Absolute Value of Negative

Theorem

Let $x \in \R$ be a real number.

Then:

$\size x = \size {-x}$

where $\size x$ denotes the absolute value of $x$.

Proof

Let $x \ge 0$.

By definition of absolute value:

$\size x = x$

We have that:

$-x < 0$

and so by definition of absolute value:

$\size {-x} = -\paren {-x} = x$

$\Box$

Now let $x < 0$.

By definition of absolute value:

$\size x = -x$

We have that:

$-x > 0$

and so: and so by definition of absolute value:

$\size {-x} = -x$

$\Box$

In both cases it is seen that:

$\size x = \size {-x}$

$\blacksquare$