The definition and construction of the surreals is due to and Found Total Happiness''. This book is a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea was first presented in a work of fiction. In his book, which takes the form of a dialogue, Knuth coined the term ''surreal numbers'' for what Conway had simply called ''numbers'' originally. Conway liked the new name, and later adopted it himself. Conway then described the surreal numbers and used them for analyzing games in his 1976 book '' On Numbers And Games ''.
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{ ''X<sub>L</sub>'' ''X<sub>R</sub>'' } the sets ''X<sub>L</sub>'' and ''X<sub>R</sub>'' are called the ''left set'' of ''x'' and ''right set'' of ''x'' respectively To avoid lots of brackets we will write { {''a'', ''b'', } { ''x'', ''y'', } } simply as { ''a'', ''b'', ''x'', ''y'', } and { {''a''} {} } as { ''a'' } and { {} {''a''} } as { ''a'' }Lower-case characters in this notation { ''a'' ''b'' } refer to individual numbers or games, while upper-case characters { ''L'' ''R'' } refer to sets of numbers or games
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{ ''X<sub>L</sub>'' ''X<sub>R</sub>'' } and ''y'' = { ''Y<sub>L</sub>'' ''Y<sub>R</sub>'' } it holds that ''x'' &le ''y'' if and only if ''y'' is less than or equal to no member of ''X<sub>L</sub>'', and no member of ''Y<sub>R</sub>'' is less than or equal to ''x''
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= { '''-1''', '''0''' } == { '''-1''', '''1''' } == { '''-1''', '''0''', '''1''' } <
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= '''-1''' <
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= { '''-1''' '''0''', '''1''' } <
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= { '''1''' } == { '''-1''' '''1''' } == '''0''' <
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= { '''-1''', '''0''' '''1''' } <
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= '''1''' <
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= { '''0''', '''1''' } == { '''-1''', '''1''' } == { '''-1''', '''0''', '''1''' }
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"{" class="copylinks" target="_blank"> '''-1''' } , '''-1''' '''0''' } , '''0''' '''1''' } , and '''1''' }
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"''X<sub>L</sub>'']" class="copylinks" target="_blank">= [''Y<sub>L</sub>'' and = [''Y<sub>R</sub>'' then ''X<sub>L</sub>'' ''X<sub>R</sub>'' } = ''Y<sub>L</sub>'' ''Y<sub>R</sub>'' }
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"''X'']" class="copylinks" target="_blank">denotes { [''x'' ''x'' in ''X'' } So the description of the ordered set that was found above can be rewritten to:
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= { -1, 0 } == { -1, 1 } == { -1, 0, 1 } <
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= -1 <
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= { -1 0, 1 } <
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= { 1 } == { -1 1 } == 0 <
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= { -1, 0 1 } <
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= 1 <
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= { 0, 1 } == { -1, 1 } == { -1, 0, 1 }
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{ ''X<sub>L</sub>'' + ''y'' &cup ''x'' + ''Y<sub>L</sub>'' ''X<sub>R</sub>'' + ''y'' &cup ''x'' + ''Y<sub>R</sub>'' }
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{ ''x'' + ''y'' ''x'' in ''X'' } and ''x'' + ''Y'' = { ''x'' + ''y'' ''y'' in ''Y'' }
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{ -''X<sub>R</sub>'' -''X<sub>L</sub>'' }
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{ -''x'' ''x'' in ''X'' }
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{ (''X<sub>L</sub>y'' + ''xY<sub>L</sub>'' - ''X<sub>L</sub>Y<sub>L</sub>'') &cup (''X<sub>R</sub>y'' + ''xY<sub>R</sub>'' - ''X<sub>R</sub>Y<sub>R</sub>'') (''X<sub>L</sub>y'' + ''xY<sub>R</sub>'' - ''X<sub>L</sub>Y<sub>R</sub>'') &cup (''X<sub>R</sub>y'' + ''xY<sub>L</sub>'' - ''X<sub>R</sub>Y<sub>L</sub>'') }
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{ ''xy'' ''x'' in ''X'' and ''y'' in ''Y'' }, ''Xy'' = ''X''{''y''} and ''xY'' = {''x''}''Y''
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{ { ''a'' / 2<sup>''b''</sup> in ''S''<sub>&omega</sub> 3''a'' < 2<sup>''b''</sup> } { ''a'' / 2<sup>''b''</sup> in ''S''<sub>&omega</sub> 3''a'' > 2<sup>''b''</sup> } }
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"http://wwwinformationdelightinfo/encyclopedia/entry/Pi" class="copylinks">&pi ''' = {3, 25/8, 201/64, , 101/32, 51/16, 13/4, 7/2, 4}
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{ 0 , 1/16, 1/8, 1/4, 1/2, 1 }
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{ &epsilon , &epsilon + 1/16, &epsilon + 1/8, &epsilon + 1/4, &epsilon + 1/2, &epsilon + 1 } and
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{ 0 &epsilon }
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{ ''S''<sub>&omega</sub> }
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1, 2, 3, 4, }
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{ &omega } and
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{ ''S''<sub>&omega</sub> &omega }
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{ &omega + 1 },
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{ &omega + 2 },
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{ ''S''<sub>&omega</sub> &omega - 1 } and
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{ ''S''<sub>&omega</sub> &omega - 2 }
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{ &omega + ''S''<sub>&omega</sub> }
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{ ''x'' + ''y'' ''y'' in ''Y'' } Just as 2&omega is bigger than &omega it can also be shown that &omega/2 is smaller than &omega because
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{ ''S''<sub>&omega</sub> &omega - ''S''<sub>&omega</sub> }
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{ ''x'' - ''y'' ''y'' in ''Y'' } Finally, it can be shown that there is a close relationship between &omega and &epsilon because it holds that
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{ ''X<sub>L</sub>'' ''X<sub>R</sub>'' } if it holds for all elements of ''X<sub>L</sub>'' and ''X<sub>R</sub>''
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{ 0, ''r'' &omega<sup>''x''<sub>L</sub></sup> ''s'' &omega<sup>''x''<sub>R</sub></sup> },
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"http://wwwinformationdelightinfo/encyclopedia/entry/fuzzy_game" class="copylinks">Fuzzy '' (incomparable with zero, such as {1-1})
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"http://wwwinformationdelightinfo/encyclopedia/entry/Go_(board_game)" class="copylinks">Go , and there are numerous connections between popular games and the surreals In this section, we will use a capitalized ''Game'' for the mathematical object {LR}, and the lowercase ''game'' for recreational games like Chess or Go
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"http://wwwinformationdelightinfo/encyclopedia/entry/fuzzy_game" class="copylinks">Fuzzy , as opposed to Positive, Negative, Or Zero An example of a fuzzy game is Star ()
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{ ''x''<sub>&alpha</sub> : &alpha < dom(''x'') &and ''x''(&alpha) = + 1 }
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{ ''x''<sub>&alpha</sub> : &alpha < dom(''x'') &and ''x''(&alpha) = - 1 },
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&sigma(''M'',''S''), where ''M'' = { ''f''(''x'') : ''x'' &isin ''L'' } and ''S'' = { ''f''(''x'') : ''x'' &isin ''R'' }
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{ ''L'' ''R'' }, where ''L'' = { ''g''(''y'') : ''y'' &isin ''L''(''x'') } and ''R'' = { ''g''(''y'') : ''y'' &isin ''R''(''x'') }
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