Editing Surreal number (section)

{{Short description|Generalization of the real numbers}}
[[File:Surreal number tree.svg|thumb|440px|A visualization of the surreal number tree]]
In [[mathematics]], the '''surreal number''' system is a [[total order|totally ordered]] [[proper class]] containing not only the [[real number]]s but also [[Infinity|infinite]] and [[infinitesimal|infinitesimal numbers]], respectively larger or smaller in [[absolute value]] than any positive real number. Research on the [[Go endgame]] by [[John Horton Conway]] led to the original definition and construction of 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'''''. 

The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an [[ordered field]].{{efn|In the original formulation using [[von Neumann–Bernays–Gödel set theory]], the surreals form a proper class, rather than a set, so the term [[field (mathematics)|field]] is not precisely correct; where this distinction is important, some authors use Field or FIELD to refer to a proper class that has the arithmetic properties of a field.  One can obtain a true field by limiting the construction to a [[Grothendieck universe]], yielding a set with the cardinality of some [[strongly inaccessible cardinal]], or by using a form of set theory in which constructions by [[transfinite recursion]] stop at some countable ordinal such as [[Epsilon numbers (mathematics)|epsilon nought]].}} If formulated in [[von Neumann–Bernays–Gödel set theory]], the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the [[rational function]]s, the [[Levi-Civita field]], the [[superreal number]]s (including the [[hyperreal number]]s) can be realized as subfields of the surreals.<ref name=bajnok>{{cite book|last=Bajnok|first=Béla|title=An Invitation to Abstract Mathematics|year=2013|publisher=Springer |isbn=9781461466369|quote=Theorem 24.29. The surreal number system is the largest ordered field|url=https://books.google.com/books?id=cNFzKnvxXoAC&q=%22surreal+numbers%22 | doi=10.1007/978-1-4614-6636-9_24 |doi-access= free| page= 362}}</ref> The surreals also contain all [[transfinite number|transfinite]] [[ordinal number]]s; the arithmetic on them is given by the [[Ordinal arithmetic#Natural operations|natural operations]]. 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.