I just now had a grade school flashback. As soon as I learned how to compute decimal expansions of rationals, I noticed that they'd have to repeat, and since .696969... = 69/99, it seemed axiomatically obvious that .999999 = 99/99. In fact, to compute, e.g., 22/7 I would divide 7 into 21.9999999..., knowing the division would periodically come out even, but never really understanding why. To this day, I have great difficulty proving things that seem obvious. --rwg On 2015-09-17 15:08, James Propp wrote:
I thought my conclusions hinge on the Archimedean Axiom.
Jim
On Thursday, September 17, 2015, Dan Asimov <asimov@msri.org> wrote:
Jim,
In your essay it appears that your conclusions depend — as they certainly should — on the definition of the real numbers.
But the main reference for that is "the number line", which does not seem to be defined. If the number line had infinitesimals, your conclusions would probably be different.
—Dan
On Sep 17, 2015, at 5:12 AM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
The final version of my essay "The One About .999..." is available at http://mathenchant.wordpress.com . I especially like the new graphic contributed by Li-Mei Lim.
I'd welcome comments in the Comments section; the blog has been too quiet. And surely the pedagogical aspects of the topic are subject to legitimate controversy. Do you think I'm wrong about the best way to deal with convergence confusion and decimal dissidence?
Jim