Editing Hyperreal number (section)

{{short description|Element of a nonstandard model of the reals, which can be infinite or infinitesimal}}
{{Redirect2|*R|R*||R* (disambiguation)}}

{{More citations needed|date=July 2023}}
[[File:Números hiperreales.png|450px|thumb|Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)]]

In [[mathematics]], '''hyperreal numbers''' are an [[Field extension|extension]] of the real numbers to include certain classes of [[Infinity|infinite]] and [[infinitesimal]] numbers.<ref name=":0">{{Cite web |last=Weisstein |first=Eric W. |title=Hyperreal Number |url=https://mathworld.wolfram.com/ |access-date=2024-03-20 |website=mathworld.wolfram.com |language=en}}</ref> A hyperreal number <math>x</math> is said to be finite if, and only if, <math>|x|<n</math> for some integer <math>n</math>.<ref name=":0" /><ref name=":1">{{Cite book |last=Robinson |first=Abraham |title=Selected papers of Abraham Robinson. 2: Nonstandard analysis and philosophy |date=1979 |publisher=Yale Univ. Press |isbn=978-0-300-02072-4 |location=New Haven |pages=67 |language=en}}</ref> <math>x</math> is said to be infinitesimal if, and only if, <math>|x|<1/n</math> for all positive integers <math>n</math>.<ref name=":0" /><ref name=":1" /> The term "hyper-real" was introduced by [[Edwin Hewitt]] in 1948.<ref>Hewitt (1948), p.&nbsp;74, as reported in Keisler (1994)</ref>

The hyperreal numbers satisfy the [[transfer principle]], a rigorous version of [[Gottfried Wilhelm Leibniz|Leibniz's]] heuristic [[law of continuity]].  The transfer principle states that true [[first-order logic|first-order]] statements about '''R''' are also valid in *'''R'''.<ref>{{Cite book |last=Dauben |first=Joseph Warren |title=Abraham Robinson: the creation of nonstandard analysis: a personal and mathematical odyssey |date=1995 |publisher=Princeton University Press |isbn=978-0-691-03745-5 |series=Princeton legacy library |location=Princeton, New Jersey |pages=474 |language=en}}</ref> For example, the [[commutative law]] of addition, {{nowrap|1=''x''&thinsp;+&thinsp;''y'' = ''y''&thinsp;+&thinsp;''x''}}, holds for the hyperreals just as it does for the reals; since '''R''' is a [[real closed field]], so is *'''R'''.  Since <math>\sin({\pi n})=0</math> for all [[integer]]s ''n'', one also has <math>\sin({\pi H})=0</math> for all [[hyperinteger]]s <math>H</math>.  The transfer principle for [[ultrapower]]s is a consequence of [[Łoś' theorem|Łoś's theorem]] of 1955.

Concerns about the [[soundness]] of arguments involving infinitesimals date back to ancient Greek mathematics, with [[Archimedes]] replacing such proofs with ones using other techniques such as the [[method of exhaustion]].<ref>Ball, p. 31</ref> In the 1960s, [[Abraham Robinson]] proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated.

The application of hyperreal numbers and in particular the transfer principle to problems of [[mathematical analysis|analysis]] is called [[nonstandard analysis]]. One immediate application is the definition of the basic concepts of analysis such as the [[derivative]] and [[integral]] in a direct fashion, without passing via logical complications of multiple quantifiers.  Thus, the derivative of ''f''(''x'') becomes <math>f'(x) = \operatorname{st}\left( \frac{f(x + \Delta x) - f(x)}{\Delta x} \right)</math> for an infinitesimal <math>\Delta x</math>, where st(&sdot;) denotes the [[standard part function]], which "rounds off" each finite hyperreal to the nearest real.  Similarly, the integral is defined as the standard part of a suitable [[infinite sum]].