TITLE: Why is $\mathbb{R}/\mathbb{Z}$ isomorphic to the complex numbers of module one?
QUESTION [8 upvotes]: Wikipedia states that the quotient group $\mathbb{R}/\mathbb{Z}$ is isomorphic to all complex numbers of module $1$. I have a hard time making sense out of this, and in particular, how complex numbers even come into play. What I am asking for, if possible, is an intuitive answer. Giving a bijective homomorphism is not considered a sufficient answer unless an intuitive explanation proves difficult.

REPLY [11 votes]: $\mathbb R/\mathbb Z$ identifies each point on the real line with a unique point in the interval $[0,1)$. We can stretch this interval by a factor of $2\pi$ and wrap it around the circumference of the unit circle in the complex plane.
Because the real number $1$ maps to $0$ in the interval $[0,1)$ we find that joining the ends together works, and each real number can be identified with a number of circuits of the unit circle.
It is like treating the whole real line as a piece of elastic and wrapping it around the unit circle so that all the integers end up in the same place.