Editing P-adic number (section)

== Motivation ==

Roughly speaking, [[modular arithmetic]] modulo a positive integer {{mvar|n}} consists of "approximating" every integer by the remainder of its [[Euclidean division|division]] by {{mvar|n}}, called its ''residue modulo'' {{mvar|n}}. The main property of modular arithmetic is that the residue modulo {{mvar|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 {{mvar|n}}. If one knows that the absolute value of the result is less than {{mvar|n/2}}, this allows a computation of the result which does not involve any integer larger than {{mvar|n}}.

For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying the [[Chinese remainder theorem]] for recovering the result modulo the product of the moduli.

Another method discovered by [[Kurt Hensel]] consists of using a prime modulus {{mvar|p}}, and applying [[Hensel's lemma]] for recovering iteratively the result modulo <math>p^2, p^3, \ldots, p^n, \ldots</math> If the process is continued infinitely, this provides eventually  a result which is a {{mvar|p}}-adic number.