Rules for modulus of a (|a|)?

Question

So I am wondering about this statement:

$|a| = a$ if $a \le 0$ and $-a$ if $a<0$

Why is it $-a$ if $a<0$? Shouldn't the modulus always be positive?

Answer 1

The modulus should never be negative. The rule $-a$ if $a<0$ ensures this is so. (Because when $a$ is negative, $-a$ is positive.)

P.S. You said, "Shouldn't the modulus always be positive?" No. The modulus can be zero.

Answer 2

It might help to not think of "$-a$" as a negative number, because its sign depends on what $a$ actually is. You might rather think of $-a$ as meaning "the opposite of $a$" or as "the additive inverse of $a$."

The opposite of $1$ is $-1$ since $1\ + \ -1 = 0$.

The opposite of $-2$ is $2$ since $-2\ + \ 2 = 0$.

The opposite of $0$ is $0$ since $0\ + \ 0 = 0$.

What $|a|$ actually is, is the distance the number $a$ is from $0$. Distance is never negative. If $a$ is negative then it lies a distance of $|a|=-a$ from $0$. If it is positive it lies a distance of $|a|=a$ from $0$.