Complex Numbers

Finishing the number system...

Since somewhere around 500 BC, beginning with the counting numbers, 1,2,3...up to very many, mathematicians have tacked on fractions, negative numbers, zero, rational numbers, irrational numbers, real numbers, imaginary numbers, complex numbers and the list goes on. What mathematicians do for a living is to continue defining larger and more complicated objects together with rules for manipulating them, similar to the rules for adding and multiplying counting numbers. Here we want to talk about real numbers, imaginary numbers and complex numbers.

The real numbers include the counting numbers (positive integers), fractions (rational numbers), irrational numbers, zero and the negative of all of these. The irrational numbers fill in all the gaps between the fractions so that the real numbers are continuous from minus as large as we can imagine to positive as large as we can imagine. We often represent this continuum of real numbers as a line called the real number line. Every real number has its place on this line. Adding or multiplying real numbers gives us another real number but there are some peculiar rules about multiplying. The product of a negative and positive real is a negative real. The product of two negative reals is a positive real. It is not just some mathematician's whim that dictates these rules but the very nature of real numbers.

There is another operation on real numbers called squaring, meaning to multiply a number by itself. The inverse of this operation, called taking the square root, means to find a number which when multiplied by itself gives back the number of which you are taking the square root. Now try to imagine a number which when multiplied by itself gives us -1. According to the rules for multiplication there is no real number which when multiplied by itself gives a negative real. Some folks were probably content to let half the real numbers go without square roots but some mathematicians felt they must have missed something in defining the real numbers. They called the thing they missed i, the square root of -1, and defined another number line that was i multiplied by the real number line. The numbers inhabiting this new number line were called imaginary numbers. Multiplication by i has the effect of rotating the object i was multiplied by 90 degrees counterclockwise. The real number line, drawn horizontal, and the imaginary number line, drawn vertical, then are perpendicular and define a plane called the complex plane.

Every point on this plane can be identified with a number in the form a+b*i where a and b are real numbers. Numbers consisting of two parts like this, one part real and one part imaginary, are called complex numbers. Real numbers are those where b=0 and imaginary numbers are those where a=0 so all numbers can be considered complex. Normally complex variables, numbers used in mathematical expressions are symbolized by the letter z, just as real variables are often symbolized by x. The discovery of complex numbers suddenly made many difficult mathematical problems easy. Problems are always harder when you are working with an incomplete tool kit.