Question

In mathematics, what are surreal numbers?

Answer 1

Surreal numbers are the most natural collection of numbers which includes both the real numbers and the infinite ordinal numbers of Georg Cantor. They were invented by John H. Conway in 1969. Every real number is surrounded by surreals, which are closer to it than any real number. Knuth (1974) describes the surreal numbers in a work of fiction.

The surreal numbers are written using the notation \(\{a \mid b\}\), where \(\{\mid\}=0,\{0 \mid\}=1\) is the simplest number greater than \(0,\{1 \mid\}=2\) is the simplest number greater than 1, etc. Similarly, \(\{\mid 0\}=-1\) is the simplest number less than 0 , etc. However, 2 can also be represented by \(\{1 \mid 3\},\{3 / 2 \mid 4\},\{1 \mid \omega\}\), etc.
Some simple games have abbreviated names that can be expressed in terms of surreal numbers. For example, \(*=\{0 \mid 0\}, 1=\{0 \mid\}\), \(n=\{n-1 \mid\}\) for an integer \(n, 1 / 2=\{0 \mid 1\}, \uparrow=\{0 \mid *\}\), and \(\downarrow=\{* \mid 0\}\). Most surreal numbers can be represented as hackenbush positions.