Editing Surreal number (section)

== Powers of ''ω'' and the Conway normal form ==

To classify the "orders" of infinite and infinitesimal surreal numbers, also known as [[archimedean property|archimedean]] classes, Conway associated to each surreal number {{mvar|x}} the surreal number
* {{math|1=''ω''{{sup|''x''}} = {{mset| 0, ''r'' ''ω''{{sup|''x''{{sub|L}}}} {{!}} ''s'' ''ω''{{sup|''x''{{sub|R}}}} }}}},
where {{mvar|r}} and {{mvar|s}} range over the positive real numbers. If {{math|''x'' < ''y''}} then {{math|''ω''{{sup|''y''}}}} is "infinitely greater" than {{math|''ω''{{sup|''x''}}}}, in that it is greater than {{math|''r'' ''ω''{{sup|''x''}}}} for all real numbers {{mvar|r}}. Powers of {{mvar|ω}} also satisfy the conditions
* {{math|1=''ω''{{sup|''x''}} ''ω''{{sup|''y''}} = ''ω''{{sup|''x''+''y''}}}},
* {{math|1=''ω''{{sup|−''x''}} = {{sfrac|1|''ω''{{sup|''x''}}}}}},
so they behave the way one would expect powers to behave.

Each power of {{mvar|ω}} also has the redeeming feature of being the ''simplest'' surreal number in its archimedean class; conversely, every archimedean class within the surreal numbers contains a unique simplest member. Thus, for every positive surreal number {{mvar|x}} there will always exist some positive real number {{mvar|r}} and some surreal number {{mvar|y}} so that {{math|''x'' − ''rω''{{sup|''y''}}}} is "infinitely smaller" than {{mvar|x}}.  The exponent {{mvar|y}} is the "base {{mvar|ω}} logarithm" of {{mvar|x}}, defined on the positive surreals; it can be demonstrated that {{math|log{{sub|''ω''}}}} maps the positive surreals onto the surreals and that
:{{math|1=log{{sub|''ω''}}(''xy'') = log{{sub|''ω''}}(''x'') + log{{sub|''ω''}}(''y'')}}.

This gets extended by transfinite induction so that every surreal number has a "normal form" analogous to the [[Ordinal arithmetic#Cantor normal form|Cantor normal form]] for ordinal numbers. This is the Conway normal form: Every surreal number {{mvar|x}} may be uniquely written as
: {{math|1=''x'' = ''r''{{sub|0}}''ω''{{sup|''y''{{sub|0}}}} + ''r''{{sub|1}}''ω''{{sup|''y''{{sub|1}}}} + ...}},
where every {{math|''r''{{sub|''α''}}}} is a nonzero real number and the {{math|''y''{{sub|''α''}}}}s form a strictly decreasing sequence of surreal numbers. This "sum", however, may have infinitely many terms, and in general has the length of an arbitrary ordinal number. (Zero corresponds of course to the case of an empty sequence, and is the only surreal number with no leading exponent.)

Looked at in this manner, the surreal numbers resemble a [[Formal power series|power series field]], except that the decreasing sequences of exponents must be bounded in length by an ordinal and are not allowed to be as long as the class of ordinals. This is the basis for the formulation of the surreal numbers as a [[#Hahn series|Hahn series]].