Signum Function is Quotient of Number with Absolute Value

Theorem
Let $x \in \R_{\ne 0}$ be a non-zero real number.

Then:
 * $\operatorname{sgn} \left({x}\right) = \dfrac x {\left\vert{x}\right\vert} = \dfrac {\left\vert{x}\right\vert} x$

where:
 * $\operatorname{sgn} \left({x}\right)$ denotes the signum function of $x$
 * $\left\vert{x}\right\vert$ denotes the absolute value of $x$.

Proof
Let $x \in \R_{\ne 0}$.

Then either $x > 0$ or $x < 0$.

Let $x > 0$.

Then:

Similarly:

Let $x < 0$.

Then:

Similarly: