The Eudoxus Real Numbers

Author name not available (Why is that?)

Publication date: 24 May 2004

Abstract: This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real numbers using Dedekind cuts, the idea is inspired by the ancient Greek theory of proportion, due to Eudoxus. However, unlike the Dedekind construction, the construction proceeds directly from the integers to the real numbers bypassing the intermediate construction of the rational numbers. The construction of the additive group of the reals depends on rather simple algebraic properties of the integers. This part of the construction can be generalised in several natural ways, e.g., with an arbitrary abelian group playing the role of the integers. This raises the question: what does the construction construct? In an appendix we sketch some generalisations and answer this question in some simple cases. The treatment of the main construction is intended to be self-contained and assumes familiarity only with elementary algebra in the ring of integers and with the definitions of the abstract algebraic notions of group, ring and field.

Has companion code repository: https://github.com/Lix0120/eudoxus

Mathematics Subject Classification ID

No records found.

This page was built for publication: The Eudoxus Real Numbers

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6474026)