The absolute value of a real number $r$ is defined to be the additive inverse of $r$ is $r < 0$, and $r$ is $r \geq 0$, where $0$ is the additive identity of the commutative group $\mathbb{R}$.
Suppose we consider the multiplicative commutative group $\mathbb{R}_{>0}$. We can play the same trick and define a "multiplicative absolute value" by $|r| == \frac{1}{r}$ if $r <1$, and $|r| = r$ is $r \geq 1$.
Does this function have a name, or a symbol? Moreover, does this concept (usefully) extend to the level of some type of ordered commutative groups?