Proof that for all real numbers a, |-a| = |a|

We can think of the absolute value of a number as its distance from 0. So, the absolute value of a, which is denoted by $latex |a|$ is always greater than 0. In this post, we are going to prove that for all real numbers a, |-a| = |a|.

There are two possible cases: $latex a \geq 0$ and $latex b < 0$. (i) For $latex a \geq 0$, $latex since - a \leq 0$ $latex |-a| = -(-a) = a$ (since a is negative, we negate it to make it positive) $latex |a| = a$ (ii) For $latex a < 0$, since $latex - a > 0$,

$latex |-a|$ = $latex -a$ (since a is negative, we negate it to make it positive)
$latex |a| = -a$

By (i) and (ii), for any real number a,

$latex |-a| = |a|$.

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