The number of dimensionless variables equivalent to a given set of original variables is given by the Pi theorem, also called Buckingham’s theorem. By the Pi theorem, the number of dimensionless variables corresponding to a number $$n$$ of original variables that describe some physical problem is equal to $$n -m$$, where $$m$$ is the number of dimensions by which the problem must be expressed. If you want to go back to the original source of the proofs (the theorem was not proved in the foregoing material, just demonstrated), see Buckingham (1914, 1915).