Difference between revisions of "Archimedean"
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A [[ring (mathematics)|ring]] '''R''' is said to be '''Archimedean''' if the ring is ordered, has a [[metric (mathematics)|metric]] <math>| |</math> and for all <math>x,y</math> in '''R''', x non-zero, there exists <math>n</math> in the natural numbers such that <math>n|x| > y </math>. Here concatentation with <math> n </math> denotes adding <math>n</math> times. Informally, a ring is Archimedean if it has no infinitely small or infinitely large elements. Examples of Archimedean rings include the [[real number]]s and the [[rational number]]s. Examples of non-Archimedean are less simple. | A [[ring (mathematics)|ring]] '''R''' is said to be '''Archimedean''' if the ring is ordered, has a [[metric (mathematics)|metric]] <math>| |</math> and for all <math>x,y</math> in '''R''', x non-zero, there exists <math>n</math> in the natural numbers such that <math>n|x| > y </math>. Here concatentation with <math> n </math> denotes adding <math>n</math> times. Informally, a ring is Archimedean if it has no infinitely small or infinitely large elements. Examples of Archimedean rings include the [[real number]]s and the [[rational number]]s. Examples of non-Archimedean are less simple. | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 20:01, September 17, 2011
A ring R is said to be Archimedean if the ring is ordered, has a metric Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | |} and for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y} in R, x non-zero, there exists in the natural numbers such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n|x| > y } . Here concatentation with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } denotes adding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} times. Informally, a ring is Archimedean if it has no infinitely small or infinitely large elements. Examples of Archimedean rings include the real numbers and the rational numbers. Examples of non-Archimedean are less simple.
