Absolute ratios

Question

I'm curious about the following idea:

suppose we have two values $P$ and $Q$, and the magnitude of the ratio $\frac{P}{Q}$ is between $0$ and $\infty$. If $P$ is smaller, then it's between $0$ and $1$. If $Q$ is smaller, it's between $1$ and $\infty$ (but the ratio $\frac{P}{Q}$ is between $0$ and $1$).

Is there a way to denote the "absolute ratio" (my term) that is always the ratio that is between $0$ and $1$ (either $\frac{P}{Q}$ or $\frac{Q}{P}$)?

As an example, the $\operatorname{absratio}(10,1) = \operatorname{absratio}(1,10) = 0.1.$

Answer 1

Because we're not supposed to start sentences with mathematical symbols, I'm presenting the answer this way: $$\displaystyle \min \biggl(\left\{\frac{P}{Q} ,\frac{Q}{P}\right\}\biggr).$$

Answer 2

Here are a couple of candidates: $$ \frac12\left(\frac PQ+\frac QP-\left|\frac PQ-\frac QP\right|\right) $$ and $$ \frac{|P+Q|-|P-Q|}{|P+Q|+|P-Q|} $$