Surreal numbers grew out of Go endgames and contain infinity itself
While studying endgames in the board game Go, John Horton Conway stumbled onto a number system that holds every real number plus infinite and infinitesimal ones. Donald Knuth gave it a name, surreal numbers, in a 1974 book written as a dialogue between two former students discovering mathematics.
Conway simply called them numbers; Knuth's label stuck, and Conway adopted it for his 1976 book On Numbers and Games. The construction starts from nothing. Each number is written as a pair of sets, a left set and a right set, with everything on the left smaller than everything on the right, and the pair names a value sitting between them. With both sides empty you get zero.
Numbers are then born day by day. On day one, minus one and one appear. On day two come a half and minus a half, since Conway's simplicity rule picks the simplest value that fits between existing ones. Finite steps produce only fractions whose denominators are powers of two, but after infinitely many stages every real number can be captured, much like a Dedekind cut. Keep going and you meet omega, bigger than every integer, and epsilon, positive yet smaller than any positive real.
Remarkably, ordinary arithmetic still works. Addition, subtraction, multiplication and division extend to these exotic values, so expressions like two omega or omega minus one make sense, and the whole collection forms an ordered field. In a suitable set theory, every other ordered field, including the rationals, the reals and the hyperreals, fits inside it.
Another path led to the same place. In 1907 Hans Hahn and Felix Hausdorff introduced ideas that Norman Alling used in 1962 to build related fields, and in 1987 he showed that extending his method to all ordinals gives something equivalent to the surreals. Alling nevertheless credited Conway, whose notion of each number's birthday became central to how the system is understood.
Source: Surreal number
