[math-fun] when and why was it agreed that 0.999...=1?
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
I'm pretty sure the shift in thinking -- at least among mathematicians -- occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821. I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress. --Dan On 2012-11-13, at 8:33 PM, Gary Antonick 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
Vi Hart dealt with all these flawed proofs that 1= 0.99999....in a video http://www.youtube.com/watch?v=wsOXvQn3JuE On Tue, Nov 13, 2012 at 9:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm pretty sure the shift in thinking -- at least among mathematicians -- occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.
I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.
--Dan
On 2012-11-13, at 8:33 PM, Gary Antonick 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
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-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
On a related point, is there any worked out finitist arithmetic in which, for example, (10^500 + 1)=10^500? Brent Meeker On Tue, Nov 13, 2012 at 9:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm pretty sure the shift in thinking -- at least among mathematicians -- occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.
I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.
--Dan
On 2012-11-13, at 8:33 PM, Gary Antonick 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
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The question has been discussed -- but not answered to my satisfaction, probably because no such system has been developed -- at http://mathoverflow.net/questions/44208/is-there-any-formal-foundation-to-ul... Charles Greathouse Analyst/Programmer Case Western Reserve University On Wed, Nov 14, 2012 at 4:15 PM, meekerdb <meekerdb@verizon.net> wrote:
On a related point, is there any worked out finitist arithmetic in which, for example, (10^500 + 1)=10^500?
Brent Meeker
On Tue, Nov 13, 2012 at 9:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm pretty sure the shift in thinking -- at least among mathematicians --
occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.
I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.
--Dan
On 2012-11-13, at 8:33 PM, Gary Antonick 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<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
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Similar is the sorties paradox<http://en.wikipedia.org/wiki/Sorites_paradox>: A heap of sand has the property that removal of a single grain leaves a heap of sand--but a single grain of sand is not a heap. I recall hearing Rohit Parikh <http://web.gc.cuny.edu/philosophy/faculty/parikh.htm> give a talk on this 30 years or so ago. As I recall he had devised a logic in which this made sense. Perhaps it was in his paper on vague predicates since vagueness is one of his listed interests. By assuming x - 1 = x for x sufficiently large you would have a similar problem. But + replacing - is perhaps a different game. On Wed, Nov 14, 2012 at 4:15 PM, meekerdb <meekerdb@verizon.net> wrote:
On a related point, is there any worked out finitist arithmetic in which, for example, (10^500 + 1)=10^500?
Brent Meeker
On Tue, Nov 13, 2012 at 9:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm pretty sure the shift in thinking -- at least among mathematicians --
occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.
I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.
--Dan
On 2012-11-13, at 8:33 PM, Gary Antonick 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<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
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Isn't that called floating point arithmetic? On 2012-11-14 16:15, meekerdb wrote:
On a related point, is there any worked out finitist arithmetic in which, for example, (10^500 + 1)=10^500?
Brent Meeker
On Tue, Nov 13, 2012 at 9:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm pretty sure the shift in thinking -- at least among mathematicians -- occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.
I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.
--Dan
On 2012-11-13, at 8:33 PM, Gary Antonick 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
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Floating point arithmetic would an example. But is there an axiomatic foundation of floating point arithmetic as a separate theory - not just as an approximation to real numbers. Brent On 11/14/2012 4:57 PM, Mike Speciner wrote:
Isn't that called floating point arithmetic?
On 2012-11-14 16:15, meekerdb wrote:
On a related point, is there any worked out finitist arithmetic in which, for example, (10^500 + 1)=10^500?
Brent Meeker
On Tue, Nov 13, 2012 at 9:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm pretty sure the shift in thinking -- at least among mathematicians -- occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.
I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.
--Dan
On 2012-11-13, at 8:33 PM, Gary Antonick 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
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="meekerdb" <meekerdb@verizon.net> Floating point arithmetic would an example. But is there an axiomatic foundation of floating point arithmetic as a separate theory - not just as an approximation to real numbers.
Heh, well, of course, sure--one can "axiomatize" any random set of facts, right? But more informally, I've always liked Gosper's formulation: "Flonums are just rationals with a 'Kick Me!' sign taped to their butts."
Do those same people think that 0.3333... doesn't equal 1/3? And if 0.3333... isn't 1/3, then how do you represent 1/3 in decimal? And if it does equal 1/3, doesn't multiplying by 3 give 0.9999... ? On 2012-11-14 00:32, Dan Asimov wrote:
I'm pretty sure the shift in thinking -- at least among mathematicians -- occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.
I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.
--Dan
On 2012-11-13, at 8:33 PM, Gary Antonick 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
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Mike, I could not tell you what "those same people" think. But maybe by asking the questions you're asking, more of them could be persuaded to see the error of their ways. (To start with, virtually none of them seem to know or care about the rigorous definition of limit.) --Dan On 2012-11-13, at 10:11 PM, Mike Speciner wrote:
Do those same people think that 0.3333... doesn't equal 1/3? And if 0.3333... isn't 1/3, then how do you represent 1/3 in decimal? And if it does equal 1/3, doesn't multiplying by 3 give 0.9999... ?
On 2012-11-14 00:32, Dan Asimov wrote:
I'm pretty sure the shift in thinking -- at least among mathematicians -- occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.
I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.
Even before you get to limits, I think you have to establish whether they can parse the phrase "two different ways of writing the same number". Some cannot. On November 14, 2012 at 2:07 AM Dan Asimov <dasimov@earthlink.net> wrote:> Mike, I could not tell you what "those same people" think.
But maybe by asking the questions you're asking, more of them could be persuaded to see the error of their ways.
(To start with, virtually none of them seem to know or care about the rigorous definition of limit.)
--Dan
On 2012-11-13, at 10:11 PM, Mike Speciner wrote:
Do those same people think that 0.3333... doesn't equal 1/3? And if 0.3333... isn't 1/3, then how do you represent 1/3 in decimal? And if it does equal 1/3, doesn't multiplying by 3 give 0.9999... ?
On 2012-11-14 00:32, Dan Asimov wrote:
I'm pretty sure the shift in thinking -- at least among mathematicians -- occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.
I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.
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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
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,
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I think Cauchy laid to rest any questions about the concept of time being an aspect of the concept of convergence. --Dan On 2012-11-14, at 9:51 PM, Gary Antonick wrote:
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
It seems odd to me to say that 0.999... = 1 is a "definition". I would certainly consider it a theorem. It can be proven in any axiomatization of the real numbers. The 19th century yielded the first clear constructions of the reals. People were summing infinite series well before Cantor. On Thu, Nov 15, 2012 at 1:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I think Cauchy laid to rest any questions about the concept of time being an aspect of the concept of convergence.
