Complex Roots of Unity in Exponential Form

Theorem
Let $n \in \Z$ be an integer such that $n > 0$.

Let $z \in \C$ be a complex number such that $z^n = 1$.

Then:
 * $U_n = \left\{{e^{2 i k \pi / n}: k \in \N_n}\right\}$

where $U_n$ is the set of $n$th roots of unity.

That is:
 * $z \in \left\{ {1, e^{2 i \pi / n}, e^{4 i \pi / n}, \ldots, e^{2 \left({n - 1}\right) i \pi / n}}\right\}$

Thus for every integer $n$, the number of $n$th roots of unity is $n$.

Setting $\omega := e^{2 i \pi / n}$, $U_n$ can then be written as:
 * $U_n = \left\{{1, \omega, \omega^2, \ldots, \omega^{n-1}}\right\}$

Proof
Let $z \in \left\{{e^{2 i k \pi / n}: k \in \N_n}\right\}$.

Then:
 * $z^n \in \left\{{e^{2 i k \pi}: k \in \N_n}\right\}$

Hence $z^n = 1$.

Now suppose $z^n = 1$. We determine the possible values of $z$ using Roots of Complex Number.

Let $z = r e^{i \theta}$.

Then $\left|{z^n}\right| = 1 \implies \left|{z}\right| = 1$.

Similarly, we have $n \theta = 0 \bmod 2 \pi$.

So $\theta = \dfrac {2 k \pi} n$ for $k \in \Z$.

Hence the result.

Also see

 * Condition for Complex Root of Unity to be Primitive