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Hyperreal number

In mathematics, the hyperreal numbers are the elements of one of several possible field extensions ∗ R {\displaystyle \ast \mathbb {R} } of the field of real numbers R {\displaystyle \mathbb {R} } , extensions that include certain classes of infinite and infinitesimal numbers. A hyperreal number x {\displaystyle x} is said to be finite when | x | < n…

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Development

UltrafilterNewtonLeibnizEulerCauchy
50 related topics

Properties

Ordered fieldMetric spaceOrder topologyVladimir KanoveiSaharon Shelah
12 related topics

Overview

MathematicsField extensionsReal numbersInfiniteInfinitesimal
26 related topics

Properties of infinitesimal and infinite numbers

Local ringValuation ringHomomorphicKernelContinuous
6 related topics

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Overview

Transfer principle

Use in analysis

Properties

Development

Properties of infinitesimal and infinite numbers

Hyperreal fields

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Hyperreal number

Nodes114
Edges113
Triples65
Avg. degree1.98
Density0.017544
Components1

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Hyperreal number

Top relations

related to From Leibniz to Robinson · 19
Hyperreal number → Abraham Robinson, Archimedean, Berkeley's, Bolzano, Cauchy, Ehrlich, Euler, George Berkeley, Ghosts, Hewitt, However, Hyper-real, In, Leibniz, Nonetheless, Robinson, Weierstrass, When, When Newton
see also · 8
Hyperreal number → Constructive, Field, Generalization, Hyperreal, Line, Mathematics, Modern, Surreal
related to An intuitive approach to the ultrapower construction · 7
Hyperreal number → Consider, Goldblatt, Let, Recall, The, These, They
related to Ultrapower construction · 7
Hyperreal number → As, For, In, Similarly, The, This, We
related to Properties · 6
Hyperreal number → Furthermore, However, Saharon Shelah, The, Unlike, Vladimir Kanovei
related to Differentiation · 3
Hyperreal number → For, Leibniz, One
related to Integration · 3
Hyperreal number → Another, For, Leibniz
related to Use in analysis · 3
Hyperreal number → As, Informal, This

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Important terminology

displaystyle hyperreal real numbers sequences number mathbb field infinitesimal one hyperreals reals infinitesimals transfer principle standard also function set mathfrak

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
the method of exhaustioninstance ofwith Archimedes replacing such proofs with ones using other techniques0.80text
the derivativeinstance ofOne immediate application is the definition of the basic concepts of analysis0.80text
integral in a direct fashioninstance ofOne immediate application is the definition of the basic concepts of analysis0.80text
without passing via logical complications of multiple quantifiersinstance ofOne immediate application is the definition of the basic concepts of analysis0.80text
functionsinstance ofor other higher-level structures0.80text
relationsinstance ofor other higher-level structures0.80text
which are typically constructed out of setsinstance ofor other higher-level structures0.80text
Eulerinstance ofthey used infinitesimals and these were still regarded as useful by later mathematicians0.80text
Cauchyinstance ofthey used infinitesimals and these were still regarded as useful by later mathematicians0.80text
Hyperreal numberrelated to An intuitive approach to the ultrapower constructionThe0.60section
Hyperreal numberrelated to An intuitive approach to the ultrapower constructionGoldblatt0.60section
Hyperreal numberrelated to An intuitive approach to the ultrapower constructionRecall0.60section

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