Editing Surreal number (section)

===Order===

The recursive definition of surreal numbers is completed by defining comparison:

Given numeric forms {{math|1=''x'' = {{mset| ''X''{{sub|''L''}} {{!}} ''X''{{sub|''R''}} }}}} and {{math|1=''y'' = {{mset| ''Y''{{sub|''L''}} {{!}} ''Y''{{sub|''R''}} }}}}, {{math|''x'' ≤ ''y''}} if and only if both:
*There is no {{math|''x''{{sub|''L''}} ∈ ''X''{{sub|''L''}}}} such that {{math|''y'' ≤ ''x''{{sub|''L''}}}}. That is, every element in the left part of {{mvar|x}} is strictly smaller than {{mvar|y}}.
*There is no {{math|''y''{{sub|''R''}} ∈ ''Y''{{sub|''R''}}}} such that {{math|''y''{{sub|''R''}} ≤ ''x''}}. That is, every element in the right part of {{mvar|y}} is strictly larger than {{mvar|x}}.
Surreal numbers can be compared to each other (or to numeric forms) by choosing a numeric form from its equivalence class to represent each surreal number.