In mathematical terms, if we have a physically meaningful equation such as f ( q 1 , q 2 , … … , q n ) = 0 , where q 1 , … … , q n are any n physical variables, and there is a maximal dimensionally independent subset of size k , then the above equation can be restated as F ( π π 1 , π π 2 , … … , π π p ) = 0 , where π π 1 , … … , π π p are dimensionless parameters constructed from the q i by p = n − − k dimensionless equations — the so-called Pi groups — of the form π π i = q 1 a 1 q 2 a 2 ⋯ ⋯ q n a n , where the exponents a i are rational numbers. (They can always be taken to be integers by redefining π π i as being raised to a power that clears all denominators.) If there are ℓ ℓ fundamental units in play, then p ≥ ≥ n − − ℓ ℓ .