Definition:Archimedean Property

Definition

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

For all $a \in S$ and for all $m \in \Z$ such that $m \ge 1$, let $m \cdot a$ be defined as:

$\forall a \in S, \forall m \in \N, m \ge 1: m \cdot a = \begin{cases} a & : m = 1 \\ a \circ \left({\left({m - 1}\right) \cdot a}\right) & : m > 1 \end {cases}$

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:

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

Archimedean Property on Ordering

Let $\prec$ be a strict ordering on $\left({S, \circ}\right)$.

Then $\prec$ satisfies the Archimedean property on $S$ iff:

$\forall a, b \in S: a \prec b \implies \exists m \in \N: b \prec m \cdot a$

Also see

• Infinitesimal