Other absolute value definitions in $\mathbb R$

Question

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.

Answer 1

First of all, what you are asking is not really notation. The word "notation" refers to how we write a particular concept. The concept of "absolute value" has really only one notation: the vertical bars. That is, $|x|$ is the standard notation for the concept "absolute value of $x$".

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What you are asking is the definitions of $|x|$, and in particular, you are listing two equivalent definitions of $|x|$.

I can think of two more equivalend definitions for $|x|$:

• $|x| = \mathrm{sign}(x) \cdot x$
• $|x| = \sqrt{x^2}$

Answer 2

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 $|1|=1$.