Why is the absolute value of a equal to -a when a < 0?

Question

I'm doing some math work involving proofs, and one of the definitions is:

|a| = -a when a < 0

Isn't the absolute value of a, positive a no matter what a is in the beginning? Am I looking at this wrong? Could use an explanation.

Answer 1

If $a < 0$ then $-a > 0$, so $-a$ is positive.

The point is that if $a = -2$ for instance, then $\lvert -2 \vert = -(-2) = 2.$

Answer 2

$|a|$ is always positive, but if $a$ is negative at the beginning, the absolute value is equal to $-a$, because negating a negative number results in a positive. This is merely a a property of $a$, not of $|a|$.

Answer 3

If $a<0$ you can consider it as $-|a|$ since $|a|$ is absolute value(like unsigned value).

So $a=-|a|$, and $|a|=-a$

Example: $a=-1$, $|a|=1=-(-1)=-a$

Answer 4

$\renewcommand\sgn{\operatorname{sgn}}$The general idea is this: For any nonzero real number $a$, we can decompose it as $s \cdot t$, where $s = +1$ or $-1$ is a sign, and $t > 0$ is a positive number. I'll call these $s = \sgn(a)$ and $t = |a|$. (Here $\sgn$ is called the signum function.)

If $a$ is positive, then put $\sgn(a) = +1$ and $|a| = a$. Then it's clear that $|a|$ is positive, and $\sgn(a) \cdot |a| = +1 \cdot a = 1$.

If $a$ is negative, then put $\sgn(a) = -1$ and $\color{red}{|a| = -a}$. Then $|a|$ is positive ($a$ is negative, so $-a$ is positive), and $\sgn(a) \cdot |a| = -1 \cdot -a = a$.

(If $a = 0$ then $\sgn(a)$ is undefined, but $|a| = 0$.)

Answer 5

"Isn't the absolute value of $a$, positive $a$ no matter what $a$ is in the beginning?"

Yes. $-a $ is a positive number.

"Am I looking at this wrong?" Yes. $-a$ is a positive number.

"Could use an explanation?"

$a < 0$. So $a$ is negative. Which means $-a > 0$ and $-a$ is positive.

Also $a = -|a|$ (an equation with 2 negative values). So if you negate both sides you get $-a = -(-|a|) = |a|$.

Or a specific example. Let a = -25. Then $|a| = 25$. $-a = -(-25) = 25$. SO $|a| = 25 = -a$.

Moral: when we say a "negative times a negative is a positive" we mean it. $-a$ is positive if $a$ is negative.