In mathematics, and chiefly number theory, the padic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of absolute value.
First described by Kurt Hensel in 1897^{[1]}, the padic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of padic analysis essentially provides an alternative form of calculus.
More formally, for a given prime p, the field Q_{p} of padic numbers is a completion of the rational numbers. The field Q_{p} is also given a topology derived from a metric, which is itself derived from an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Q_{p}. This is what allows the development of calculus on Q_{p}, and it is the interaction of this analytic and algebraic structure which gives the padic number systems their power and utility.
The p in padic is a variable and may be replaced with a constant (yielding, for instance, "the 2adic numbers") or another placeholder variable (for expressions such as "the ladic numbers").
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