--Dan
On 2012-11-14, at 9:51 PM, Gary Antonick wrote:
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
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Thanks, Allan. Am trying to clarify Cantor's contribution. Aristotle and Newton could sum infinite series, but if I've got it right they thought of infinity as a very large number and not something in a separate category. Am wondering how Zeno, Aristotle, Cauchy and Cantor would answer the following two questions. *1. Do these lines meet?* y = 1/2 + 1/4 + 1/8 ... Y = 2 2. Is an odometer displaying 0.999... indicating the same thing as an odometer displaying 1? - - - - I'm thinking the answers would be First question Zeno: No. y doesn't touch Y Aristotle: Yes. y touches Y eventually ("at infinity") but unclear how this happens. Cauchy: Yes, effectively. y touches Y effectively (can get as close as you want) Cantor: No. y will never touch Y. However: y', which is the infinite-x version of y, is identical to Y Second question: Zeno: no. Aristotle: you can make the display very, very long, until the display equals 1 Cauchy: you can make the gap between the odometer and 1 as close to zero as you want, so the odometer display essentially equals 1. Cantor: the display as is will never equal 1. You need a different design eg an odometer that circles around on itself. Once you have this, there is no gap between the display and 1. They indicate the same thing. On Thu, Nov 15, 2012 at 8:47 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It seems odd to me to say that 0.999... = 1 is a "definition". I would certainly consider it a theorem. It can be proven in any axiomatization of the real numbers. The 19th century yielded the first clear constructions of the reals. People were summing infinite series well before Cantor.
On Thu, Nov 15, 2012 at 1:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I think Cauchy laid to rest any questions about the concept of time being an aspect of the concept of convergence.
--Dan
On 2012-11-14, at 9:51 PM, Gary Antonick wrote:
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
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I think I can now be more clear about what I'm after, thanks to a couple direct emails. I'm wondering when "..." was defined to mean "the infinite sum (or the limit of the infinite sequence of partial sums)." Using this definition, "0.999..." and "1" are considered to be representations of the same real number (commonly represented as "1"). There have been different answers: 1824, ~1890, 1904. Am wondering which it is. On Thu, Nov 15, 2012 at 9:08 AM, Gary Antonick <gantonick@post.harvard.edu>wrote:
Thanks, Allan. Am trying to clarify Cantor's contribution. Aristotle and Newton could sum infinite series, but if I've got it right they thought of infinity as a very large number and not something in a separate category.
Am wondering how Zeno, Aristotle, Cauchy and Cantor would answer the following two questions.
*1. Do these lines meet?* y = 1/2 + 1/4 + 1/8 ... Y = 2
2. Is an odometer displaying 0.999... indicating the same thing as an odometer displaying 1?
- - - - I'm thinking the answers would be
First question Zeno: No. y doesn't touch Y Aristotle: Yes. y touches Y eventually ("at infinity") but unclear how this happens. Cauchy: Yes, effectively. y touches Y effectively (can get as close as you want) Cantor: No. y will never touch Y. However: y', which is the infinite-x version of y, is identical to Y
Second question: Zeno: no. Aristotle: you can make the display very, very long, until the display equals 1 Cauchy: you can make the gap between the odometer and 1 as close to zero as you want, so the odometer display essentially equals 1. Cantor: the display as is will never equal 1. You need a different design eg an odometer that circles around on itself. Once you have this, there is no gap between the display and 1. They indicate the same thing.
On Thu, Nov 15, 2012 at 8:47 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It seems odd to me to say that 0.999... = 1 is a "definition". I would certainly consider it a theorem. It can be proven in any axiomatization of the real numbers. The 19th century yielded the first clear constructions of the reals. People were summing infinite series well before Cantor.
On Thu, Nov 15, 2012 at 1:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I think Cauchy laid to rest any questions about the concept of time being an aspect of the concept of convergence.
--Dan
On 2012-11-14, at 9:51 PM, Gary Antonick wrote:
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
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Apparently Euler included a demonstration of 9.999... = 10 in his _Elements of Algebra_, around 1770. (I looked at an online English translation and could not find the proof, whose existence is alleged in the Wikipedia article on "999...".) On Thu, Nov 15, 2012 at 3:57 PM, Gary Antonick <gantonick@post.harvard.edu>wrote:
I think I can now be more clear about what I'm after, thanks to a couple direct emails.
I'm wondering when "..." was defined to mean "the infinite sum (or the limit of the infinite sequence of partial sums)."
Using this definition, "0.999..." and "1" are considered to be representations of the same real number (commonly represented as "1").
There have been different answers: 1824, ~1890, 1904. Am wondering which it is.
On Thu, Nov 15, 2012 at 9:08 AM, Gary Antonick <gantonick@post.harvard.edu>wrote:
Thanks, Allan. Am trying to clarify Cantor's contribution. Aristotle and Newton could sum infinite series, but if I've got it right they thought of infinity as a very large number and not something in a separate category.
Am wondering how Zeno, Aristotle, Cauchy and Cantor would answer the following two questions.
*1. Do these lines meet?* y = 1/2 + 1/4 + 1/8 ... Y = 2
2. Is an odometer displaying 0.999... indicating the same thing as an odometer displaying 1?
- - - - I'm thinking the answers would be
First question Zeno: No. y doesn't touch Y Aristotle: Yes. y touches Y eventually ("at infinity") but unclear how this happens. Cauchy: Yes, effectively. y touches Y effectively (can get as close as you want) Cantor: No. y will never touch Y. However: y', which is the infinite-x version of y, is identical to Y
Second question: Zeno: no. Aristotle: you can make the display very, very long, until the display equals 1 Cauchy: you can make the gap between the odometer and 1 as close to zero as you want, so the odometer display essentially equals 1. Cantor: the display as is will never equal 1. You need a different design eg an odometer that circles around on itself. Once you have this, there is no gap between the display and 1. They indicate the same thing.
On Thu, Nov 15, 2012 at 8:47 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It seems odd to me to say that 0.999... = 1 is a "definition". I would certainly consider it a theorem. It can be proven in any axiomatization of the real numbers. The 19th century yielded the first clear constructions of the reals. People were summing infinite series well before Cantor.
On Thu, Nov 15, 2012 at 1:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I think Cauchy laid to rest any questions about the concept of time being an aspect of the concept of convergence.
--Dan
On 2012-11-14, at 9:51 PM, Gary Antonick wrote:
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
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On Thu, Nov 15, 2012 at 12:08 PM, Gary Antonick <gantonick@post.harvard.edu> wrote:
Thanks, Allan. Am trying to clarify Cantor's contribution. Aristotle and Newton could sum infinite series, but if I've got it right they thought of infinity as a very large number and not something in a separate category.
