In [[non-standard analysis]], the '''standard part function''' is a function from the limited (finite) [[Hyperreal number|hyperreal]] numberss 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}} [http://www.springerlink.com/content/tj7j2810n8223p43/] See [http://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]].
It can also be thought of as a mathematical implementation of [[Gottfried Wilhelm Leibniz|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 the derivative and the integral, in [[non-standard 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'''.
