Let me put in my 9.9999999... cents worth, though I'm not sure it's suitable for the classroom. Define the reals as (infinite) negative continued fractions. Each real has a unique expansion. The rationals are those with only finitely many nontwos. There's no trouble of the 1.000000... =? 0.9999999... kind. Proofs of Cantor's theorems are very simple. Incidentally, if you want to pursue the theory, you don't have to worry about whether it's (-1)^n or (-1)^{n+1} -- the signs are always the same. R. On Fri, 29 May 2009, James Propp wrote:
Rich Schroeppel writes:
My favorite construction for the reals is to match the implicit "infinte decimals" that we grew up with. It's the ground state, hence "intuitive". There's no harm in mentioning the alternatives, nor in emphasizing there are a bunch of equivalents.
Does anyone know of a good write-up of the construction of the reals and their arithmetic via decimal expansions?
The only example I can think of is F. Faltin, N. Metropolis, B. Ross and G.-C. Rota, The real numbers as a wreath product, Advances in Math., 16(1975), 278--304.
Jim Propp
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