In [[non-standardnonstandard analysis]], the '''standard part function''' is a function from the limited (finite) [[hyperreal number]]s to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal <math>x</math>, the unique real <math>x_0</math> infinitely close to it, i.e. <math>x-x_0</math> is [[infinitesimal]]. As such, it is a mathematical implementation of the historical concept of [[adequality]] introduced by [[Pierre de Fermat]],<ref>Karin Usadi Katz and [[Mikhail Katz|Mikhail G. Katz]] (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. [[Foundations of Science]]. {{doi|10.1007/s10699-011-9223-1}} [https://doi.org/10.1007%2Fs10699-011-9223-1] See [https://arxiv.org/abs/1104.0375 arxiv]. The authors refer to the Fermat-Robinson standard part.</ref> as well as [[Leibniz]]'s [[Transcendental law of homogeneity]].
The standard part function was first defined by [[Abraham Robinson]] who used the notation <math>{}^{\circ}x</math> for the standard part of a hyperreal <math>x</math> (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in [[non-standardnonstandard analysis]]. The latter theory is a rigorous formalisation of calculations with [[infinitesimal]]s. The standard part of ''x'' is sometimes referred to as its '''shadow'''.
==Definition==
[[File:Standard part function with two continua.svg|360px|thumb|right|The standard part function "rounds off" a finite hyperreal to the nearest real number. The "infinitesimal microscope" is used to view an infinitesimal neighborhood of a standard real.]]
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 [[ordered field]] 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-standardnonstandard 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 ''x<sub>0</sub>'' which is infinitely close to it. The relationship is expressed symbolically by writing
