# surreal number

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### Etymology

Coined by Donald Knuth in his 1974 novelette Surreal Numbers: How Two Ex-Students 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.

### Noun

surreal number (plural surreal numbers)

1. (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 Inductive-Inductive Definitions, Ulrich Berger, Hannes Diener, Peter Schuster, Monika Seisenberger (editors), Logic, Construction, Computation, Ontos Verlag, page 263,
The class2 of surreal numbers is defined inductively, together with an order relation on surreal numbers wich is also defined inductively:
• A surreal number $X=(X_{L},X_{R})$ consists of two sets $X_{L}$ and $X_{R}$ of surreal numbers, such that no element from $X_{L}$ is greater than any element from $X_{R}$ .
• A surreal number $Y=(Y_{L},Y_{R})$ is greater than another surreal number $X=(X_{L},X_{R})$ , $X\leq Y$ , if and only if
− there is no $x\in X_{L}$ such that $Y\leq x$ , and
− there is no $y\in Y_{R}$ such that $y\leq X$ .
• 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 $L$ and $R$ be two sets of numbers. Assume that no member of $L$ is greater than or equal to any member of $R$ . Then $\{L\vert R\}$ is a surreal number. All surreal numbers are constructed in this fashion.