Am wondering how Zeno, Aristotle, Cauchy and Cantor would answer the following two questions.
*1. Do these lines meet?* y = 1/2 + 1/4 + 1/8 ... Y = 2
I'll assume you meant either "y = 1 + 1/2 + 1/4 + 1/8 ..." or "Y = 1"; otherwise the answer is "of course not". I think Zeno and Aristotle would have answered "I'm unfamiliar with your notation; can you explain it to me?" Zeno and Aristotle would not have been familiar with the = sign, much less the "...". The answer would no doubt depend on details of the explanation you gave of what ... means. If you gave the the modern definition of "...", incorporating the modern epsilon-delta definition of limit, Im sure they would say that y = Y; they may not have had that notion of limit before, but they were bright guys, and would have figured it out if you had explained it to them. Cauchy and Cantor would certainly have agreed that y = Y. I think they would have all been confused by the reference to lines meeting; frankly, so am I. If you mean these as equations of two vertical lines in the x-y plane (if so, why use y in one and Y in the other?) then I think they all would agree (after you explained Cartesian coordinates to Zeno and Aristotle) that these are the same line, so that they of course "meet". If you mean something else, I don't know what it is. It's not so much that before Cauchy people had a different, precise, definition of the sum of an infinite series than we have now, so that you could ask the question "using the old definition, did .9 + .09 + .009 + .0009... sum to 1"? They had the idea that some infinite series had sums, but not a precise definition of the sum of an infinite series, or the idea that such a precise definition was important or even useful, though one way to read Zeno is as saying "If you don't have a better definition of the sum of an infinite series than we have now, it's going to get you in trouble; you're going to end up with paradoxes and contradictions". People worked with infinite series and their sums before they had a rigorous definion. Euler in 1760 had the formula for summing a geometric series, so he believed .9 + .09 + .009 + .0009 ... was = 1. He also believed that 1 + 2 + 4 + 8 + ... = -1, and 1 - 1 + 1 - 1 + 1 ... = 1/2, and 1 - 2 + 3 - 4 + 5... = - 1/4. Though he also said “Nobody doubts that the geometric series 1 + 1/2 + 1/4 + 1/8 +... converges to 2. As more terms are added, the sum approaches 2, and if 100 terms are added, the difference between the sum and 2 is a fraction with 30 digits in its denominator and a 1 in its numerator. The series 1 + 1 + 1 + 1 + 1 + etc. and 1 + 2 + 3 + 4 + 5 + 6 + etc. whose terms do not tend toward zero, will grow to infinity and are divergent.” So he also had some notion of a divergent series. See http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2032%20divergent%2... for more about how Euler viewed divergent series and their sums. I think the notion that people used to think .99999... was different from 1, and then learned that it was the same, is incorrect, so asking when the change happened is the wrong question. While some, including implicitly Zeno, certainly questioned whether the sum of an infinite series had a meaning at all, anyone who thought it was a meaningful operation felt that this series converged to 1. The idea that it converges to some number just slightly less than 1 is not an old notion of old incorrect mathematicians, but a new notion of people misled by notation and a confusion between numbers and numerals leading to an implicit belief that different numerals must always denote different numbers.
2. Is an odometer displaying 0.999... indicating the same thing as an odometer displaying 1?
Well, all of the above's answer would of course be "what's an odometer?" If you asked me, I'd ask you what you meant by a physical object "indicating" a number. If you gave a precise definition, and then asked me about two hypothetical odometers, I'd say they were indicating the same number if you gave any reasonable meaning to an odometer "indicating" a number. If instead of describing hypothetical odometers, you confronted me with two physical objects, one of them stretching out into the distance beyond my field of vision, I would assume that it was nontheless a finite object and answer "no". If you assured me that it was in fact infinite, and all its dials read 9, I wouldn't believe you, but I suppose that if I pretended to believe you, I would answer "yes".
- - - - I'm thinking the answers would be
First question Zeno: No. y doesn't touch Y Aristotle: Yes. y touches Y eventually ("at infinity") but unclear how this happens. Cauchy: Yes, effectively. y touches Y effectively (can get as close as you want) Cantor: No. y will never touch Y. However: y', which is the infinite-x version of y, is identical to Y
Second question: Zeno: no. Aristotle: you can make the display very, very long, until the display equals 1 Cauchy: you can make the gap between the odometer and 1 as close to zero as you want, so the odometer display essentially equals 1. Cantor: the display as is will never equal 1. You need a different design eg an odometer that circles around on itself. Once you have this, there is no gap between the display and 1. They indicate the same thing.
On Thu, Nov 15, 2012 at 8:47 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It seems odd to me to say that 0.999... = 1 is a "definition". I would certainly consider it a theorem. It can be proven in any axiomatization of the real numbers. The 19th century yielded the first clear constructions of the reals. People were summing infinite series well before Cantor.
On Thu, Nov 15, 2012 at 1:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I think Cauchy laid to rest any questions about the concept of time being an aspect of the concept of convergence.
--Dan
On 2012-11-14, at 9:51 PM, Gary Antonick wrote:
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
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-- Andy.Latto@pobox.com
Andy Latto - very helpful. First of all, I did mean the series 1+1/2+1/4... Second - the odometer wasn't a very good analogy. Good point. Third - the bit about the line wasn't well stated. I'll try again below. Anyway, your note is great. Perhaps the big distinction historically is whether or not the sum of an infinite series was a meaningful operation. And I do have a couple questions. Regarding "0.999...", by "anyone who thought it was a meaningful operation felt that this series converged to 1" you're saying that within this group it's always been thought (notation issues aside) that the expressions "0.999..." and "1/2+1/4+1/8..." and "1" all referred to the same number? And are they interchangeable terms? If so, you could say 1 converges to 0.999... Someone might argue that 1 can't converge. But neither can 0.999... the ellipses, if I understand it correctly, mean an infinite number of nines, so the expression already refers to the number 1. Similarly - Euler: "Nobody doubts that the geometric series 1 + 1/2 + 1/4 + 1/8 +... converges to 2." Does Euler mean "1 + 1/2 + 1/4 + 1/8 +..." and "2" are referring to the same number? There is a bit of a difference between "converged" and "converges," and am wondering if the distinction makes a difference. Related to convergence -- let's take the the hyperbola xy=1. Would mathematicians consider this hyperbola to touch the x axis at x=infinity? If not, how is this convergence different from the convergence above? If so, has this always been considered to be the case among mathematicians when defining hyperbolas? On Thu, Nov 15, 2012 at 1:53 PM, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Nov 15, 2012 at 12:08 PM, Gary Antonick <gantonick@post.harvard.edu> wrote:
Thanks, Allan. Am trying to clarify Cantor's contribution. Aristotle and Newton could sum infinite series, but if I've got it right they thought of infinity as a very large number and not something in a separate category.
