More formally, for a given prime , the field of -adic numbers is a completion of the rational numbers. The field is also given a topology derived from a metric, which is itself derived from the -adic order, an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in . This is what allows the development of calculus on , and it is the interaction of this analytic and algebraic structure that gives the -adic number systems their power and utility.