Surreal Numbers and Games
Surreal numbers are a fascinating mathematical structure, built from a few simple rules but giving rise to marvellous complexity. The surreal numbers contain all the real numbers with which we are familiar, as well as an infinitude of new quantities. We will discover surreal numbers that are greater than any positive integer, and ones that are infinitesimally small. Concepts like the square root and the reciprocal of infinite quantities will not only be defined, but we will find that they show logical and beautiful behaviour.
The Surreal numbers were invented by mathematician John H. Conway as part of an investigation into endgames in the game of Go, an oriental board game that also produces complex behaviour from a small set of simple rules. They were presented to the world in the form of a small novelette by Donald E. Knuth, in which a young couple on holiday discover a rock inscribed with Conway's rules and proceed to derive the entire theory.
We will begin the same way, beginning with the initial axioms and working our way up to the entire vast structure. Along the way we will prove that all the familiar properties of real numbers (such as the transitive law of inequality, and the commutative law of addition) all hold. A basic familiarity with set theory is assumed; for a refresher, see Set Theory.