Am wondering how Zeno, Aristotle, Cauchy and Cantor would answer the following two questions.
*1. Do these lines meet?* y = 1/2 + 1/4 + 1/8 ... Y = 2
I'll assume you meant either "y = 1 + 1/2 + 1/4 + 1/8 ..." or "Y = 1"; otherwise the answer is "of course not".
I think Zeno and Aristotle would have answered "I'm unfamiliar with your notation; can you explain it to me?" Zeno and Aristotle would not have been familiar with the = sign, much less the "...". The answer would no doubt depend on details of the explanation you gave of what ... means. If you gave the the modern definition of "...", incorporating the modern epsilon-delta definition of limit, Im sure they would say that y = Y; they may not have had that notion of limit before, but they were bright guys, and would have figured it out if you had explained it to them. Cauchy and Cantor would certainly have agreed that y = Y.
I think they would have all been confused by the reference to lines meeting; frankly, so am I. If you mean these as equations of two vertical lines in the x-y plane (if so, why use y in one and Y in the other?) then I think they all would agree (after you explained Cartesian coordinates to Zeno and Aristotle) that these are the same line, so that they of course "meet". If you mean something else, I don't know what it is.
It's not so much that before Cauchy people had a different, precise, definition of the sum of an infinite series than we have now, so that you could ask the question "using the old definition, did .9 + .09 + .009 + .0009... sum to 1"? They had the idea that some infinite series had sums, but not a precise definition of the sum of an infinite series, or the idea that such a precise definition was important or even useful, though one way to read Zeno is as saying "If you don't have a better definition of the sum of an infinite series than we have now, it's going to get you in trouble; you're going to end up with paradoxes and contradictions".
People worked with infinite series and their sums before they had a rigorous definion. Euler in 1760 had the formula for summing a geometric series, so he believed .9 + .09 + .009 + .0009 ... was = 1. He also believed that 1 + 2 + 4 + 8 + ... = -1, and 1 - 1 + 1 - 1 + 1 ... = 1/2, and 1 - 2 + 3 - 4 + 5... = - 1/4. Though he also said
“Nobody doubts that the geometric series 1 + 1/2 + 1/4 + 1/8 +... converges to 2. As more terms are added, the sum approaches 2, and if 100 terms are added, the difference between the sum and 2 is a fraction with 30 digits in its denominator and a 1 in its numerator. The series 1 + 1 + 1 + 1 + 1 + etc. and 1 + 2 + 3 + 4 + 5 + 6 + etc. whose terms do not tend toward zero, will grow to infinity and are divergent.”
So he also had some notion of a divergent series. See
http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2032%20divergent%2... for more about how Euler viewed divergent series and their sums.
I think the notion that people used to think .99999... was different from 1, and then learned that it was the same, is incorrect, so asking when the change happened is the wrong question. While some, including implicitly Zeno, certainly questioned whether the sum of an infinite series had a meaning at all, anyone who thought it was a meaningful operation felt that this series converged to 1. The idea that it converges to some number just slightly less than 1 is not an old notion of old incorrect mathematicians, but a new notion of people misled by notation and a confusion between numbers and numerals leading to an implicit belief that different numerals must always denote different numbers.
2. Is an odometer displaying 0.999... indicating the same thing as an odometer displaying 1?
Well, all of the above's answer would of course be "what's an odometer?" If you asked me, I'd ask you what you meant by a physical object "indicating" a number. If you gave a precise definition, and then asked me about two hypothetical odometers, I'd say they were indicating the same number if you gave any reasonable meaning to an odometer "indicating" a number. If instead of describing hypothetical odometers, you confronted me with two physical objects, one of them stretching out into the distance beyond my field of vision, I would assume that it was nontheless a finite object and answer "no". If you assured me that it was in fact infinite, and all its dials read 9, I wouldn't believe you, but I suppose that if I pretended to believe you, I would answer "yes".
- - - - I'm thinking the answers would be
First question Zeno: No. y doesn't touch Y Aristotle: Yes. y touches Y eventually ("at infinity") but unclear how
this
happens. Cauchy: Yes, effectively. y touches Y effectively (can get as close as you want) Cantor: No. y will never touch Y. However: y', which is the infinite-x version of y, is identical to Y
Second question: Zeno: no. Aristotle: you can make the display very, very long, until the display equals 1 Cauchy: you can make the gap between the odometer and 1 as close to zero as you want, so the odometer display essentially equals 1. Cantor: the display as is will never equal 1. You need a different design eg an odometer that circles around on itself. Once you have this, there is no gap between the display and 1. They indicate the same thing.
On Thu, Nov 15, 2012 at 8:47 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It seems odd to me to say that 0.999... = 1 is a "definition". I would certainly consider it a theorem. It can be proven in any axiomatization of the real numbers. The 19th century yielded the first clear constructions of the reals. People were summing infinite series well before Cantor.
On Thu, Nov 15, 2012 at 1:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I think Cauchy laid to rest any questions about the concept of time being an aspect of the concept of convergence.
--Dan
On 2012-11-14, at 9:51 PM, Gary Antonick wrote:
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
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-- Andy.Latto@pobox.com
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Since this was originally an historical question perhaps as an arithmetical post-modernist punk I should hold my piece, but perhaps those that hold that ...0.999... = 1 will also equably agree that ...999.0... = -1? After all, are we not told (presented with precisely the same corroboration as given for 0.999...=1), that the geometrical series with term ratio r=10: 9 10^0 + 9 10^1 + 9 10^2 + ... must be equal to 9/(1-r) = 9/(1-10) = 9/-9 = -1? And also therefore, surely, the apparently maximally heavy-weight ...999.999... = -1 + 1 = 0? Of course, as those correspondents here who've surrounded 0.999... with "'s implicitly intimate, the real issue is around what '=' means ("what your definition of 'is' is" as someone-or-other put it). As every type-whipped programmer would note, on the left-hand of the = sign we are placing strings, while on the right-hand we are placing real numbers named by those strings, that is "=" doesn't mean "is identical with" but rather should be read as "is an alias name for" the same (Platonic) entity. (More precisely, '"0.999..."' is our finite name for an infinite string, whilst '1' is our discrete name for a real number (neither of which are ultimately finitely "knowable"). So I guess we are merely declaring that the referents of these names is the same (Platonic) entity--but anyway, we shave barbers...). What the string 0.999... "means" depends on what interpreter we feed these (infinite) objects into. Agreeing that 0.999... = 1 is just confessing to an agreeably shared delusion. Fine, but it's only "true" with respect to certain models. There are also perfectly good arithmetics where this postulate is negated.
