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- Hyperreal_number abstract "The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the formSuch a number is infinite, and its reciprocal is infinitesimal. The term "hyper-real" was introduced by Edwin Hewitt in 1948.The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic Law of Continuity. The transfer principle states that true first order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since for all integers n, one also has for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' 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. 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 which Robinson delineated.The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis. One immediate application is the definition of the basic concepts of analysis such as derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes for an infinitesimal , where st(·) 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.".
- Hyperreal_number wikiPageExternalLink analysis_hyperreals.html.
- Hyperreal_number wikiPageExternalLink calc.
- Hyperreal_number wikiPageExternalLink InfsmlCalc.htm.
- Hyperreal_number wikiPageExternalLink calc.html.
- Hyperreal_number wikiPageID "51429".
- Hyperreal_number wikiPageRevisionID "600303119".
- Hyperreal_number hasPhotoCollection Hyperreal_number.
- Hyperreal_number subject Category:Field_theory.
- Hyperreal_number subject Category:Infinity.
- Hyperreal_number subject Category:Mathematical_analysis.
- Hyperreal_number subject Category:Mathematics_of_infinitesimals.
- Hyperreal_number subject Category:Non-standard_analysis.
- Hyperreal_number subject Category:Real_closed_field.
- Hyperreal_number comment "The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the formSuch a number is infinite, and its reciprocal is infinitesimal. The term "hyper-real" was introduced by Edwin Hewitt in 1948.The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic Law of Continuity.".
- Hyperreal_number label "Hyperreal number".
- Hyperreal_number label "Hyperreelle Zahl".
- Hyperreal_number label "Nombre hyperréel".
- Hyperreal_number label "Numero iperreale".
- Hyperreal_number label "Número hiper-real".
- Hyperreal_number label "Número hiperreal".
- Hyperreal_number label "Гиперреальное число".
- Hyperreal_number label "عدد حقيقي فائق".
- Hyperreal_number label "超实数 (非标准分析)".
- Hyperreal_number label "超実数".
- Hyperreal_number sameAs Hyperreálné_číslo.
- Hyperreal_number sameAs Hyperreelle_Zahl.
- Hyperreal_number sameAs Número_hiperreal.
- Hyperreal_number sameAs Nombre_hyperréel.
- Hyperreal_number sameAs Numero_iperreale.
- Hyperreal_number sameAs 超実数.
- Hyperreal_number sameAs 초실수.
- Hyperreal_number sameAs Número_hiper-real.
- Hyperreal_number sameAs m.0dknc.
- Hyperreal_number sameAs Q5958543.
- Hyperreal_number sameAs Q5958543.
- Hyperreal_number wasDerivedFrom Hyperreal_number?oldid=600303119.
- Hyperreal_number isPrimaryTopicOf Hyperreal_number.