Standard part function: Difference between revisions

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{{Short description|Function from the limited hyperreal to the real numbers}}

[[Image:Gottfried Wilhelm von Leibniz.jpg|thumb|200px|right|''[[Gottfried Wilhelm Leibniz]]'' Inventor of infinitesimal calculus]]

In [[nonstandard 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>{{cite journal |last1=Katz |first1=Karin Usadi |last2=Katz |first2=Mikhail G. |title=A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography |journal=Foundations of Science |date=March 2012 |volume=17 |issue=1 |pages=51–89 |doi=10.1007/s10699-011-9223-1 |url=https://link.springer.com/article/10.1007/s10699-011-9223-1 |postscript=The authors refer to the Fermat-Robinson standard part.|arxiv=1104.0375 }}</ref> as well as [[Leibniz]]'s [[Transcendental law of homogeneity]].

In [[non-standard analysis]], the '''standard part function''' "st" is the key ingredient in [[Abraham Robinson]]'s resolution of the paradox of Leibniz's definition of the derivative as the ratio of two infinitesimals

:<math>\frac{dy}{dx}</math>,

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 [[nonstandard analysis]]. The latter theory is a rigorous formalization of calculations with [[infinitesimal]]s. The standard part of ''x'' is sometimes referred to as its '''shadow'''.<ref>{{cite journal |last1=Bascelli |first1=Tiziana |last2=Bottazzi |first2=Emanuele |last3=Herzberg |first3=Frederik |last4=Kanovei |first4=Vladimir |last5=Katz |first5=Karin U. |last6=Katz |first6=Mikhail G. |last7=Nowik |first7=Tahl |last8=Sherry |first8=David |last9=Shnider |first9=Steven |title=Fermat, Leibniz, Euler, and the Gang: The True History of the Concepts of Limit and Shadow |journal=Notices of the American Mathematical Society |date=1 September 2014 |volume=61 |issue=8 |pages=848 |doi=10.1090/noti1149 |url=https://community.ams.org/journals/notices/201408/rnoti-p848.pdf}}</ref>

see more at [[non-standard calculus]].

==Definition==

[[File:Standard part function with two continua.svg|360px|thumb|right|The standard part function associates"rounds tooff" a [[finite]] [[hyperreal number|hyperreal]] ''x'',to the standardnearest real ''x₀''number. The "infinitesimal microscope" infinitelyis closeused to it,view soan thatinfinitesimal weneighborhood canof writea standard real.]]

Nonstandard analysis deals primarily with the pair <math>\R \subseteq {}^*\R</math>, where the [[hyperreal number|hyperreal]]s <math>{}^*\R</math> are an [[ordered field]] extension of the reals <math>\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 (nonstandard 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''₀ that is infinitely close to it. The relationship is expressed symbolically by writing

:<math>\,\mathrm{st}(x)=x_0</math>.

:<math>\,\mathrmoperatorname{st}(x) = x_0.</math>.

The existence of the standard part function is a consequence of the [[completeness of the reals]] or the fact that [[finite]] [[closed interval]]s of the reals are [[compact]].

The standard part of any [[infinitesimal]] is 0. Thus if ''N'' is an infinite [[hypernatural]], then 1/''N'' is infinitesimal, and {{nowrap|1=st(1/''N'') = 0.}}

The standard part function "st" is not an [[internal set|internal object]].

If a hyperreal <math>u</math> is represented by a Cauchy sequence <math>\langle u_n:n\in\mathbb{N} \rangle</math> in the [[ultrapower]] construction, then

:<math>\operatorname{st}(u) = \lim_{n\to\infty} u_n.</math>

More generally, each finite <math>u \in {}^*\R</math> defines a [[Dedekind cut]] on the subset <math>\R\subseteq{}^*\R</math> (via the total order on <math>{}^{\ast}\R</math>) 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 <math>\R\subseteq {}^*\R</math>, which is not internal; in fact every internal set in <math>{}^*\R</math> that is a subset of <math>\R</math> is necessarily ''finite''.<ref>{{cite book |last1=Goldblatt |first1=Robert |title=Lectures on the Hyperreals: An Introduction to Nonstandard Analysis |series=Graduate Texts in Mathematics |date=1998 |volume=188 |publisher=Springer |___location=New York |doi=10.1007/978-1-4612-0615-6 |isbn=978-0-387-98464-3 |url=https://link.springer.com/book/10.1007/978-1-4612-0615-6 |language=en}}</ref>

==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

:<math>f'(x) = \operatorname{st}\left(\frac {f(x+h)-f(x)}h\right).</math>

Alternatively, if <math>y=f(x)</math>, one takes an infinitesimal increment <math>\Delta x</math>, and computes the corresponding <math>\Delta y=f(x+\Delta x)-f(x)</math>. One forms the ratio <math display="inline">\frac{\Delta y}{\Delta x}</math>. The derivative is then defined as the standard part of the ratio:

:<math>\frac{dy}{dx}=\operatorname{st}\left( \frac{\Delta y}{\Delta x} \right) .</math>

===Integral===

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

===Limit===

Given a sequence <math>(u_n)</math>, its limit is defined by <math display="inline">\lim_{n\to\infty} u_n = \operatorname{st}(u_H)</math> where <math>H \in {}^*\N \setminus \N</math> 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 <math>f</math> is continuous at a real point <math>x</math> if and only if the composition <math>\operatorname{st}\circ f</math> is ''constant'' on the [[halo (mathematics)|halo]] of <math>x</math>. See [[microcontinuity]] for more details.

==See also==

*[[Adequality]]

*[[non-standardNonstandard calculus]]

==References==

{{Reflist}}

==Further reading==

*[[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.)

*[[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. {{isbn|0-691-04490-2}}

{{Infinitesimals}}

[[Category:Model theoryCalculus]]

[[Category:Nonstandard analysis]]

[[Category:Real closed field]]