Surreal numbers are the most natural collection of numbers which includes both the real numbers and the infiniteordinal 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, where , is the simplest number greater than 0, is the simplest number greater than 1, etc. Similarly,
is the simplest number less than
0. However, 2 can also be represented by , , , etc.
The same notation is used for more general Conway games, which need not be surreal numbers. For example, the games , , and are not surreal numbers (Conway 1976).
, where 
, 
is the simplest number greater than 0, 
is the simplest number greater than 1, etc. Similarly,
is the simplest number less than
0. However, 2 can also be represented by 
, 
, 
, etc.
, 
,
while for a negative integer 
, 
.
Another example is 
.
Most surreal numbers can be represented as hackenbush
positions.
, 
, and 
are not surreal numbers (Conway 1976).
