Standard part function: Difference between revisions

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Revision as of 12:43, 28 December 2013

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==Definition==

[[File:Standard part function.jpgsvg|thumb480px|upright=2thumb|right|The standard part function associates to every finite hyperreal, the unique real number infinitely close to it. The bottom line represents the "thin" real continuum. The line at top represents the "thick" hyperreal continuum. The "infinitesimal microscope" is used to view an infinitesimal neighborhood of 0standard reals.]]

Nonstandard analysis deals primarily with the pair <math>\mathbb{R}\subset{}^{\ast}\mathbb{R}</math>, where the [[hyperreal number|hyperreal]]s <math>{}^{\ast}\mathbb{R}</math> are an extension of the reals <math>\mathbb{R}</math>, and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers (called a [[monad (non-standard analysis)|monad]], or '''halo''') of hyperreals infinitely close to it. The standard part function associates to a [[Wikt:finite|finite]] [[hyperreal number|hyperreal]] ''x'', the unique standard real number ''x₀'' which is infinitely close to it. The relationship is expressed symbolically by writing