Construction, Operations, and Applications of the Surreal Numbers

Abstract

The surreal numbers are a unifying, non-Archimedean ordered eld, the existence of which was speculated at by Hahn and Hausdor and later formalized by Alling and Conway. In this expository thesis we see in Section 1.1 how the surreals can be constructed via Dedekind-like cuts, as well as how they arise naturally in quantifying game positions in Section 1.2. Section 2 describes the operations unique to the surreals, allowing us to manipulate strange numbers such as 5 3 p ! 􀀀 ". The predominant application of surreal numbers thus far has been to combinatorial game theory, and to that end in Section 3.2 we will evaluate a Go game using the program developed by Berlekamp and Wolfe. In Section 4 we see the strengthening currents of a cultural shift; where many have chided the surreals for lacking hard applications, in the past decade alone it has been shown that the surreals are isomorphic to the maximal realization of the hyperreals, that surreal analysis is a eld in its own right riddled with open questions, and excitingly that the surreals have a transseries structure with a derivation.