Suppose I have a unit circle which is also a clock with a single moving hand. Is it possible to rotate the hand around the clock (clockwise) over and over again without hitting the same point twice? How many times can this be done?
I was thinking that if we let $\theta$ be such that $\theta/2\pi$ is irrational, and we start with the hand pointing at 12.00, then I can keep rotating by $\theta$ without hitting the same point twice.
Yes, that is possible. You can do that "forever" in such a way that the function $f:\mathbb{N}\to \mathbb{C}$ defined by $f(n):= e^{-2\pi in\theta}$ is injective if $\theta\in \mathbb{R}\setminus \mathbb{Q}$.
Assume it wasn't injective. Then you could find distinct $n,m\in \mathbb{N}$ s.t. $f(n)=f(m)$, i.e. $$e^{-2\pi i\theta(n-m)}=1$$ which would happen if and only if $\theta (n-m)$ is an integer. But that can't happen since $\theta$ is not rational.