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: $X_L = X_R = \{\}$.