Make Complex Numbers Real

This post is continuation of this rant.

We do such a bad job with complex numbers that students often assume that the word Imaginary in front of Number is an adjective that i is not really a number. It is some sort of imaginary thing.

One important reason that students find this easy to believe is the fact that it is not possible to order the complex numbers. Ordering is a basic and fundamental concept for numbers. Ordering is first thing that students are to learn about numbers. We stress the idea of ordering of the numbers, we use the ordering of numbers to help explain addition and subtraction. So if we can neither tell if i is neither bigger or smaller than 0, then it might not really exist.

So makes imaginary and complex numbers real. What makes them real is that can represent things that happen in the real world. Real numbers can represent length and direction (direction in one dimension, left or right, up or down). Complex numbers can represent length and direction in two dimensions. So 3+4i can represent 3 steps to the right and 4 steps up. You can represent movement in two dimensions with complex numbers. But that is not different than vectors. What makes complex numbers so different is the existence of a coherent multiplication rule, one that even allows a coherent definition of division.

So it is multiplication that makes complex numbers so special. What does multiplication in complex numbers do. It does two things, it is capable of doing what multiplication of real numbers does, which is dilation. Multiplication by a positive real number performs a dilation on the points of the complex plane. Multiplication by a negative real number performs a dilation and a 180° rotation. In particular, multiplication by -1 is just a 180° rotation. Now how about multiplication by i. If you take any complex number and multiply it by i the result is a 90° rotation about 0. And from all that we know about i, this makes tremendous sense. We know that i is the square root of negative one, so that multiplying by i does half of what multiplying by -1 does. We know that i^4=1 so that multiplying by i four times does nothing to a number, just as rotating by 90° four times returns things to where they started. It turns out then that multiplying by complex numbers models rotation! This is what is important. Turns are everywhere. Everywhere in life there are rotations of all sorts and these numbers allow you to do computations with coordinates that affect the desired rotations. Want to turn 90°, multiply by i. Want to turn 45°, multiply by sqrt(i). This is why polar form for a complex number exists, and it is the importance of DeMoivre's Theorem.

As a final note tonight. Let's bring it back to algebra. Let's say we have to solve x^6=1. Then we are looking for 6 solutions. Well then these are complex numbers that multiplication is the same as rotating 0°, 60°, 120°, 180°, 240°, and 300°. In other words 1, 1/2 +sqrt(3)/2 i, -1/2 +sqrt(3)/2 i, -1, -1/2 -sqrt(3)/2 i, 1/2 -sqrt(3)/2 i. But what is better is seeing it on the complex plane.



That is right the solutions to this equation are vertices of regular polygon, and more this generalizes. So that the regular polygons can be associated to the polynomials x^n-1.