surreal number

plural surreal numbers

1. mathematicsAny element of a certain 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.

   3 more examples

   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.

   • 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<Y, if and only if − there is no x∈X_L such that Y<x, and − there is no y∈Y_R such that y<X.

   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

   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 #92;#123;L#92;vertR#92;#125; is a surreal number. All surreal numbers are constructed in this fashion.

   2018, Steven G. Krantz, Essentials of Mathematical Thinking, Taylor & Francis (Chapman & Hall/CRC Press), page 247