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 $$n$$.

First Root of Unity
The root $$e^{2 i \pi / n}$$ is known as the first $$n$$th root of unity.

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 $$z^n = 1$$.

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

Hence the result.