In [[non-standard analysis]], the '''standard part function''' "st" is a mathematicalfunction implementationthat provides a method of [[Pierreof detranslating Fermat]]'scalculations in the non-standard real numbers back to the real numbers. The standard part function was first defined by [[adequalityAbraham Robinson]]{{Citation needed|date=May 2011}}. as It is thea key ingredient in [[Abraham Robinson]]'s formalisationdefinitions of Leibniz'slimits [[infinitesimal]] definition (seein [[ghostsnon-standard of departed quantitiesanalysis]]), ofwhich rigorously formalizes the derivativecalculations aswith theinfinitely ratiosmall ofquantities twocalled [[infinitesimals]].
:<math>\frac{dy}{dx}</math>,
see more at [[non-standard calculus]].
==Definition==
TheIn nonstandard analysis deals primarily with the [[hyperreal number|hyperreal]] line, is an extension of the real line that allows for infinitely small quantities. Thus,In the hyperreal line every real number isaccompanied byhas a clustercollection of numbers (called a ([[monad (mathematics)|monad]]) of hyperreals infinitely close, or [[adequality|adequal]], to it. The standard part function associates to a [[Wikt:finite|finite]] [[hyperreal number|hyperreal]] ''x'', the unique standard real number ''x<sub>0</sub>'' which infinitely close to it,. sothatIn werelationship canis expressed symbolically by writewritting
:<math>\,\mathrm{st}(x)=x_0</math>.
The standard part of any [[infinitesimal]] is 0. Thus if N is an infinite [[hypernatural]], then (0.1)<sup>/N</sup> is infinitesimal, and st(0.1<sup>N</sup>N)=0.
TheSome authors develop the theory of non-standard analysis using [[Internal Set Theory]]. In this setting the standard part function "st" is not defined by an [[internal set]] {{Citation needed|date=May 2011}}.
