Signum Function is Quotient of Number with Absolute Value

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

Then:
 * $\map \sgn x = \dfrac x {\size x} = \dfrac {\size x} x$

where:
 * $\map \sgn x$ denotes the signum function of $x$
 * $\size x$ 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: