# P-adic metric

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The p -adic metric, with respect to a given prime number p, on the field Q of rational numbers is a metric which is a valuation on the field.

## Definition

Every non-zero rational number may be written uniquely in the form ${\displaystyle p^{n}.r/s}$ where r and s are integers coprime to p and n is an integer. We define the p-adic valuation ${\displaystyle |\cdot |_{p}}$ on Q by

${\displaystyle |0|_{p}=0,\,}$
${\displaystyle \left|p^{n}.{\frac {r}{s}}\right|_{p}=p^{-n}.\,}$

The p-adic metric is then defined by

${\displaystyle d_{p}(x,y)=|x-y|_{p}.\,}$

## Properties

The p-adic metric on Q is not complete: the p-adic numbers are the corresponding completion.

## Ostrowksi's Theorem

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.