very helpful. ok. I'll try to summarize. If I've got it right, two things that were clarified in the 1800s that enabled the expression 0.999...=1 to be valid: 1. a convergent series 2. an infinite series The concepts work together but were given their modern notions at separate times. Cauchy created the basis for defining a convergent series (with the epsilon-delta definition of limit 1821 (Dan's note) )... but, unless I'm missing something, there would still be the following problems: Zeno's arrow will never actually hit the tree (eg 1/2 + 1/4 + 1/8 will never quite get to 1) a polygon with more and more sides will not quite ever become a circle 9/10 + 9/100 + 9/1000 etc will not quite ever equal 1 Then Cantor (1890 or so) came up with the concept of infinity. Only then would the following hold: 1/2+1/4+1/8+...=1 a circle is a polygon with infinite sides 9/10 + 9/100 + 9/1000 ... =1 There is no proof for these: they're definitions. On Wed, Nov 14, 2012 at 6:06 AM, James Buddenhagen <jbuddenh@gmail.com>wrote:
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun