Quotient Group of Reals by Integers is Circle Group

Theorem
Let $\left({\Z, +}\right)$ be the additive group of integers.

Let $\left({\R, +}\right)$ be the additive group of real numbers.

Let $K$ be the circle group.

Then the quotient group of $\left({\R, +}\right)$ by $\left({\Z, +}\right)$ is isomorphic to $K$.

Proof
Define $\phi: \R / \Z \to K$ by:


 * $\phi \left({x + \Z}\right) = \exp \left({2\pi i x}\right)$

Then $\phi$ is well-defined.

For, if $x + \Z = y + \Z$, then $y = x + n$ for some $n \in \Z$, and:


 * $\exp \left({2\pi i \left({x + n}\right)}\right) = \exp \left({2\pi i x}\right)$

by Complex Exponential Function has Imaginary Period.

Moreover, by Exponent of Sum:


 * $\phi \left({x + y + \Z}\right) = \exp \left({2 \pi i \left({x + y}\right)}\right) = \exp \left({2\pi i x}\right) \exp \left({2\pi i y}\right)$

meaning $\phi$ is a group homomorphism.

By Euler's Formula:


 * $\exp \left({2\pi i x}\right) = \cos \left({2\pi i x}\right) + i \sin \left({2\pi i x}\right)$

so that, by Sine and Cosine are Periodic on Reals:


 * $\phi \left({x}\right) = 1$ iff $x + \Z = 0 + \Z$.

Hence, by Kernel is Trivial iff Monomorphism, $\phi$ is a monomorphism.

From polar form for complex numbers, it follows that all $z \in \C$ with $\left\vert{z}\right\vert = 1$ are of the form:


 * $z = \exp \left({2\pi i x}\right)$

for some $x \in \R$.

Hence, $\phi$ is also an epimorphism.

Thus $\phi: \R / \Z \to K$ is a group isomorphism.