# 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 where *r* and *s* are integers coprime to *p* and *n* is an integer. We define the *p*-adic valuation on **Q** by

The *p*-adic metric is then defined by

## 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.