surreal number

Etymology
Coined by 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 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 .

Noun

 * 1)  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).
 * 2) * 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 $$y\le X$$.
 * • 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 $$y\le X$$.

Translations

 * French: nombre surréel
 * Italian: numero surreale
 * Portuguese: número surreal