Putting Numbers on a Circle

Question

If I have a circle and I start numbering points along the circumference with all the natural numbers: 1, 2, 3, 4, and so on, such that the length of the arc between two consecutive numbers is constant, what angle should be enclosed between the two radii of such consecutive points so that as I continue numbering points, I never have to number the same point twice?

Answer 1

Suppose that the angle separating consecutive numbers is $\alpha$. Without loss of generality, we can start numbering at $0$ and make it the reference angle $0$. Then $n$ would be at angle $n\alpha$, modulo $2\pi$. You never want to label the same point twice, which means that we require $$n\alpha \not\equiv m\alpha \pmod{2\pi}$$ for all $n\neq m$. Equivalently, the numbers $m$ and $n$ will be labelled at the same point if and only if there is some integer $k$ such that $$(n-m)\alpha = 2\pi k.$$ In other words, $\alpha$ is a rational multiple of $\pi$. It follows that any choice of $\alpha$ which is not a rational multiple of $\pi$ will never see the same point labelled twice.

Answer 2

Hint: Think about when you will have to number the same point twice.

For example, say your angle is $x\cdot 2\pi$, where $x\in[0,1]$ (i.e., $x$ tells you what proportion of the 360 degre angle your angle is). That means you are numbering the points at angles $2\pi x, 2\pi\cdot 2x, 2\pi\cdot 3x,\dots$

Now, what is the condition that must be met in order for you to number the same point twice?