So I'm working on this problem, and I'm having trouble understanding what it actually is asking me:
Rotate the unit circle C by a fixed angle $\alpha$, say $R: C \rightarrow C$ (In polar coordinates, this transformation R sends $(1, \theta)$ to $(1, \theta + \alpha)$. If $\alpha / \pi$ is rational, show that each orbit of R is a finite set.
What does "each orbit of R" mean?
I would guess that the orbit of $ x $ under $ R $ is the set $ \{ R^n x | n \in \mathbb{Z} \} $, where $ R^n $ is just the repeated action of $ R $ on $ x $, ie. $ R^n $ sends $ (1,\theta) $ to $ (1, \theta + \alpha n ) $. If $ \alpha/\pi $ is rational, then $ R^n $ is identity for some $ n > 0 $, which implies the orbit is finite.