Multiplicative Group of Complex Numbers is not Isomorphic to Additive Group of Real Numbers

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

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

Then $\struct {\C_{\ne 0}, \times}$ is not isomorphic to $\struct {\R, +}$.