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. | + | 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 numbers and the rationals. Examples of non-Archimedean are less simple. |
Revision as of 07:02, February 22, 2007
A ring R is said to be Archimedean if the ring is ordered, has a metric and for all in R, x non-zero, there exists in the natural numbers such that . Here concatentation with denotes adding 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 rationals. Examples of non-Archimedean are less simple.
