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surreal number

A member of a mind-bogglingly vast class of numbers that includes all of the real numbers, all of Georg Cantor's infinite ordinal numbers (different kinds of infinity), a set of infinitesimals (infinitely small numbers) produced from these ordinals, and strange numbers that previously lived outside the known realm of mathematics. Each real number, it turns out, is surrounded by a "cloud" of surreals that lie closer to it than do any other real number. One of these surreal clouds occupies the curious space between zero and the smallest real number greater than zero and is made up of the infinitesimals.

Surreal numbers were invented or discovered (depending on your philosophy) by John Conway to help with his analysis of certain kinds of games. The idea came to him after watching the British Go champion playing in the mathematics department at Cambridge. Conway noticed that endgames in Go tend to break up into a sum of games, and that some positions behaved like numbers. He then found that, in the case of infinite games, some positions behaved like a new kind of number – the surreals. The name "surreal" was introduced by Donald Knuth in his 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness.1 This novelette is notable as being the only instance where a major mathematical idea has been first presented in a work of fiction. Conway went on to describe the surreal numbers and their use in analyzing games in his 1976 book On Numbers and Games.2 The surreals are similar to the hyperreal numbers, but they are constructed in a very different way and the class of surreals is larger and contains the hyperreals as a subset.


  1. Knuth, Donald. Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. Reading, Mass.: Addison-Wesley, 1974.
  2. Conway, John Horton. On Numbers and Games. New York: Academic Press, 1976.

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