Different ways of defining Absolute Value Calculus I presents this definition of absolute value:  
$$f(x)=y=|x|=\left\{\begin{array}{}\;\;\;x&\text{ if}\,\,\,x\geq 0\\-x&\text{ if}\,\,\,x<0\end{array}\right.$$
But you can also define absolute value as $$|x|= \sqrt{(\pm x)^2} $$
I can't think of any other ways to define it.
What are some other ways of defining the Absolute Value?
Please include what I might not have learned yet, i.e., definitions encountered in advanced classes. Thank you. 
 A: The supremum of two elements $x,y$ is usually written as $x \uparrow y$ and is characterized as follows:
$$\boxed{\forall z :: x \uparrow y \leq z \ \equiv \ x \leq z \land y \leq z}$$
That is, $z$ is an upper-bound precicely when it's greater than the sup (ie the least-upper-bound).
Now the definition of absolute-value is $$\boxed{|x| = x \uparrow -x}$$
This definition shows its worth in calculations.
Here are some examples...

Theorem: $x \leq |x|$, (and $-x \leq |x|$)
Proof: In the characterization of $\uparrow$, take $y := -x$ and take $z := |x|$. That's a pretty-easy proof :)

Theorem: $|-x| = |x|$
Proof: $|-x| = -x \uparrow (--x) = -x \uparrow x = x \uparrow -x = |x|$,
where we used the easily-proven property $a \uparrow b = b \uparrow a$.

Theorem: $|x+y| \leq |x| + |y|$ (The Triangle-Inequality)
Proof: Exercise! Try using this definition of absolute-value!

Theorem: $|x \cdot y| = |x| \cdot |y|$
Proof: Exercise! Try using this definition of absolute-value!


Hope this helps!
A: $|x|=x\text{sgn}(x) $,
where $\text{sgn}(x) = 1$ if $x>0$, $0$ if $x=0$, $-1$ if $x<0$.