Of course ...9999.0 = -1. That's just ten's complement arithmetic, as any old programmer would know. --ms On 2012-11-16 00:52, Marc LeBrun wrote:
Since this was originally an historical question perhaps as an arithmetical post-modernist punk I should hold my piece, but perhaps those that hold that
...0.999... = 1
will also equably agree that
...999.0... = -1?
After all, are we not told (presented with precisely the same corroboration as given for 0.999...=1), that the geometrical series with term ratio r=10:
9 10^0 + 9 10^1 + 9 10^2 + ...
must be equal to 9/(1-r) = 9/(1-10) = 9/-9 = -1?
And also therefore, surely, the apparently maximally heavy-weight
...999.999... = -1 + 1 = 0?
Of course, as those correspondents here who've surrounded 0.999... with "'s implicitly intimate, the real issue is around what '=' means ("what your definition of 'is' is" as someone-or-other put it).
As every type-whipped programmer would note, on the left-hand of the = sign we are placing strings, while on the right-hand we are placing real numbers named by those strings, that is "=" doesn't mean "is identical with" but rather should be read as "is an alias name for" the same (Platonic) entity.
(More precisely, '"0.999..."' is our finite name for an infinite string, whilst '1' is our discrete name for a real number (neither of which are ultimately finitely "knowable"). So I guess we are merely declaring that the referents of these names is the same (Platonic) entity--but anyway, we shave barbers...).
What the string 0.999... "means" depends on what interpreter we feed these (infinite) objects into.
Agreeing that 0.999... = 1 is just confessing to an agreeably shared delusion. Fine, but it's only "true" with respect to certain models.
There are also perfectly good arithmetics where this postulate is negated.
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On Fri, Nov 16, 2012 at 12:52 AM, Marc LeBrun <mlb@well.com> wrote:
Since this was originally an historical question perhaps as an arithmetical post-modernist punk I should hold my piece, but perhaps those that hold that
...0.999... = 1
will also equably agree that
...999.0... = -1?
After all, are we not told (presented with precisely the same corroboration as given for 0.999...=1), that the geometrical series with term ratio r=10:
9 10^0 + 9 10^1 + 9 10^2 + ...
must be equal to 9/(1-r) = 9/(1-10) = 9/-9 = -1?
This isn't post-modern, it's pre-classical. This is exactly the way Euler reasoned, and he came up with the same sort of conclusions. But the reasons I, and a modern mathematician, say that .99999.... = 1 not just because of the proof of the formula for the sum of a geometric series (which, if you examine it closely, shows only that *if* the series has a limit, *then* the limit is 1), but because of the epsilon-definition of a limit, and a proof I can provide, using that definition, that this series indeed does sum to 1. If you try to provide a similar epsilon-delta proof that the series ...9999.0 sums to -1, you will fail, and I can provide an epsilon-delta proof that this series does *not* have a limit.
As every type-whipped programmer would note, on the left-hand of the = sign we are placing strings, while on the right-hand we are placing real numbers named by those strings,
Nonsense. 1 is a string, and so is .999999... And for that matter, so is 1 = .9999999..., a string which represents a proposition, that says that two of its substrings represent the same number.
(More precisely, '"0.999..."' is our finite name for an infinite string, whilst '1' is our discrete name for a real number (neither of which are ultimately finitely "knowable"). So I guess we are merely declaring that the referents of these names is the same (Platonic) entity--but anyway, we shave barbers...).
There's no need to invoke Platonism here; in a formalist view, I can provide a nice, finite, proof from finitely many axioms that the string 1 = .99999.... is a theorem.
What the string 0.999... "means" depends on what interpreter we feed these (infinite) objects into.
Agreeing that 0.999... = 1 is just confessing to an agreeably shared delusion. Fine, but it's only "true" with respect to certain models.
Exactly; just like 2 + 3 = 5. An agreeably shared delusion, not shared by those who make the equally good choice to use the squiggle "3" to represent the integer between 7 and 9. I also think the reference to infinite strings is a complete blind alley and confusion here. I'm using .9999..... to mean exactly the same thing as the finite string while on the right-hand we are placing real numbers named by those strings, sum_{i=1}^{i=infinity} 9 * (10)^(-i). The latter is just an imprecise shorthand for the latter. If someone asked me about what number was denoted by .12345....., I'd have to say "I don't know; you might mean sum_{i=1}^{i=infinity} rem(i, 10) * (10)^(-i). (that is, .1234567890123456789...) or you might mean sum_{i=1}^{i=infinity} i * (10)^(-i). (that is, .12345679012345..., where you put the next integer in each position and carry), or even .12345678910111213141516..., which I could write in summation notation, but it would be a bit more complex. If someone said about .9999..... "I don't understand; does this continue for 25 more 9's, or 100, or what? You haven't told me want the 734th digit is", I'd use the summation notation, which is better because it unambiguously gives the value for every decimal place, rather than playing the mug's game of "guess the pattern". Or to put it another way, I know of no sensible system that attaches different meanings to .9999.... and sum_{i=1}^{i=infinity} 9 * (10)^(-i), and the latter is a finite string, so any paradox or difficulty isn't related to the finiteness or infiniteness of strings.
There are also perfectly good arithmetics where this postulate is negated.
Depends on what you mean by "perfectly good". The arithmetic where "9" is used to mean the the number between 4 and 6 is of course a perfectly good arithmetic where te postulate is negated. But if you look at the proof of the sum of a geometric series (not the epsilon-delta part, just the proof that "if there is a sum, the sum is 1", then any system in which this statement is not a theorem (I know of no system where it is a 'postulate') must give up one of the axioms used in the proof, which makes it in my mind less than "perfectly good". By contrast, the same proof shows that ......999999.0 = -1, meaning that if the series has a sum, the sum is -1, but I know of no perfectly good arithmetic where this is in fact true. But the same proof also shows that in base 7, ....66666666.0 = -1, and there *is* a perfectly good arithmetic (unless your definition of "perfectly good" includes "Archimedean" where this is a theorem, namely the 7-adics.
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My question appears to be partly about notation. Am wondering what the following symbols mean. "=" Does x=y mean 1. "x" and "y" are different symbols or expressions for the same thing 2. "x" and "y" are interchangeable in any mathematical expression "converge" Does converge mean gets close but never touches? "limit" Could it be said that something that converges always converges toward what could be called a limit? Is the x axis, therefore, a limit for y=1/x? "..." Does ... mean, for a series that converges, infinite steps in that series, and, by definition, the limit of that series? On Fri, Nov 16, 2012 at 6:32 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Fri, Nov 16, 2012 at 12:52 AM, Marc LeBrun <mlb@well.com> wrote:
Since this was originally an historical question perhaps as an arithmetical post-modernist punk I should hold my piece, but perhaps those that hold that
...0.999... = 1
will also equably agree that
...999.0... = -1?
