differential equations, was making great headway. Certain functions, including the trigonometric functions and exponential functions, appear in solutions to integrals and differential equations. Euler (1707–1783) made the observation, here written in modern notation, that

eix = cos x + i sin x

where i denotes √–1. This is an equation which allows you to interpret the exponentiation of an imaginary number ix as having a real part, cos x, and an imaginary part, i sin x. This was an especially useful observation in the solution of differential equations. Because of this and other uses of i, it became quite acceptable for use in mathematics. Euler, a very influential mathematician, recommended the general use of these imaginary numbers in his Introduction to Algebra.

By the end of the 18th century numbers of the form x + yi were in fairly common use by research mathematicians, and it became common to represent them as points in the plane. The standard convention now in use to display them is to place the real numbers, that is, those numbers of the form x + 0i, on the horizontal x-axis, with positive numbers to the right and negative ones to the left. Also, imaginary numbers, that is, those numbers of the form 0 + yi, on the vertical y-axis, where positive values of y are up, and negative ones down. Thus, i is located one unit above 0 (the origin, where the axes meet), and –i is located one unit below 0.

This particular display of numbers of the form x + yi is attributed to various individuals including Wessel, Argand, and Gauss. It was easy to come by, since the usual (x,y)-coordinates for the plane had been used for over a century. Nonetheless, it is a very useful way to understand these numbers.

The Fundamental Theorem of Algebra–proved!

Still, at nearly the end of the 18th century, it wasn’t yet known what form all the solutions of a polynomial equation might take. Gauss published in 1799 his first proof that an nth degree equation has n roots each of the form a + bi, for some real numbers a and b. Once he had done that, it was known that complex numbers (in the sense of solutions to algebraic equations) were the numbers a + bi, and it was appropriate to call the xy-plane the “complex plane”.

In some sense all the historical discussion before Gauss was prehistory of complex numbers. But that’s just the history that is useful in understanding the need for complex numbers. Although there are other concepts of numbers that either go beyond complex numbers or include something other than complex numbers, we know that at least no other “numbers” are needed to solve polynomial equations. The use of complex numbers pervades all of mathematics and its applications to science.

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