I have a vivid memory of performing this construction as an exercise, I think when I was a senior in high school. I was motivated by annoyance at the Cauchy-sequence construction, and was certain that the formal-decimal-expansion approach would prove much more straightforward. I don't remember whether it was, in fact. On Fri, Feb 12, 2010 at 8:38 AM, Scott Beaver <sfbeaver@minetfiber.com>wrote:
Chapter 16 of Ken Ross' "Elementary Analysis: The Theory of Calculus" doesn't construct them, but does provide an elementary proof that there is a bijection between real numbers and decimal expansions (after the choice of 9-tail or 0-tail is made if necessary for a given real number).
Scott
On 2/11/2010 11:15 PM, James Propp wrote:
Do any of you know of books or articles that construct the real numbers (and their arithmetic) in terms of their binary (or decimal) expansions?
I know of just two: the article "The real numbers as a wreath product" by Faltin, Metropolis, Ross, and Rota; and Gowers' web-screed "Real numbers as infinite decimals", aka "What is so wrong with thinking of real numbers as infinite decimals?". But I suspect there are others.
Do you know of anyone else who has done anything along these lines? or even anyone who has given a conscientious definition of what's really involved when you add two infinite decimals and perform carries? Note that for this kind of analysis, 0.999... is not the same as 1.000..., even though both represent the same real number. (Here's a fun fact I haven't seen anywhere: if you add two (positive) infinite decimals, the only way their sum can end in infinitely 0's is if both summands do. Have you seen this before?)
Any references or suggestions you can offer will be appreciated!
Jim Propp
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