---
tag: 0HQARMKEV5
kind: definition
name: Archimedean ordered field
msc: [12J15]
defines: 0XH79S06J3
uses: [010JCV9J4E]
sources:
  - url: https://en.wikipedia.org/wiki/Archimedean_property
    locator: Definition for linearly ordered groups
  - url: https://en.wikipedia.org/wiki/Archimedean_property
    locator: Ordered fields
---

## Definition

An [[ordered field|010JCV9J4E]] $F$ is Archimedean when, for all $x$ and $y$ in $F$ with $y > 0$, there is a natural number $n$ with $n y > x$.
Here $n y$ means $y + y + \dots + y$, with $n$ terms.

## Remarks

With $y = 1$, the definition gives the [[Archimedean property|0A74SX55QP]]: for each $x$, some natural number exceeds $x$. Conversely, this property implies the definition.
An equivalent condition: for each $\varepsilon > 0$ in $F$, there is a natural number $n \ge 1$ with $1/n < \varepsilon$.
The field of rational functions $\mathbb{R}(t)$, with the order in which $t$ is larger than every real number, is not Archimedean.

## Explanations

### L1 human

You can reach any distance with enough small steps of a fixed size.
In an Archimedean field there is no number so large that no multiple of a given positive number can pass it.