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Roots of Unity

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© Copyright 1999, Jim Loy

In this article, we will be dealing with complex numbers, as described in my article Imaginary Numbers.

This is called the Cyclotomic Equation:

xⁿ=1

From algebra, we know that such an equation has n solutions (roots). In general, some of the n roots of an nth degree equation may be the same. In this case, they are all different. Of course x=1 is one of the roots, for every n. But it is not the only root. The first two are easy: x¹=1 has one root (x=1), and x²=1 has two roots (x=±1). And x³=1 has 3 roots: x³-1=(x-1)(x²+x+1)=0. The (x-1) gives a solution of x=1, and we can use the quadratic formula on x²+x+1 to get x=(-1±sqr(3))/2. The four roots of x⁴=1 are ±1 and ±i.

Let's plot all of the above roots on the complex plane (diagram at the left). The complex plane is used to plot complex numbers, not functions. The horizontal axis gives the real component and the vertical axis gives the imaginary component. Here all of the above roots (for instance i is one of them, as is (-1+sqr(3))/2) are plotted as points. I have drawn line segments from a couple of these roots to the origin (0+0i). I have also drawn a unit circle, to show that all of these roots are on the unit circle. It is possible to show that all roots of xⁿ=1, for any n, are on the unit circle. Also we see that the three roots of x³=1 are approximately equally spaced around the circle. If you figure out the angles, you will see that they are equally spaced, three angles of exactly 120 degrees. Likewise, the two roots of x²=1 are equally spaced, as are the four roots of x⁴=1. One might guess that the roots for a general xⁿ=1 are equally spaced around the circle.

Let's try an example, x⁵=1 (diagram on the right). I have calculated the position of the next root, counter-clockwise from the root x=1. It has a fairly unfriendly looking value. If I take that value to the 5th power, I hope to get 1. First I have to square it. This is fairly difficult. I won't show you the process, just the result: -(sqr(5)+1)/4+i(sqr(10-2sqr(5))/4. It took me most of a day to get that result. This turns out to be the next root, counter-clockwise. Perhaps squaring one of the roots of xⁿ=1 doubles the central angle. Let's prove that, instead of dealing with these messy roots of x⁵=1.

This diagram shows two complex numbers on the unit circle. They are at the left ends of the two central angles a and b. And we want to multiply these two complex numbers. What are these two complex numbers (as I have not labeled them)? The first one A (with the central angle a) is cos a + i sin a. B (the other one) is cos b + i sin b. These are just the coordinates in the form x+yi on the complex plane (as deduced from their central angles). This is not polar coordinates. Well the product AB is:

=(cosa+isina)(cosb+isinb)
=(cosa cosb - sina sinb) + i(sina cosb + cosa sinb)

These look familiar to me. They are trig functions of sums of angles, as in my article Sin(x+y) & Cos(x+y). We can simplify the last line to:

=cos(a+b) + i sin(a+b)

So if we want to multiply two complex number on the unit circle, we just add the angles. We can deduce a more general result, about multiplying complex numbers that are not on the unit circle. But that won't be necessary in this article. From the above result, we can also deduce that squaring a complex number on the unit circle just doubles the angle. And it also follows that: Taking the nth power of a complex number on the unit circle just multiplies the angle by n. Aⁿ=cos na + i sin na (for a complex number A on the unit circle with central angle a).

That result clears up our original problem, xⁿ=1. If xⁿ=1, that means that the sin(na)=0, which means that na=360 degrees, and a=360/n. That, in turn, means that: The roots of xⁿ=1 are equally spaced around the unit circle, as a regular n-gon.

We can use this to calculate the roots for any n. But these may be difficult to find exactly, in most cases. An estimate is readily available from a calculator or from sine and cosine tables.

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