Other absolute value definitions in $\mathbb R$ I know these definitions for the absolute value (or module): given a real number $x$, then
$$\bbox[yellow]
{|x|=\begin{cases}x & \text{if } x\geq 0\\ -x& \text{if } x< 0\end{cases}}$$
or
$$\bbox[yellow]
{|x|=\max\{x,-x\}}$$
Are there other definitions in $\mathbb R$ (for example using $\text{sgn}\, x$)?
PS: The question is referred to high school students.

Here's some I could think of:

*

*$|x|$ can be defined as the (unsigned) distance of $x$ from the origin.

*It's the even extension of $f:[0,\infty)\to\mathbb R$ where $f(x):=x$.

*$|x|$ is the unique norm on $\mathbb R$ with
$|1|=1$.

