The purpose of this thesis is to explore the Surreal Numbers from an elementary, con- structivist point of view, with the intention of introducing the numbers in a palatable way for a broad audience with minimal background in any specific mathematical field. Created from two recursive definitions, the Surreal Numbers form a class that contains a copy of the real numbers, transfinite ordinals, and infinitesimals, combinations of these, and in- finitely many numbers uniquely Surreal. Together with two binary operations, the surreal numbers form a field. The existence of the Surreal Numbers is proven, and the class is constructed from nothing, starting with the integers and dyadic rationals, continuing into the transfinite ordinals and the remaining real numbers, and culminating with the infinitesimals and uniquely surreal numbers. Several key concepts are proven regarding the ordering and containment properties of the numbers. The concept of a surreal continuum is introduced and demonstrated. The binary operations are explored and demonstrated, and field properties are proven, using many methods, including transfinite induction.
| Identifer | oai:union.ndltd.org:vcu.edu/oai:scholarscompass.vcu.edu:etd-3934 |
| Date | 05 December 2012 |
| Creators | Hostetler, Joshua |
| Publisher | VCU Scholars Compass |
| Source Sets | Virginia Commonwealth University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Theses and Dissertations |
| Rights | © The Author |
Page generated in 0.002 seconds