Definition:Archimedean Property
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Definition
Let $\struct {S, \circ}$ 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:
- $m \cdot a = \begin{cases} a & : m = 1 \\ a \circ \paren {\paren {m - 1} \cdot a} & : m > 1 \end {cases}$
Archimedean Property on Norm
Let $n: S \to \R$ be a norm on $S$.
Then $n$ satisfies the Archimedean property on $S$ if and only if:
- $\forall a, b \in S: n \paren a < n \paren b \implies \exists m \in \N: n \paren {m \cdot a} > n \paren b$
Using the more common symbology for a norm:
- $\forall a, b \in S: \norm a < \norm b \implies \exists m \in \Z_{>0}: \norm {m \cdot a} > \norm b$
Archimedean Property on Ordering
Let $\preceq$ be an ordering on $S$.
Then $\preceq$ satisfies the Archimedean property on $S$ if and only if:
- $\forall a, b \in S: a \prec b \implies \exists m \in \Z_{>0}: b \prec m \cdot a$
Also see
Source of Name
This entry was named for Archimedes of Syracuse.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Archimedean property