P-adic number

In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number p rather than ten, and extending (possibly infinitely) to the left rather than to the right. Formally, given a prime number p, a p-adic number can be defined as a series where k is an integer (possibly negative), and each is a integer such that A p-adic integer is a p-adic number such that In general the series that represents a p-adic number is not convergent in the usual sense, but it is convergent for the p-adic absolute value where k is the least integer i such that (if all are zero, one has the zero p-adic number, which has 0 as its p-adic absolute value). Every rational number can be uniquely expressed as the sum of a series as above, with respect to the p-adic absolute value. This allows considering rational numbers as special p-adic numbers, and alternatively defining the p-adic numbers as the completion of the rational numbers for the p-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value. p-adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using p-adic numbers. Roughly speaking, modular arithmetic modulo a positive integer n consists of "approximating" every integer by the remainder of its division by n, called its residue modulo n. The main property of modular arithmetic is that the residue modulo n of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo n. If one knows that the absolute value of the result is less than n/2, this allows a computation of the result which does not involve any integer larger than n.