查看“︁Surreal number”︁的源代码

==英語==
{{wikipedia|lang=en}}

===詞源===
1974年由{{w|高德納}}在其小說 ''Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness'' 中提出。此概念後來由英國數學家{{w|約翰·何頓·康威}}應用在對圍棋博弈論的研究中。最初康威只稱其為 ''numbers''，但在1976年的 ''{{w|On Numbers and Games}}'' 中採用了高德納的用詞。

===名詞===
{{en-noun}}

# {{lb|en|數學}} [[超現實數]]
#: {{ux|en|Conway's construction of '''surreal numbers''' relies on the use of [[transfinite induction]].|康威對'''超現實數'''的建構基於超限歸納法的使用。}}
#: {{ux|en|Conway's approach was to build numbers from scratch using a construction inspired by his game theory research; the resulting class of '''surreal numbers''' proved much larger than the class of real numbers.}}
#* '''1986''', Harry Gonshor, ''[https://books.google.com.au/books?id=fCjLwvRhKocC&printsec=frontcover&dq=%22surreal+number%22%7C%22surreal+numbers%22&hl=en&sa=X&ved=0ahUKEwjBip-Rj77iAhU27nMBHYICAnEQ6AEILDAB#v=onepage&q=%22surreal%20number%22%7C%22surreal%20numbers%22&f=false An Introduction to the Theory of '''Surreal Numbers''']'', {{w|劍橋大學出版社|Cambridge University Press}}, 1987, Paperback, {{ISBN|9780521312059}}.
#* '''2012''', Fredrik Nordvall Forsberg, Anton Setzer, ''A Finite Axiomatisation of Inductive-Inductive Definitions'', Ulrich Berger, Hannes Diener, Peter Schuster, Monika Seisenberger (editors), ''Logic, Construction, Computation'', Ontos Verlag, [https://books.google.com.au/books?id=axA-uS3vwDoC&pg=PA263&dq=%22surreal+number%22%7C%22surreal+numbers%22&hl=en&sa=X&ved=0ahUKEwjBip-Rj77iAhU27nMBHYICAnEQ6AEIVjAK#v=onepage&q=%22surreal%20number%22%7C%22surreal%20numbers%22&f=false page 263],
#*: The class<sup>2</sup> of '''surreal numbers''' is defined inductively, together with an order relation on '''surreal numbers''' wich is also defined inductively:
#*:: • A '''surreal number''' <math>X=(X_L,X_R)</math> consists of two sets <math>X_L</math> and <math>X_R</math> of '''surreal numbers''', such that no element from <math>X_L</math> is greater than any element from <math>X_R</math>.
#*:: • A '''surreal number''' <math>Y=(Y_L,Y_R)</math> is greater than another '''surreal number''' <math>X=(X_L,X_R)</math>, <math>X\le Y</math>, if and only if
#*::: − there is no <math>x\in X_L</math> such that <math>Y\le x</math>, and
#*::: − there is no <math>y\in Y_R</math> such that <math>y\le X</math>.
#* '''2018''', Steven G. Krantz, ''Essentials of Mathematical Thinking'', Taylor & Francis (Chapman & Hall/CRC Press), [https://books.google.com.au/books?id=NGhQDwAAQBAJ&pg=PA247&dq=%22surreal+number%22%7C%22surreal+numbers%22&hl=en&sa=X&ved=0ahUKEwjBip-Rj77iAhU27nMBHYICAnEQ6AEIWzAL#v=onepage&q=%22surreal%20number%22%7C%22surreal%20numbers%22&f=false page 247],
#*: Here we shall follow Conway's exposition rather closely. Let <math>L</math> and <math>R</math> be two sets of numbers. Assume that no member of <math>L</math> is greater than or equal to any member of <math>R</math>. Then <math>\{ L\vert R\}</math> is a '''surreal number'''. All '''surreal numbers''' are constructed in this fashion.

===延伸閱讀===
* {{pedialite|Hahn series|lang=en}}
* {{pedialite|Hyperreal number|lang=en}}
* {{pedialite|Non-standard analysis|lang=en}}
* [http://mathworld.wolfram.com/SurrealNumber.html Surreal Number] on {{w|MathWorld|Wolfram MathWorld}}
* [https://www.encyclopediaofmath.org/index.php/Surreal_numbers Surreal numbers] on {{w|數學百科全書|Encyclopedia of Mathematics}}
[[Category:英語 數字]]