Relationship between circlular $i^n$ property and that $e^i{\theta}$ traces the unit circle.

Question

That $e^{i\theta}$ traces a circle $\theta \in R$ has been well discussed elsewhere.

However, I was always curious with it's relation to the following property of $i$:

$i^0$ = 1

$i^1$ = i

$i^2$ = -1

$i^3$ = -i

$i^4$ = 1

These properties seem like they should be related, but I can't map them too eachother.

Answer 1

Since $i=e^{i\pi/2}$, you see that taking powers of $i$ is equivalent to making successive rotations of $\pi/2$ around the unit circle. This gives exactly the periodic sequence of $4$ elements you describe.