Standard part function: Difference between revisions

Revision as of 22:52, 26 December 2020

Revision as of 09:44, 15 March 2021

Line 1:

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>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 [[nonstandard analysis]]. The latter theory is a rigorous formalisationformalization 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} \subseteq {}^{*\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 (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>\,\operatorname{st}(x) = x_0.</math>

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'')&nbsp; =&nbsp; 0.}}

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

More generally, each finite <math>u \in {}^{*\ast}\mathbb{R}</math> defines a [[Dedekind cut]] on the subset <math>\mathbb{R}\subseteq{}^{*\ast}\mathbb{R}</math> (via the total order on <math>{}^{\ast}\mathbb{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>\mathbb{R}\subseteq {}^*\mathbb{R}</math>, which is not internal; in fact every internal set in <math>{}^*\ast\mathbb{R}</math> that is a subset of <math>\mathbb{R}</math> is necessarily ''finite'', see (Goldblatt, 1998).

==Applications==

All the traditional notions of calculus arecan 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''&prime;′(''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 = \textoperatorname{st}(u_H)</math> where <math>H \in {}^*\ast\mathbb{N} \setminus \mathbb{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>\textoperatorname{st}\circ f</math> is ''constant'' on the [[halo (mathematics)|halo]] of <math>x</math>. See [[microcontinuity]] for more details.

==See also==