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Can someone explain me why does the unit circle can be expressed as the following quotient : $ \mathbb{R} / \mathbb{Z} $ ?

Thanks !

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@NilsMatthes: How on earth do you get two circles? – Harald Hanche-Olsen Apr 28 '12 at 8:03
@Harald, yeah sorry, fixed that. Too early in the morning here in Germany for me ;) – Nils Matthes Apr 28 '12 at 8:05
up vote 2 down vote accepted

Well, consider the map $x \mapsto \mathrm{e}^{2 \pi i x}$ from $[0,1)$ to $\mathbb{S}^1$. Prov that is is bijective. This may give you a hint.

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Short answer: Think of the unit circle as residing in the complex plane, and consider the map $x\mapsto e^{2\pi ix}$. Now realize that $x_1$ and $x_2$ map to the same value if and only if $x_1-x_2\in\mathbb{Z}$. Or, in more group theoretic language: The map is a continuous group homomorphism of the additive group $\mathbb{R}$ onto the multiplicative group of the circle, and the kernel is $\mathbb{Z}$.

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