---
tag: 1H5F1BP6XS
kind: concept
name: Non-Archimedean absolute value
msc: [12J25, 12J10]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_value_(algebra)
---

## Summary

A non-Archimedean absolute value on a field is an absolute value that satisfies the strong triangle inequality $\abs{x + y} \le \max(\abs{x}, \abs{y})$, see the [[definition|8XDYNK6TV0]].
It generalizes the usual [[absolute value|0FFXKTKKQC]] of real numbers in a different direction: small elements stay small under addition.
The [[p-adic absolute value|WGN82TCEF2]] is the main example, and the induced metric is an [[ultrametric|A6DPR620B5]].
These absolute values are the same as real-valued [[valuations|B85T5VKR3M]], up to a choice of base for the exponent.

## Explanations

### L1 human

The usual absolute value makes $1 + 1 + \cdots + 1$ grow without end.
A non-Archimedean absolute value never lets a sum be larger than its largest term.
For the $2$-adic absolute value, a number is small when it is divisible by a high power of $2$: $\abs{8}_2 = 1/8$ and $\abs{3}_2 = 1$.
Then $\abs{8 + 16}_2 = \abs{24}_2 = 1/8$, which is not more than the larger of $1/8$ and $1/16$.
In such a geometry every triangle is isosceles, and every point of a disc is a centre of it.