After all, are we not told (presented with precisely the same corroboration as given for 0.999...=1), that the geometrical series with term ratio r=10:
9 10^0 + 9 10^1 + 9 10^2 + ...
must be equal to 9/(1-r) = 9/(1-10) = 9/-9 = -1?
This isn't post-modern, it's pre-classical. This is exactly the way Euler reasoned, and he came up with the same sort of conclusions. But the reasons I, and a modern mathematician, say that .99999.... = 1 not just because of the proof of the formula for the sum of a geometric series (which, if you examine it closely, shows only that *if* the series has a limit, *then* the limit is 1), but because of the epsilon-definition of a limit, and a proof I can provide, using that definition, that this series indeed does sum to 1. If you try to provide a similar epsilon-delta proof that the series ...9999.0 sums to -1, you will fail, and I can provide an epsilon-delta proof that this series does *not* have a limit.
As every type-whipped programmer would note, on the left-hand of the = sign we are placing strings, while on the right-hand we are placing real numbers named by those strings,
Nonsense. 1 is a string, and so is .999999... And for that matter, so is 1 = .9999999..., a string which represents a proposition, that says that two of its substrings represent the same number.
(More precisely, '"0.999..."' is our finite name for an infinite string, whilst '1' is our discrete name for a real number (neither of which are ultimately finitely "knowable"). So I guess we are merely declaring that the referents of these names is the same (Platonic) entity--but anyway, we shave barbers...).
There's no need to invoke Platonism here; in a formalist view, I can provide a nice, finite, proof from finitely many axioms that the string 1 = .99999.... is a theorem.
What the string 0.999... "means" depends on what interpreter we feed
these
(infinite) objects into.
Agreeing that 0.999... = 1 is just confessing to an agreeably shared delusion. Fine, but it's only "true" with respect to certain models.
Exactly; just like 2 + 3 = 5. An agreeably shared delusion, not shared by those who make the equally good choice to use the squiggle "3" to represent the integer between 7 and 9.
I also think the reference to infinite strings is a complete blind alley and confusion here. I'm using .9999..... to mean exactly the same thing as the finite string while on the right-hand we are placing real numbers named by those strings, sum_{i=1}^{i=infinity} 9 * (10)^(-i). The latter is just an imprecise shorthand for the latter. If someone asked me about what number was denoted by .12345....., I'd have to say "I don't know; you might mean sum_{i=1}^{i=infinity} rem(i, 10) * (10)^(-i). (that is, .1234567890123456789...)
or you might mean sum_{i=1}^{i=infinity} i * (10)^(-i). (that is, .12345679012345..., where you put the next integer in each position and carry), or even
.12345678910111213141516..., which I could write in summation notation, but it would be a bit more complex. If someone said about .9999..... "I don't understand; does this continue for 25 more 9's, or 100, or what? You haven't told me want the 734th digit is", I'd use the summation notation, which is better because it unambiguously gives the value for every decimal place, rather than playing the mug's game of "guess the pattern".
Or to put it another way, I know of no sensible system that attaches different meanings to .9999.... and sum_{i=1}^{i=infinity} 9 * (10)^(-i), and the latter is a finite string, so any paradox or difficulty isn't related to the finiteness or infiniteness of strings.
There are also perfectly good arithmetics where this postulate is
negated.
Depends on what you mean by "perfectly good". The arithmetic where "9" is used to mean the the number between 4 and 6 is of course a perfectly good arithmetic where te postulate is negated. But if you look at the proof of the sum of a geometric series (not the epsilon-delta part, just the proof that "if there is a sum, the sum is 1", then any system in which this statement is not a theorem (I know of no system where it is a 'postulate') must give up one of the axioms used in the proof, which makes it in my mind less than "perfectly good".
By contrast, the same proof shows that ......999999.0 = -1, meaning that if the series has a sum, the sum is -1, but I know of no perfectly good arithmetic where this is in fact true. But the same proof also shows that in base 7,
....66666666.0 = -1, and there *is* a perfectly good arithmetic (unless your definition of "perfectly good" includes "Archimedean" where this is a theorem, namely the 7-adics.
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= 1. The symbol = for equals means that the left side and the right side represent the same element of the set under consideration. In this discussion, that set has usually been the real numbers -- and a few times, the surreal numbers. (Though technically speaking, the surreals are not a set but a class.) 2. If the mathematical expression is all about real numbers, then yes: the two sides of = are interchangeable (assuming the equation is a true one!). (This doesn't really pertain to this discussion, but of course if the mathematical expression is about the set of mathematical expressions, then the two sides of = will be interchangeable only if they are the same expression.) ----- converge To say that a sequence x: Z+ -> R converges to a real number c means that for any real number eps > 0, there is an integer N (depending on eps) such that, for all k >= N, it holds that |x_k - c| < eps. I.e., for any desired closeness eps > 0 to c, then all the terms of the sequence after a certain point are that close or closer to the number c. This does not require the sequence to always get closer to c, as long as it eventually does. Nor does it require the terms of the sequence to be unequal to c. E.g., the sequence 0, 1, 2, 0, ½, 1, 0, ¼, ½, 0, ⅛, ¼, . . . (where each successive three terms are exactly half of the last one) exemplifies these things, since by the above definition it converges to 0. ... The ellipsis does not have a formal definition, but instead is an informal way of saying "continue in the same pattern", with the assumption that readers will know which pattern is referred to. But after a few terms of a sequence -- when the pattern is clear -- the ellipsis just (generally) signifies only the rest of the terms of that sequence. Not its limit (if any). --Dan On 2012-11-16, at 10:17 AM, Gary Antonick wrote:
My question appears to be partly about notation. Am wondering what the following symbols mean.
"=" Does x=y mean 1. "x" and "y" are different symbols or expressions for the same thing 2. "x" and "y" are interchangeable in any mathematical expression
"converge" Does converge mean gets close but never touches?
"limit" Could it be said that something that converges always converges toward what could be called a limit? Is the x axis, therefore, a limit for y=1/x?
"..." Does ... mean, for a series that converges, infinite steps in that series, and, by definition, the limit of that series?
