surreal number
Noun
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Noun
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 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, 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\le Y, if and only if
- − there is no x\in X_L such that Y\le x, and
- − there is no y\in Y_R such that y\le X.
- The class2 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 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.
- French: nombre surréel
- Italian: numero surreale
- Portuguese: número surreal
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