Editing Surreal number (section)

===Arithmetic closure===

For each [[natural number]] (finite ordinal) {{mvar|n}}, all numbers generated in {{math|''S''{{sub|''n''}}}} are [[dyadic fraction]]s, i.e., can be written as an [[irreducible fraction]] {{math|{{sfrac|''a''|2{{sup|''b''}}}}}}, where {{mvar|a}} and {{mvar|b}} are [[integer]]s and {{math|0 ≤ ''b'' < ''n''}}.

The set of all surreal numbers that are generated in some {{math|''S''{{sub|''n''}}}} for finite {{mvar|n}} may be denoted as <math display=inline>S_* = \bigcup_{n \in N}  S_n</math>.  One may form the three classes
<math display=block>\begin{align}
S_{0} &= \{ 0 \} \\
S_{+} &= \{ x \in S_*: x > 0 \} \\
S_{-} &= \{ x \in S_*: x < 0 \}
\end{align}</math>
of which {{math|''S''{{sub|''∗''}}}} is the union.  No individual {{math|''S''{{sub|''n''}}}} is closed under addition and multiplication (except {{math|''S''{{sub|0}}}}), but {{math|''S''{{sub|∗}}}} is; it is the subring of the rationals consisting of all dyadic fractions.

There are infinite ordinal numbers {{mvar|β}} for which the set of surreal numbers with birthday less than {{mvar|β}} is closed under the different arithmetic operations.<ref name=vdDE2001>{{cite journal | last1 = van den Dries | first1 = Lou | last2 = Ehrlich | first2 = Philip | author2-link = Philip Ehrlich | title = Fields of surreal numbers and exponentiation | journal = Fundamenta Mathematicae | volume = 167 | issue = 2 | pages = 173–188 | publisher = Institute of Mathematics of the Polish Academy of Sciences | location = Warszawa | date = January 2001 | issn = 0016-2736 | doi = 10.4064/fm167-2-3 | doi-access = free }}</ref> For any ordinal {{mvar|α}}, the set of surreal numbers with birthday less than {{math|1=''β'' = ''ω''{{sup|''α''}}}} (using [[#Powers of ω|powers of {{mvar|ω}}]]) is closed under addition and forms a group; for birthday less than {{mvar|ω{{sup|ω{{sup|α}}}}}} it is closed under multiplication and forms a ring;{{efn|1=The set of dyadic fractions constitutes the simplest non-trivial group and ring of this kind; it consists of the surreal numbers with birthday less than {{math|1=''ω'' = ''ω''{{sup|1}} = ''ω''{{sup|''ω''{{sup|0}}}}.}}}} and for birthday less than an (ordinal) [[Epsilon number (mathematics)|epsilon number]] {{mvar|ε{{sub|α}}}} it is closed under multiplicative inverse and forms a field. The latter sets are also closed under the exponential function as defined by Kruskal and Gonshor.<ref name=vdDE2001 /><ref name=G1986>{{cite book | last=Gonshor | first=Harry | title=An Introduction to the Theory of Surreal Numbers | year=1986 | publisher=Cambridge University Press | series=London Mathematical Society Lecture Note Series | volume=110 | isbn= 9780521312059 | doi=10.1017/CBO9780511629143 }}</ref>{{rp|at=ch. 10}}<ref name=vdDE2001 />

However, it is always possible to construct a surreal number that is greater than any member of a set of surreals (by including the set on the left side of the constructor) and thus the collection of surreal numbers is a [[proper class]]. With their ordering and algebraic operations they constitute an [[ordered field]], with the caveat that they do not form a [[Set (mathematics)|set]]. In fact it is the biggest ordered field, in that every ordered field is a subfield of the surreal numbers.<ref name=bajnok/> The class of all surreal numbers is denoted by the symbol <math display=inline>\mathbb{No}</math>.