---
tag: BT6T702C28
kind: concept
name: p-adic number
msc: [12J25, 11S99]
sources:
  - url: https://en.wikipedia.org/wiki/P-adic_number
same_as:
  wikidata: Q311627
  wikipedia: P-adic number
---

## Summary

For a [[prime number|01826KMG2H]] $p$, the $p$-adic numbers form the field $\Q_p$, the completion of $\Q$ for the [[p-adic absolute value|WGN82TCEF2]].
In $\Q_p$, a number is small when it is divisible by a high power of $p$.
The field $\Q_p$ is a [[local field|J85FZEQG38]] of characteristic $0$.
Questions about congruences modulo all powers of $p$ become questions about equations in $\Q_p$.

## Explanations

### L1 human

A real number can be written with infinitely many digits after the decimal point.
A $p$-adic number is written in base $p$ with infinitely many digits to the left instead, and only finitely many to the right.
Two numbers are close when their difference is divisible by a large power of $p$.
For example, in $\Q_5$ the powers $5, 25, 125, \ldots$ get smaller and smaller and tend to $0$.
The sum $4 + 4 \cdot 5 + 4 \cdot 5^2 + \cdots$ converges in $\Q_5$, and its value is $-1$.