Are the surreal numbers at all relevant to this discussion? That is, the surreal numbers include numbers that are, for all finite n, greater than sum_{i=0}^{i=n}(9 * (10)^(-i). They are "infinitesimally less" than 1. But that doesn't mean that .99999.... is infinitesimally less than 1! I don't know if there is a standard or useful definition of the sum of an infinite series in the surreal numbers. But if there is, then every step in the chain of equalities (letting S denote the sum of the series in question, if it exists) 9 S = 10 S - S = 9.9999999 - .99999999 = 9 is still valid in the surreal numbers, as is the deduction that if 9S= 9, S = 1. The bottom line is that this proof that if .999999... has a meaning at all, it means 1, works not only in the standard real numbers, but in many other systems. If you want a system where .999999.... is a well-defined number less than 1, you need a system where the axioms are weakened in some significant way so that one of the equalities above isn't true. They are all true in the surreal numbers, so if .999... has a value at all, it's still 1, not one of the numbers infinitesimally less than 1. Andy On Fri, Nov 16, 2012 at 4:58 PM, Dan Asimov <dasimov@earthlink.net> wrote:
=
1. The symbol = for equals means that the left side and the right side represent the same element of the set under consideration. In this discussion, that set has usually been the real numbers -- and a few times, the surreal numbers. (Though technically speaking, the surreals are not a set but a class.)
2. If the mathematical expression is all about real numbers, then yes: the two sides of = are interchangeable (assuming the equation is a true one!).
(This doesn't really pertain to this discussion, but of course if the mathematical expression is about the set of mathematical expressions, then the two sides of = will be interchangeable only if they are the same expression.) -----
converge
To say that a sequence x: Z+ -> R converges to a real number c means that for any real number eps > 0, there is an integer N (depending on eps) such that, for all k >= N, it holds that |x_k - c| < eps.
I.e., for any desired closeness eps > 0 to c, then all the terms of the sequence after a certain point are that close or closer to the number c.
This does not require the sequence to always get closer to c, as long as it eventually does. Nor does it require the terms of the sequence to be unequal to c.
E.g., the sequence 0, 1, 2, 0, ½, 1, 0, ¼, ½, 0, ⅛, ¼, . . . (where each successive three terms are exactly half of the last one) exemplifies these things, since by the above definition it converges to 0.
...
The ellipsis does not have a formal definition, but instead is an informal way of saying "continue in the same pattern", with the assumption that readers will know which pattern is referred to.
But after a few terms of a sequence -- when the pattern is clear -- the ellipsis just (generally) signifies only the rest of the terms of that sequence. Not its limit (if any).
--Dan
On 2012-11-16, at 10:17 AM, Gary Antonick wrote:
My question appears to be partly about notation. Am wondering what the following symbols mean.
"=" Does x=y mean 1. "x" and "y" are different symbols or expressions for the same thing 2. "x" and "y" are interchangeable in any mathematical expression
"converge" Does converge mean gets close but never touches?
"limit" Could it be said that something that converges always converges toward what could be called a limit? Is the x axis, therefore, a limit for y=1/x?
"..." Does ... mean, for a series that converges, infinite steps in that series, and, by definition, the limit of that series?
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I'm not very familiar with the n-adic numbers, but I think that ...999.0... = -1 is true in the 10-adic numbers, where the reasoning Marc uses can be justified rigorously. (Alternatively, just add 1 to ...999.0... to get 0.) --Dan On 2012-11-16, at 6:32 AM, Andy Latto wrote:
On Fri, Nov 16, 2012 at 12:52 AM, Marc LeBrun <mlb@well.com> wrote:
Since this was originally an historical question perhaps as an arithmetical post-modernist punk I should hold my piece, but perhaps those that hold that
...0.999... = 1
will also equably agree that
...999.0... = -1?
After all, are we not told (presented with precisely the same corroboration as given for 0.999...=1), that the geometrical series with term ratio r=10:
9 10^0 + 9 10^1 + 9 10^2 + ...
must be equal to 9/(1-r) = 9/(1-10) = 9/-9 = -1?
This isn't post-modern, it's pre-classical. This is exactly the way Euler reasoned, and he came up with the same sort of conclusions. But the reasons I, and a modern mathematician, say that .99999.... = 1 not just because of the proof of the formula for the sum of a geometric series (which, if you examine it closely, shows only that *if* the series has a limit, *then* the limit is 1), but because of the epsilon-definition of a limit, and a proof I can provide, using that definition, that this series indeed does sum to 1. If you try to provide a similar epsilon-delta proof that the series ...9999.0 sums to -1, you will fail, and I can provide an epsilon-delta proof that this series does *not* have a limit.
People often don't like to talk about n-adic numbers when n isn't prime, because the resulting structure fails to be a field. I think that's the problem, anyway -- I don't remember too clearly. On Fri, Nov 16, 2012 at 6:48 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm not very familiar with the n-adic numbers, but I think that
...999.0... = -1
is true in the 10-adic numbers, where the reasoning Marc uses can be justified rigorously.
(Alternatively, just add 1 to ...999.0... to get 0.)
--Dan
On 2012-11-16, at 6:32 AM, Andy Latto wrote:
On Fri, Nov 16, 2012 at 12:52 AM, Marc LeBrun <mlb@well.com> wrote:
Since this was originally an historical question perhaps as an arithmetical post-modernist punk I should hold my piece, but perhaps those that hold that
...0.999... = 1
will also equably agree that
...999.0... = -1?
After all, are we not told (presented with precisely the same corroboration as given for 0.999...=1), that the geometrical series with term ratio r=10:
9 10^0 + 9 10^1 + 9 10^2 + ...
must be equal to 9/(1-r) = 9/(1-10) = 9/-9 = -1?
This isn't post-modern, it's pre-classical. This is exactly the way Euler reasoned, and he came up with the same sort of conclusions. But the reasons I, and a modern mathematician, say that .99999.... = 1 not just because of the proof of the formula for the sum of a geometric series (which, if you examine it closely, shows only that *if* the series has a limit, *then* the limit is 1), but because of the epsilon-definition of a limit, and a proof I can provide, using that definition, that this series indeed does sum to 1. If you try to provide a similar epsilon-delta proof that the series ...9999.0 sums to -1, you will fail, and I can provide an epsilon-delta proof that this series does *not* have a limit.
