My �construction�of the reals, referred to by Ross and Mike, came fom an observation of Tate about bilinear maps many years earlier, which essentially goes back to Eudoxus. I take the task to be relating the continuous to the discrete, rather than constructing the former from the latter, but never mind. R is the ring of endomorphisms E(T(L))of the group T(L) of translations (rigid maps at finite distance from the identity) of the line L. If we start instead with a discrete line D (a line of dots), then T(D) is isomorphic to Z (additive group without preferred generator). E(T(D) is the ring Z, but instead take A(T(D)), �almost homomorphisms� Z to Z. A(T(D)) is an additive group with a �multiplication� by composition, but not a ring, since one distributive law and commutativity of multiplication fail; but A(T(D)) modulo bounded maps (much as in Mike�s description) is R. Steve Schanuel