Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers

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

Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers.

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