14.8 The roots of unity

The nth roots of unity are the nth roots of one. How can there be more than one ($1$) nth root of one? Well, some of them are complex, of course!

Note that ⁸⁸ 8 In this document the natural numbers include zero!

This is because of Euler’s formula.

Therefore, if we take the $n$th roots of both sides, we get that

Which we can use to compute the $n$th roots of unity. Note that by the fundamental theorem of algebra for the $nth$ roots of unity, there are $n$ different values.

Note that we often use the letter $\omega$ (the Greek “omega”) to denote $e^{2\pi/n}$.

Proof: I don’t think you need to know this for A Level Mathematics, but an easy-ish proof is to consider

and note that if we multiply through by $\omega_{n}$ we actually get the same value back! That is

The only complex number for which $ax=x$ is $0$ and therefore the sum of the roots of unity is $0$.

Using Euler’s formula ¹⁰¹⁰ 10 $e^{\theta i}=\cos(\theta)+i\sin(\theta)$, see above for more we can write the sum of $\cos(\frac{2\pi}{9})+\cos(\frac{4\pi}{9})+\cos(\frac{6\pi}{9})+\cos(\frac{8\pi}{9})$ as

This is actually the sum of eight of the nine roots of unity in disguise! Note that if we add $2\pi$ to anything in the form $e^{ai}$, this has no effect (again, Euler’s formula and the fact that $2\pi$ radians is a full rotation). Therefore, the previous expression is the same as

which can be re-ordered as

which are all the ninth roots of unity (except $1$).

Thus we can write that

which means that