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- Surreal_number abstract "In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. (Strictly speaking, the surreals are not a set, but a proper class.) If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals. It has also been shown (in Von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are the largest ordered field. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations.In 1907 Hahn introduced Hahn series as a generalization of formal power series, and Hausdorff introduced certain ordered sets called ηα-sets for ordinals α and asked if it was possible to find a compatible ordered group or field structure. In 1962 Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α, and taking α to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.Research on the go endgame by John Horton Conway led to a simpler definition and construction of the surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. In his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book On Numbers and Games.".
- Surreal_number thumbnail Surreal_number_tree.svg?width=300.
- Surreal_number wikiPageExternalLink infinityplusonea599.
- Surreal_number wikiPageExternalLink surreal-numbers.
- Surreal_number wikiPageExternalLink Hack.
- Surreal_number wikiPageExternalLink sn.html.
- Surreal_number wikiPageExternalLink surreal.html.
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- Surreal_number wikiPageRevisionID "680628260".
- Surreal_number wikiPageWikiLink Absolute_value.
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- Surreal_number wikiPageWikiLink Algebraic_closure.
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- Surreal_number wikiPageWikiLink Algebraically_closed_field.
- Surreal_number wikiPageWikiLink Algebraically_independent.
- Surreal_number wikiPageWikiLink Antisymmetric_relation.
- Surreal_number wikiPageWikiLink Archimedean_property.
- Surreal_number wikiPageWikiLink Axiom_of_global_choice.
- Surreal_number wikiPageWikiLink Category:Combinatorial_game_theory.
- Surreal_number wikiPageWikiLink Category:Infinity.
- Surreal_number wikiPageWikiLink Category:Mathematical_logic.
- Surreal_number wikiPageWikiLink Category:Real_closed_field.
- Surreal_number wikiPageWikiLink Chess.
- Surreal_number wikiPageWikiLink Class_(set_theory).
- Surreal_number wikiPageWikiLink Dedekind_cut.
- Surreal_number wikiPageWikiLink Determinism.
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- Surreal_number wikiPageWikiLink Discover_(magazine).
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- Surreal_number wikiPageWikiLink Domain_of_a_function.
- Surreal_number wikiPageWikiLink Donald_Knuth.
- Surreal_number wikiPageWikiLink Dyadic_fraction.
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- Surreal_number wikiPageWikiLink Equivalence_class.
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- Surreal_number wikiPageWikiLink Field_(mathematics).
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- Surreal_number wikiPageWikiLink Go_(board_game).
- Surreal_number wikiPageWikiLink Go_(game).
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- Surreal_number wikiPageWikiLink Go_strategy_and_tactics.
- Surreal_number wikiPageWikiLink Go_terms.
- Surreal_number wikiPageWikiLink Hahn_series.
- Surreal_number wikiPageWikiLink Hyperreal_field.
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- Surreal_number wikiPageWikiLink Inductive_definition.
- Surreal_number wikiPageWikiLink Infinitesimal.
- Surreal_number wikiPageWikiLink Infinity.
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- Surreal_number wikiPageWikiLink Irreducible_fraction.
- Surreal_number wikiPageWikiLink Isomorphic_(mathematics).
- Surreal_number wikiPageWikiLink Isomorphism.
- Surreal_number wikiPageWikiLink John_Horton_Conway.
- Surreal_number wikiPageWikiLink Levi-Civita_field.
- Surreal_number wikiPageWikiLink Limit_ordinal.
- Surreal_number wikiPageWikiLink Linear_continuum.
- Surreal_number wikiPageWikiLink Linear_order.
- Surreal_number wikiPageWikiLink List_of_Go_terms.
- Surreal_number wikiPageWikiLink Martin_Gardner.
- Surreal_number wikiPageWikiLink Mathematical_class.
- Surreal_number wikiPageWikiLink Mathematical_induction.
- Surreal_number wikiPageWikiLink Mathematics.
- Surreal_number wikiPageWikiLink Natural_number.
- Surreal_number wikiPageWikiLink Negative_number.
- Surreal_number wikiPageWikiLink Non-standard_analysis.
- Surreal_number wikiPageWikiLink On_Numbers_and_Games.
- Surreal_number wikiPageWikiLink Order_theory.
- Surreal_number wikiPageWikiLink Ordered_field.
- Surreal_number wikiPageWikiLink Ordered_pair.
- Surreal_number wikiPageWikiLink Ordinal_arithmetic.
- Surreal_number wikiPageWikiLink Ordinal_number.
- Surreal_number wikiPageWikiLink Partial_order.
- Surreal_number wikiPageWikiLink Partially_ordered_set.
- Surreal_number wikiPageWikiLink Philip_Ehrlich.
- Surreal_number wikiPageWikiLink Positive_number.
- Surreal_number wikiPageWikiLink Proper_class.
- Surreal_number wikiPageWikiLink Range_(mathematics).
- Surreal_number wikiPageWikiLink Rational_function.
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- Surreal_number wikiPageWikiLink Real_number.
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- Surreal_number wikiPageWikiLink Recursion.
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- Surreal_number wikiPageWikiLink Ring_(mathematics).
- Surreal_number wikiPageWikiLink Set_(mathematics).
- Surreal_number wikiPageWikiLink Sign_(mathematics).
- Surreal_number wikiPageWikiLink Standard_analysis.
- Surreal_number wikiPageWikiLink Star_(game_theory).
- Surreal_number wikiPageWikiLink Subset.
- Surreal_number wikiPageWikiLink Superreal_number.
- Surreal_number wikiPageWikiLink Total_order.
- Surreal_number wikiPageWikiLink Transcendental_(mathematics).
- Surreal_number wikiPageWikiLink Transcendental_number.
- Surreal_number wikiPageWikiLink Transfinite_induction.