# Definition of Real Absolute Value

This definition is correct:

"let $A,B \in \mathbb{R}$, $B$ is absolute value of $A$, $B \triangleq|A|$, if $B=\begin{cases} A, & \mbox{if }A \geq0 \\ (-A), & \mbox{if }A \leq 0 \end{cases}$"

??

You probably mean $-(A)$ where you wrote $(-A)$, even though it is the same thing. – Git Gud Aug 12 '13 at 10:23
@GarnakOlegovitc Yes. That's why your definition, the way you wrote it, doesn't work. What you looking for is something like: given $A\in \Bbb R$, define the absolute value of $A$ and denote it by $|A|$ as $$|A|=\begin{cases} A, & \mbox{if }A \geq0 \\ (-A), & \mbox{if }A \leq 0 \end{cases}.$$ – Git Gud Aug 12 '13 at 10:36
@GarnakOlegovitc Yes, $|A|\in \Bbb R$, but that's by definition. You should think of absolute value as a function which is written in sort of an infix notation: $$|\cdot |\colon \Bbb R\to \Bbb R, A\mapsto \begin{cases} A, & \mbox{if }A \geq0 \\ (-A), & \mbox{if }A \leq 0 \end{cases}$$ – Git Gud Aug 12 '13 at 10:45
@GarnakOlegovitc Yes, that's correct and that's an alternative way to define absolute value. Bu the use of $\iff$ is very important. You didn't use it in your question. – Git Gud Aug 12 '13 at 10:52