Complex Roots of Unity/Examples/4th Roots

Example of Complex Roots of Unity
The complex $4$th roots of unity are the elements of the set:
 * $U_n = \set {z \in \C: z^4 = 1}$

They are:

Proof
By definition, the first complex $4$th root of unity $\alpha$ is given by:

We have that:
 * $e^{0 i \pi / 4} = e^0 = 1$

which gives us, as always, the zeroth complex $n$th root of unity for all $n$.

The remaining complex $4$th roots of unity can be expressed as $e^{4 i \pi / 4} = e^{i \pi}$ and $e^{6 i \pi / 4} = e^{3 i \pi / 2}$, but it is simpler to calculate them as follows: