Surreal numbers are something of an extension to the theory of
real numbers,
relating somehow to Leibniz's theories of infintesimals.

Surreal numbers were invented by John Conway, and include
all natural numbers, negative numbers, fractions, irrational
numbers, numbers bigger than infinity, and smaller than
the smallest fraction.

There are a number of ways of looking at surreal numbers.

### Set-pairs

One way to look at a surreal number is as the pair
of sets $( X_L | X_R )$. $X_L$ and $X_R$ are possibly
empty sets of other surreal numbers called the left and
right sets. The only constraint on these sets is that
no member of $X_R$ is less than or equal to any member
of $X_L$. The first and simplest surreal number is
zero, and is defined as having empty left and right
sets: $X_L = X_R = \{\}$.