  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.
