Complex Numbers & Polar Coordinates

Complex numbers: The very name sounds difficult. However, these numbers are based on just one very simple idea. Or, rather, they are based on one very simple need : To describe the things for which real numbers are inadequate.

Historically, complex numbers arose from a need to give accurate descriptions of the roots of polynomials.

For example, the polynomial:

has two real roots that can be found easily with the quadratic formula:

Change a sign, however, to:

and the quadratic formula fails us:

The sticking point is the square root of the negative number. Since grade school you have been told that such a thing is impossible, since the square of any number is always non-negative. This is quite true: There are no real numbers with a square of  –4 . Thus, we say that the function  g  has no roots. Indeed, a plot of this function shows that it never crosses the x-axis.

Let us, however, imagine... some other number system in which there are values for the square roots of negative numbers. We might give such "imaginary" numbers suggestive names, like:

In this "imaginary" number system we would then write:

There are two roots, just as before, except now the roots are a complex combination of both real and imaginary numbers. The word "complex", in this setting, means "composite", not "difficult".

The question, of course, is why we would choose to imagine such numbers in the first place. The previous conclusion — that the polynomial has no real roots — seemed both accurate and definitive. Why "imagine" things that can not be?

That is a legitimate question. However, it could also be asked, at any place and any time, of anyone who cares to imagine something different than the status quo. People told the Wright brothers that they "could not" fly.

Complex numbers, once conceived, transformed mathematics. By giving a higher vantage point from which to view familiar numbers and the functions, complex numbers clarified almost everything related to the study of calculus. Old "complexities" didn't look so complex after all. Many new connections became apparent, even as they looped away from the "real" and into the "imaginary".