# Definition:Archimedean Property

## Contents

## Definition

Let $\left({S, \circ}\right)$ be a closed algebraic structure on which there exists either an ordering or a norm.

Let $\cdot: \Z_{>0} \times S \to S$ be the operation defined as:

- $m \cdot a = \begin{cases} a & : m = 1 \\ a \circ \left({\left({m - 1}\right) \cdot a}\right) & : m > 1 \end {cases}$

### Archimedean Property on Norm

Let $n: S \to \R$ be a norm on $S$.

Then $n$ satisfies the **Archimedean property on $S$** iff:

- $\forall a, b \in S: n \left({a}\right) < n \left({b}\right) \implies \exists m \in \N: n \left({m \cdot a}\right) > n \left({b}\right)$

Using the more common symbology for a norm:

- $\forall a, b \in S: \left \Vert{a}\right \Vert < \left \Vert{b}\right \Vert \implies \exists m \in \Z_{>0}: \left \Vert{m \cdot a}\right \Vert > \left \Vert{b}\right \Vert$

### Archimedean Property on Ordering

Let $\preceq$ be an ordering on $S$.

Then $\preceq$ satisfies the **Archimedean property on $S$** iff:

- $\forall a, b \in S: a \prec b \implies \exists m \in \Z_{>0}: b \prec m \cdot a$

## Source of Name

This entry was named for Archimedes of Syracuse.