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In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal x, the unique real x_0 infinitely close to it, i.e. x-x_0 is
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
. As such, it is a mathematical implementation of the historical concept of adequality introduced by
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
,Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography.
Foundations of Science ''Foundations of Science'' is a peer-reviewed interdisciplinary academic journal focussing on methodological and philosophical topics concerning the structure and the growth of science. It is the official journal of the Association for Found ...
.

Se
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The authors refer to the Fermat-Robinson standard part.
as well as
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
's Transcendental law of homogeneity. The standard part function was first defined by Abraham Robinson who used the notation ^x for the standard part of a hyperreal x (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 nonstandard analysis. The latter theory is a rigorous formalization of calculations with
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
s. The standard part of ''x'' is sometimes referred to as its shadow.


Definition

Nonstandard analysis deals primarily with the pair \R \subseteq ^*\R, where the hyperreals ^*\R are an ordered field extension of the reals \R, and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers (called a monad, or halo) of hyperreals infinitely close to it. The standard part function associates to a finite hyperreal ''x'', the unique standard real number ''x''0 that is infinitely close to it. The relationship is expressed symbolically by writing :\operatorname(x) = x_0. The standard part of any
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
is 0. Thus if ''N'' is an infinite hypernatural, then 1/''N'' is infinitesimal, and If a hyperreal u is represented by a Cauchy sequence \langle u_n:n\in\mathbb \rangle in the ultrapower construction, then :\operatorname(u) = \lim_ u_n. More generally, each finite u \in ^*\R defines a Dedekind cut on the subset \R\subseteq^*\R (via the total order on ^\R) and the corresponding real number is the standard part of ''u''.


Not internal

The standard part function "st" is not defined by an internal set. There are several ways of explaining this. Perhaps the simplest is that its domain L, which is the collection of limited (i.e. finite) hyperreals, is not an internal set. Namely, since L is bounded (by any infinite hypernatural, for instance), L would have to have a least upper bound if L were internal, but L doesn't have a least upper bound. Alternatively, the range of "st" is \R\subseteq ^*\R, which is not internal; in fact every internal set in ^*\R that is a subset of \R is necessarily ''finite'', see (Goldblatt, 1998).


Applications

All the traditional notions of calculus can be expressed in terms of the standard part function, as follows.


Derivative

The standard part function is used to define the derivative of a function ''f''. If ''f'' is a real function, and ''h'' is infinitesimal, and if ''f''′(''x'') exists, then :f'(x) = \operatorname\left(\frac h\right). Alternatively, if y=f(x), one takes an infinitesimal increment \Delta x, and computes the corresponding \Delta y=f(x+\Delta x)-f(x). One forms the ratio \frac. The derivative is then defined as the standard part of the ratio: :\frac=\operatorname\left( \frac \right) .


Integral

Given a function f on ,b/math>, one defines the integral \int_a^b f(x)\,dx as the standard part of an infinite Riemann sum S(f,a,b,\Delta x) when the value of \Delta x is taken to be infinitesimal, exploiting a hyperfinite partition of the interval 'a'',''b''


Limit

Given a sequence (u_n), its limit is defined by \lim_ u_n = \operatorname(u_H) where H \in ^*\N \setminus \N is an infinite index. Here the limit is said to exist if the standard part is the same regardless of the infinite index chosen.


Continuity

A real function f is continuous at a real point x if and only if the composition \operatorname\circ f is ''constant'' on the halo of x. See microcontinuity for more details.


See also

* Adequality * Nonstandard calculus


Notes


References

* H. Jerome Keisler. '' Elementary Calculus: An Infinitesimal Approach''. First edition 1976; 2nd edition 1986. (This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html.) * Goldblatt, Robert. ''Lectures on the
hyperreals In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains number ...
''. An introduction to nonstandard analysis.
Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard ...
, 188. Springer-Verlag, New York, 1998. * Abraham Robinson. Non-standard analysis. Reprint of the second (1974) edition. With a foreword by Wilhelmus A. J. Luxemburg. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1996. xx+293 pp. {{Infinitesimals Calculus Nonstandard analysis Real closed field