Complex Numbers as External Direct Product

Theorem
Let $\left({\C_{\ne 0}, \times}\right)$ be the group of non-zero complex numbers under multiplication.

Let $\left({\R_{> 0}, \times}\right)$ be the group of positive real numbers under multiplication.

Let $\left({K, \times}\right)$ be the circle group.

Then:
 * $\left({\C_{\ne 0}, \times}\right) \cong \left({\R_{> 0}, \times}\right) \times \left({K, \times}\right)$

Proof
Identify $r e^{i \theta}\in \C_{\ne 0}$ with the ordered pair $\left({r, e^{i \theta} }\right): r \in \R_{> 0}, e^{i \theta} \in K$.

The result follows immediately from the associativity and commutativity of complex multiplication.