Every non-zero rational number may be written uniquely in the form where r and s are integers coprime to p and n is an integer. We define the p-adic valuation on Q by
The p-adic metric is then defined by
The p-adic metrics and the usual absolute value on Q are mutually inequivalent. Ostrowkski's theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.