Function that maps the "pureness" of a rational number? By pureness I mean a number that shows how much the numerator and denominator are small.
E.g. $\frac{1}{1}$ is purest, $\frac{1}{2}$ is less pure (but the same as $\frac{2}{1}$), $\frac{2}{3}$ is less pure than the previous examples, $\frac{53}{41}$ is worse, .... $\pi$ isn't pure at all (as well as e...).

I have often used the sum of the terms in the Continued Fraction. This is


*

*finite for rational numbers

*the same for $x$ and $\frac1x$

*for $0\lt x\lt 1$, the same for $x$ and $1-x$


$$
1=(1)\to1
$$
$$
\frac12=(0,2)\to2\quad\text{and}\quad2=(2)\to2
$$
$$
\frac23=(0,1,2)\to3
$$
$$
\frac{53}{41}=(1,3,2,2,2)\to10
$$
etc.
