A Complex Number

A Complex Number

Why we need a new kind of number. Consider the equation $x^2+1=0$, i.e. $x^2=-1$. This has no solution among the real numbers, because the square of every real number — positive, negative, or zero — is never negative. To let equations like this have a solution at all, mathematicians extended the real numbers to a larger number system that includes such solutions.

The imaginary unit. We introduce a new symbol $i$ (the Greek letter iota) defined by $i=\sqrt{-1}$, so that $i^2=-1$. The Swiss mathematician Leonard Euler (1707-1783) was the first to introduce this symbol with exactly this property. With $i$ in hand, $x^2+1=0$ now has the two solutions $x=i$ and $x=-i$, since $i^2+1=-1+1=0$.

This one new symbol, together with the ordinary real numbers, is enough to build the whole system of complex numbers covered in this chapter — numbers of the form (real number) $+\ i\times$(real number), covered next in Section 1.1(a)-(b).