TITLE: Show that $\mathbb{R}/\mathbb{Z}$ is isomorphic to $\{e^{i\theta} : 0 \le \theta \le 2\pi \}$
QUESTION [9 upvotes]: This question is asking to prove that the quotient group $\mathbb{R}/\mathbb{Z}$ is isomorphic to the group of complex numbers with modulus 1 (under multiplication).   It's hard for me to visualize the structure of $\mathbb{R}/\mathbb{Z}$, so I can't think of an isomorphism. I feel like the first isomorphism theorem can help? Help is appreciated.

REPLY [4 votes]: You can see the problem this way : $\mathbb{R}/\mathbb{Z}=[0,1)$. Then, at each point in this interval correspond an unique point in the circle $\{ e^{2\pi i \theta} : \theta \in [0,1)  \}$.
About the group operations, when you add two numbers in $[0,1)$, you multiply them in the set representing the circle (adding the argument).