Decimal numbers were invented by Stevin in 1585, but he only considered finite decimals. So saying that .9999... = 1 was accepted by Stevin, as you seem to below, may be stretching it. Wallis, 100 years after Stevin only used finite decimals. Thus the number 1/3 was not yet identified with an infinite decimal for Stevin or Wallis. For me, a more interesting question is when was it realized that infinitely long decimal strings (not ending in 0's) can be (or should be) identified with the real numbers. Maybe with Cauchy (?). Some interesting history is here: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Real_numbers_1.html http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Real_numbers_2.html (you may have to cut and paste if gmail scarfs those links). On Tue, Nov 13, 2012 at 10:33 PM, Gary Antonick <gantonick@post.harvard.edu> wrote:
Hi all,
I'm wondering if anyone knows when and why it was agreed that 0.999...=1.
That is, the expression meant its limit of 1 and not something that got closer and closer to 1.
I've asked several people and have gotten some big pieces but not quite the whole story: Keith Devlin (last week): before Cantor mathematicians considered "0.999..." to mean a *growing sequence* of 9's after the decimal. After Cantor it was decided that the expression meant the *limit* of this sequence: an infinite number of 9's after the decimal all at once. Steven Strogatz (this afternoon) suggested talking to John Stillwell John Stillwell (a couple hours ago) seemed to indicate the shift in perspective happened gradually, with Zeno arguing for the growing sequence (which would never get to 1) and everyone after the Axiom of Choice agreeing that the expression was referring to its limit.
-400 Zeno: 1/2 + 1/4 + 1/8 etc will never get to 1 -350 Aristotle: 1/2 + 1/4 + 1/8 etc eventually gets to 1 -300 Euclid: 1/4 + 1/4^2 + 1/4^3 + etc eventually gets to 1/3 1585 Stevin: 0.999... = 1 1671 Newton: 0.999... = 1 1858 Dedekind cuts 1880? Cantor 1904 Zermelo: Axiom of Choice
Does anyone have more precise timing for this shift from thinking?
All the best,
Gary _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun