Homomorphism from Reals to Circle Group/Corollary

Theorem
Let $\struct {\R, +}$ be the additive group of real numbers.

Let $\struct {C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.

Let $\phi: \struct {\R, +} \to \struct {C_{\ne 0}, \times}$ be the mapping defined as:
 * $\forall x \in \R: \map \phi x = \cos x + i \sin x$

Then $\phi$ is a (group) homomorphism.

Proof
By Euler's Identity, $\phi$ can also be expressed as:


 * $\forall x \in \R: \map \phi x = e^{i x}$

From Homomorphism from Reals to Circle Group, $\phi$ is a homomorphism from $\struct {\R, +}$ to the circle group $\struct {K, \times}$.

From Circle Group is Infinite Abelian Group, we note that $\struct {K, \times}$ is a subgroup of the multiplicative group of complex numbers.