Roots of unity

From AoPSWiki

The Roots of unity are a topic closely related to trigonometry. Roots of unity come up when we examine the complex roots of the polynomial $x^n=1$ .

Solving the Equation

First, we note that since we have an nth degree polynomial, there will be n complex roots.

Now, we can convert everything to polar form by letting $x = re^{i\theta}$ , and noting that $1 = e^{2\pi ik}$ for $k\in \mathbb{Z}$ , to get $r^ne^{ni\theta} = e^{2\pi ik}$ . The magnitude of the RHS is 1, making $r^n=1\Rightarrow r=1$ (magnitude is always expressed as a positive real number). This leaves us with $e^{ni\theta} = e^{2\pi ik}$ .

Taking the natural logarithm of both sides gives us $ni\theta = 2\pi ik$ . Solving this gives $\theta=\frac{2\pi k}n$ . Additionally, we note that for each of $k=0,1,2,\ldots,n-1$ we get a distinct value for $\theta$ , but once we get to $k > n-1$ , we start getting coterminal angles.

Thus, the solutions to $x^n=1$ are given by $x = e^{2\pi k i/n}$ for $k=0,1,2,\ldots,n-1$ . We could also express this in trigonometric form as $x=\cos\left(\frac{2\pi k}n\right) + i\sin\left(\frac{2\pi k}n\right) = \mathrm{cis }\left(\frac{2\pi k}n\right).$

Geometry

All of the roots of unity lie on the unit circle in the complex plane. This can be seen by considering the magnitudes of both sides of the equation $x^n = 1$ . If we let $x = re^{i\theta}$ , we see that $r^n = 1$ , since the magnitude of the RHS of $x^n=1$ is 1, and for two complex numbers to be equal, both their magnitudes and arguments must be equivalent.

Additionally, we can see that when the nth roots of unity are connected in order (more technically, we would call this their convex hull), they form a regular n-sided polygon. This becomes even more evident when we look at the arguments of the roots of unity.

Properties

Listed below is a quick summary of important properties of roots of unity.

They occupy the vertices of a regular n-gon in the complex plane.
For $n>1$ , the sum of the nth roots of unity is 0. More generally, if $\zeta$ is a primitive nth root of unity (i.e. $\zeta^m\neq 1$ for $1\le m\le n-1$ ), then $\sum_{k=0}^{n-1} \zeta^{km}=\begin{cases} n & {n\mid m}, \\ 0 & \mathrm{otherwise.}\end{cases}$
- This is an immediate result of Vieta's formulas on the polynomial $x^n-1 = 0$ and Newton sums.
If $\zeta$ is a primitive nth root of unity, then the roots of unity can be expressed as $1, \zeta, \zeta^2,\ldots,\zeta^{n-1}$ .
Also, don't overlook the most obvious property of all! For each $n$ th root of unity, $\zeta$ , we have that $\zeta^n=1$

Uses

Roots of unity show up in many surprising places. Here, we list a few:

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