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. |