# 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:

 $\displaystyle e^{0 i \pi / 4}$ $=$ $\displaystyle 1$ $\displaystyle e^{i \pi / 2}$ $=$ $\displaystyle i$ $\displaystyle e^{i \pi}$ $=$ $\displaystyle -1$ $\displaystyle e^{3 i \pi / 2}$ $=$ $\displaystyle -i$

## Proof

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

 $\displaystyle \alpha$ $=$ $\displaystyle e^{2 i \pi / 4}$ $\displaystyle$ $=$ $\displaystyle e^{i \pi / 2}$ $\displaystyle$ $=$ $\displaystyle \cos \frac \pi 2 + i \sin \frac \pi 2$ $\displaystyle$ $=$ $\displaystyle 0 + i \times 1$ Cosine of $\dfrac \pi 2$, Sine of $\dfrac \pi 2$ $\displaystyle$ $=$ $\displaystyle i$

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:

 $\displaystyle \alpha^2$ $=$ $\displaystyle i^2$ $\displaystyle$ $=$ $\displaystyle -1$ Definition of Imaginary Unit

 $\displaystyle \alpha^3$ $=$ $\displaystyle \alpha^2 \times \alpha$ $\displaystyle$ $=$ $\displaystyle \paren {-1} \times i$ $\displaystyle$ $=$ $\displaystyle -i$

$\blacksquare$