Rational numbers are countable as shown by the usual table here: https://aminsaied.wordpress.com/2012/05/21/diagonal-arguments/
So, counting in the zig-zag manner as shown in the table, $1/1$ is the first rational, $3/2$ is the eighth, $1/4$ is the tenth etc. I don't know what the technical name is for “first”, “eighth” and “tenth” - maybe “position” or "rank". My question is, given the integers $a$ and $b$ how can I calculate the position or rank of the rational $\frac{a}{b}$ (without physically counting the numbers)?
This is from G. H. Hardy, A Course of Pure Mathematics, page 1 of Chapter 1. http://www.gutenberg.org/files/38769/38769-pdf.pdf?session_id=4495e3437d1f87369cf842af766691752f0981be
You are probably looking for the more complicated formula ...