Dense on the unit circle

Question

I am reading:

"It is sufficient to show that the points $z_n = e^{2\pi in \xi}$ $\:\:n = (1, 2, 3...)$ are dense on the unit circle. ( $\xi$ is an irrational number)"

How is this possible? Can anyone give me an intuitive explanation of this? (Not a solution)

Thank you.

Answer 1

Intuitive explanation. Let $\alpha$ be an irrational angle - not a rational multiple of $2\pi$. Then if you start at $(1,0)$ and step around the unit circle in steps of size $\alpha$ the set of points you reach will come as close as you like to any other point. That follows from the fact that you can never get back to your starting point in a finite number of steps (the irrationality).