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Combinatorial Games (Part II): Different Moves for Left and Right

Feature Column Archive

5. Surreal numbers

The surreal numbers were developed by John Conway,

 A photo of John Conway

but the name universally used for these numbers, surreal, was coined by the Stanford computer scientist and mathematician Donald Knuth

A photo of Donald Knuth

in his book Surreal Numbers (1974). In an oversimplified way one can view the real numbers as a way of filling in the "spaces" between the integers. There are many approaches to this, including Richard Dedekind's idea of Dedekind Cuts which, starting with the rational numbers, constructs the real numbers. A new number {L | R} emerged from two sets L and R of rational numbers with the property that no element of L was greater than any element in R.

Conway's surreal numbers fill in the "spaces" between the ordinal numbers in a way that builds on their notation and is related to Dedekind's approach. The idea is that {a, b, c, d, .... | e, f, g, h, .... } where all the elements to the left of the vertical bar are always less than the elements to the right of the vertical bar. However, exactly what number is given by this symbolism?

Let us put our toes into the water: we know the ordinal numbers are those where there are no numbers to the right of the vertical bar. Recall that { | } is the ordinal number 0, {0 | } is the ordinal number 1, { 0, 1 | } is the ordinal number 2 and so on. So, in the surreal numbers, we identify these ordinal numbers with the surreal number notation by having them be 0, 1, 2, etc. By symmetry we can construct the negatives of these whole surreal numbers: { | 0} is -1,   { | 0, 1} is -2, and { | 0, 1, 2 } is -3. Conway's clever idea was to have other numbers be the "simplest" number associated with values on both sides of the vertical bar. Thus, he took { 0 | 1 } to be the number 1/2.

  1. Introduction
  2. Hex
  3. Hackenbush
  4. Counting and sets
  5. Surreal numbers
  6. Surreal numbers and games
  7. References