Editing Surreal number (section)

==Description==

===Notation===
In the context of surreal numbers, an [[ordered pair]] of sets {{mvar|L}} and {{mvar|R}}, which is written as {{math|(''L'', ''R'')}} in many other mathematical contexts, is instead written {{math|{{mset| ''L'' {{!}} ''R'' }}}} including the extra space adjacent to each brace.  When a set is empty, it is often simply omitted.  When a set is explicitly described by its elements, the pair of braces that encloses the list of elements is often omitted.  When a union of sets is taken, the operator that represents that is often a comma.  For example, instead of {{math|(''L''{{sub|1}} ∪ ''L''{{sub|2}} ∪ {{mset|0, 1, 2}}, ∅)}}, which is common notation in other contexts, we typically write {{math|{{mset| ''L''{{sub|1}}, ''L''{{sub|2}}, 0, 1, 2 {{!}} }}}}.

===Outline of construction===
In the Conway construction,<ref name="Con01">{{Cite book|url=https://books.google.com/books?id=tXiVo8qA5PQC|title=On Numbers and Games|edition=2|last=Conway|first=John H.|date=2000-12-11|orig-year=1976|publisher=CRC Press|isbn=9781568811277|language=en}}</ref> the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers {{mvar|a}} and {{mvar|b}}, {{math|''a'' ≤ ''b''}} or {{math|''b'' ≤ ''a''}}.  (Both may hold, in which case {{mvar|a}} and {{mvar|b}} are equivalent and denote the same number.)  Each number is formed from an ordered pair of subsets of numbers already constructed:  given subsets {{mvar|L}} and {{mvar|R}} of numbers such that all the members of {{mvar|L}} are strictly less than all the members of {{mvar|R}}, then the pair {{math|{{mset| ''L'' {{!}} ''R'' }}}} represents a number intermediate in value between all the members of {{mvar|L}} and all the members of {{mvar|R}}.

Different subsets may end up defining the same number:  {{math|{{mset| ''L'' {{!}} ''R'' }}}} and {{math|{{mset| ''L&prime;'' {{!}} ''R&prime;'' }}}} may define the same number even if {{math|''L'' ≠ ''L&prime;''}} and {{math|''R'' ≠ ''R&prime;''}}.  (A similar phenomenon occurs when [[rational numbers]] are defined as quotients of integers:  {{sfrac|1|2}} and {{sfrac|2|4}} are different representations of the same rational number.)  So strictly speaking, the surreal numbers are [[equivalence class]]es of representations of the form {{math|{{mset| ''L'' {{!}} ''R'' }}}} that designate the same number.

In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set:  {{math|{{mset| {{!}} }}}}.  This representation, where {{mvar|L}} and {{mvar|R}} are both empty, is called 0.  Subsequent stages yield forms like

:{{math|1={{mset| 0 {{!}} }} = 1}}

:{{math|1={{mset| 1 {{!}} }} = 2}}

:{{math|1={{mset| 2 {{!}} }} = 3}}

and

:{{math|1={{mset| {{!}} 0 }} = −1}}

:{{math|1={{mset| {{!}} −1 }} = −2}}

:{{math|1={{mset| {{!}} −2 }} = −3}}

The integers are thus contained within the surreal numbers.  (The above identities are definitions, in the sense that the right-hand side is a name for the left-hand side. That the names are actually appropriate will be evident when the arithmetic operations on surreal numbers are defined, as in the section below.) Similarly, representations such as

:{{math|1={{mset| 0 {{!}} 1 }} = {{sfrac|1|2}}}}

:{{math|1={{mset| 0 {{!}} {{sfrac|1|2}} }} = {{sfrac|1|4}}}}

:{{math|1={{mset| {{sfrac|1|2}} {{!}} 1 }} = {{sfrac|3|4}}}}

arise, so that the [[dyadic rational]]s (rational numbers whose denominators are powers of 2) are contained within the surreal numbers.

After an infinite number of stages, infinite subsets become available, so that any [[real number]] {{mvar|a}} can be represented by {{math|{{mset| ''L''{{sub|''a''}} {{!}} ''R''{{sub|''a''}} }},}}
where {{math|''L''{{sub|''a''}}}} is the set of all dyadic rationals less than {{mvar|a}} and 
{{math|''R''{{sub|''a''}}}} is the set of all dyadic rationals greater than {{mvar|a}} (reminiscent of a [[Dedekind cut]]).  Thus the real numbers are also embedded within the surreals.

There are also representations like

:{{math|1={{mset| 0, 1, 2, 3, ... {{!}}  }} = ''ω''}}

:{{math|1={{mset| 0 {{!}} 1, {{sfrac|1|2}}, {{sfrac|1|4}}, {{sfrac|1|8}}, ... }} = ε}}

where {{mvar|ω}} is a transfinite number greater than all integers and {{mvar|ε}} is an infinitesimal greater than 0 but less than any positive real number.  Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about {{math|2''ω''}} or {{math|''ω'' − 1}} and so forth.