Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers/Proof 1

Proof
Consider the element $-1 \in \struct {\R_{\ne 0}, \times}$.

We have that:
 * $-1 \times -1 = 1$

From Real Multiplication Identity is One it follows that $-1$ is of order $2$ in $\struct {\R_{\ne 0}, \times}$.

From Order of Isomorphic Image of Group Element, it is sufficient to demonstrate that there exists no element of order $2$ in $\struct {\R, +}$.

From Real Addition Identity is Zero, that means finding $x$ such that $x + x = 0$.

But:
 * $x + x = 0 \implies x = 0$

and as $0$ is the identity of $\struct {\R, +}$, there is indeed no element of order $2$ in $\struct {\R, +}$.

The result follows from Order of Isomorphic Image of Group Element.