Showing that a subgroup $Gz_0$ of $S^1$ is dense and that the quotient $S^1/G$ is a topological group.

Question

Let $G = \langle e^{i2\pi r}\rangle$ where r is an irrational number and let $Gz_0 = \{ zz_0 | z \in G \}$ where $z_0 \in S^1$. I want to show that $Gz_0$ is dense in $S^1$ and that the function $f:(S^1/G) \times(S^1/G) \to S^1/G$ such that $f(x,y) = xy^{-1}$ is continuous.