surreal number
Contents
EnglishEdit
EtymologyEdit
Coined by Donald Knuth in his 1974 novelette Surreal Numbers: How Two ExStudents Turned on to Pure Mathematics and Found Total Happiness. The concept had been developed by British mathematician John Conway for his game theoretic research of the board game go. Conway had simply called them numbers, but subsequently adopted Knuth's term and used it in his 1976 book On Numbers and Games.
NounEdit
surreal number (plural surreal numbers)
 (mathematics) Any element of a field equivalent to the real numbers augmented with infinite and infinitesimal numbers (respectively larger and smaller (in absolute value) than any positive real number).
 Conway's construction of surreal numbers relies on the use of transfinite induction.
 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, An Introduction to the Theory of Surreal Numbers, Cambridge University Press, 1987, Paperback, →ISBN.
 2012, Fredrik Nordvall Forsberg, Anton Setzer, A Finite Axiomatisation of InductiveInductive Definitions, Ulrich Berger, Hannes Diener, Peter Schuster, Monika Seisenberger (editors), Logic, Construction, Computation, Ontos Verlag, page 263,
 The class^{2} of surreal numbers is defined inductively, together with an order relation on surreal numbers wich is also defined inductively:
 • A surreal number consists of two sets and of surreal numbers, such that no element from is greater than any element from .
 • A surreal number is greater than another surreal number , , if and only if
 − there is no such that , and
 − there is no such that .
 The class^{2} of surreal numbers is defined inductively, together with an order relation on surreal numbers wich is also defined inductively:
 2018, Steven G. Krantz, Essentials of Mathematical Thinking, Taylor & Francis (Chapman & Hall/CRC Press), page 247,
 Here we shall follow Conway's exposition rather closely. Let and be two sets of numbers. Assume that no member of is greater than or equal to any member of . Then is a surreal number. All surreal numbers are constructed in this fashion.
TranslationsEdit
element of an extension of the real numbers that includes infinite numbers and infinitesimals


Further readingEdit
 Hahn series on Wikipedia.Wikipedia
 Hyperreal number on Wikipedia.Wikipedia
 Nonstandard analysis on Wikipedia.Wikipedia
 Surreal Number on Wolfram MathWorld
 Surreal numbers on Encyclopedia of Mathematics