Editing Surreal number (section)

====Addition and multiplication====

The sum {{math|''x'' + ''y''}} of two numbers {{mvar|x}} and {{mvar|y}} is defined by induction on {{math|dom(''x'')}} and {{math|dom(''y'')}} by {{math|1=''x'' + ''y'' = ''σ''(''L'',{{px2}}''R'')}}, where
* {{math|1=''L'' = {{mset| ''u'' + ''y'' : ''u'' ∈ ''L''(''x'') }} ∪ {{mset| ''x'' + ''v'' : ''v'' ∈ ''L''(''y'') }}}},
* {{math|1=''R'' = {{mset| ''u'' + ''y'' : ''u'' ∈ ''R''(''x'') }} ∪ {{mset| ''x'' + ''v'' : ''v'' ∈ ''R''(''y'') }}}}.

The additive identity is given by the number {{math|1=0 = {{(}} {{)}}}}, i.e. the number {{math|0}} is the unique function whose domain is the ordinal {{math|0}}, and the additive inverse of the number {{mvar|x}} is the number {{math|−''x''}}, given by {{math|1=dom(−''x'') = dom(''x'')}}, and, for {{math|''α'' < dom(''x'')}}, {{math|1=(−''x'')(''α'') = −1}} if {{math|1=''x''(''α'') = +1}}, and {{math|1=(−''x'')(''α'') = +1}} if {{math|1=''x''(''α'') = −1}}.

It follows that a number {{mvar|x}} is [[Positive number|positive]] if and only if {{math|1=0 < dom(''x'')}} and {{math|1=''x''(0) = +1}}, and {{mvar|x}} is [[negative number|negative]] if and only if {{math|1=0 < dom(''x'')}} and {{math|1=''x''(0) = −1}}.

The product {{mvar|xy}} of two numbers,  {{mvar|x}} and {{mvar|y}}, is defined by induction on {{math|dom(''x'')}} and {{math|dom(''y'')}} by {{math|1=''xy'' = ''σ''(''L'',{{px2}}''R'')}}, where
* {{math|1=''L'' = {{mset| ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''L''(''x''), ''v'' ∈ ''L''(''y'') }} ∪ {{mset| ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''R''(''x''), ''v'' ∈ ''R''(''y'') }}}}
* {{math|1=''R'' = {{mset| ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''L''(''x''), ''v'' ∈ ''R''(''y'') }} ∪ {{mset| ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''R''(''x''), ''v'' ∈ ''L''(''y'') }}}}

The multiplicative identity is given by the number {{math|1=1 = {{mset| (0, +1) }}}}, i.e. the number {{math|1}} has domain equal to the ordinal {{math|1}}, and {{math|1=1(0) = +1}}.