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Since I am wandering away from the topic (specifically, when did 0.9999... = 1 become accepted), I thought I would change the subject line slightly. The reason I brought up the surreals was because I knew that under the standard interpretation of 0.9999... as an abbreviation for SUM(n = 1..inf; 9 * 10^-n), and given the standard model of real numbers as an ordered field satisfying the least upper bound property (continuous), then 0.9999... = 1 is provable. I don't know when this was first proved or when it became a well-known mathematical fact. I asked about surreals because I do not believe they satisfy the least upper bound property or continuity, and I wondered whether, under reasonable interpretations of these numerals as surreal number, it might not be possible to shoehorn infinitesimals between them, and thus provide an example of a reasonable interpretations of 0.9999... and 1 that represent different values. It occurred to me that the standard proof of 0.9999... = 1 depends on the continuity of the reals numbers, which I believe the surreals lack. I did also want to make an observation. Consider a double-ended sequence a of real numbers (a: Z -> R) with formal "sum" z sum(a) = sum({a_i}) = ... + a_-3 + a_-2 + a_-1 + a_0 + a_1 + a_2 + a_3 + ... = z. These sequences form a linear space with respect to termwise sum and scalar multiplication. We would want obviously the mapping sum: S -> R to be a linear mapping: If m in R and b_i = m a_i for all i in Z, then sum(b) = m sum(a). If c_i = a_i + b_i for all i in Z, then sum(c) = sum(a) + sum(b). We would want sums to be preserved by shifting the sequence left or right, e.g: If k in Z and b_i = a_{i+k} for all i in Z, then sum(b) = sum(a). Now consider a geometric double-ended sequence: {a_i} = {r * s^i} The sum is sum(a) = sum({r * s^i}) = sum({r * s^(i+1)}) (shifting elements) = sum({r * s * s^i}) = s * sum({r * s^i}) (sum is linear) = s * sum(a) From which we conclude that for geometric sequence a with ratio s, sum(a) = 0 if s is not 1. Thus we would have ... 900 + 90 + 9 + .9 + .09 + .009 + ... = 0 Any right suffix of this particular sum is convergent when interpreted as a standard infinite sum, i.e: .9 + .09 + .009 + ... = 1. So we might allow ... 0 + 0 + 0 + .9 + .09 + .009 + ... = 1 whence subtraction would yield ... 900 + 90 + 9 + 0 + 0 + 0 + ... = -1. Thus we might assign the value ... 900 + 90 + 9 = -1 to what would be a divergent standard infinite sum. I don't know if this line of inquiry will hold up if pursued to the its logical limit. But if it did, it might be the germ of a theory for evaluating some divergent sequences.
On Fri, Nov 16, 2012 at 11:57 PM, David Wilson <davidwwilson@comcast.net> wrote:
It occurred to me that the standard proof of 0.9999... = 1 depends on the continuity of the reals numbers, which I believe the surreals lack.
I don't know what you mean by "the continuity of the reals"; do you mean completeness? I know what it means for a function to be continuous, but not a set. But I don't understand why you think the proof that .9999... = 1 requires continuity. It seems to me that if we write this as a two-way-infinite series, ... + 0 + 0 + 0 + .9 + .09 + .009 + ..., with a_1 = .9 Then all we need to assume about such sums to prove that if it has a sum, that sum must be 1 is exactly the three properties you describe below:
If m in R and b_i = m a_i for all i in Z, then sum(b) = m sum(a). If c_i = a_i + b_i for all i in Z, then sum(c) = sum(a) + sum(b).
We would want sums to be preserved by shifting the sequence left or right, e.g:
If k in Z and b_i = a_{i+k} for all i in Z, then sum(b) = sum(a).
If the sum is S, the first property tells us that the sum of ... + 0 + 0 + 0 + 9 + .9 + .09 +... with a_1 = 9, is 10 * S The third property says that the sum of this is still 10 * S when we shift it over by 1, so that a_0 = 9 and a_1 = .9. Now the second property (subtracting rather than adding, but we can use the first property to multiply the first sequence by -1) says that the sum of the sequence ...0 + 0 + 0 + 9 + 0 + 0 + 0 + ... So any notion of summability that assigns the sum 9 to this series, and satisfies your three properties, must assign the sum 0 to the sequence ... 0 + 0 + 0 + .9 + .09 + .009 + ....., with no assumption of "continuity" needed. Andy
On Saturday 17 November 2012 05:10:23 Andy Latto wrote:
But I don't understand why you think the proof that .9999... = 1 requires continuity. [...] Then all we need to assume about such sums to prove that if it has a sum, that sum must be 1 is exactly the three properties you describe below:
If m in R and b_i = m a_i for all i in Z, then sum(b) = m sum(a). If c_i = a_i + b_i for all i in Z, then sum(c) = sum(a) + sum(b).
We would want sums to be preserved by shifting the sequence left or right,
e.g: If k in Z and b_i = a_{i+k} for all i in Z, then sum(b) = sum(a).
The second of those properties doesn't hold even in R without the proviso that all three sums converge; otherwise you'd have 1-1+1-1+1-1... being zero and similarly for -1+1-1+1-1+1..., and with those in hand plus the principle that a0 + sum{1..oo}ai = sum{0..oo}ai you can prove 0=1. So perhaps the right question is: is there actually any such thing as the value of .9 + .09 + .009 + ... in the surreal numbers? Well, what's the right way to understand infinite sums generally, or infinite decimals in particular, in the surreal numbers? In R, you can say e.g. that an infinite sum of nonnegative terms is the l.u.b. of the finite partial sums, but you can't do that in the surreals because you don't have lubs. You could handle infinite decimals specially, as per Scott's suggestion that .abcd... means {.a, .ab, .abc, .abcd, ... | .A, .aB, .abC, .abcD, ... } where I hope my notation is obvious, and indeed that will get you plausible answers for infinite decimals, but all that's doing is defining your notation so that infinite decimals denote (something like) the same real numbers as they usually do; it doesn't seem to shine much light on infinite series of surreal numbers in general. I think that if you reinterpret .9 + .09 + .009 + ... as a sum over *all the ordinals* then we can rightly say that it converges to 1. But what if it's just a sum over the finite integers? For any infinitesimal h, all the finite partial sums are < 1-h; so it certainly shouldn't converge to 1; it can't converge to any ordinary real number < 1 because some of the partial sums are bigger; if it converges to 1-h for some infinitesimal h, why not 1-2h or 1-h/2? Well, if h is 1/omega then maybe 1-h is the *simplest* thing satisfying some convenient property. As Scott says, that's what you get from his interpretation of infinite decimals in this case. (On the other hand, one can tweak Scott's definition a little -- e.g., add 2 rather than 1 to each final digit -- so that all infinite decimals, even terminating ones, "converge" to their conventional values, and that might be a better convention.) I'm not sure how you'd make that work for infinite series generally, but I bet it would end up violating the first of those properties: you'd get cases where two series converge to a-1/omega and b-1/omega and their termwise sum converges to a+b-1/omega instead of a+b-2/omega. -- g
On Sat, Nov 17, 2012 at 6:25 AM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
I think that if you reinterpret .9 + .09 + .009 + ... as a sum over *all the ordinals* then we can rightly say that it converges to 1.
I'm very skeptical that one could get a consistent definition for sums over classes that are that large. Even the first step of defining raising 10 to ordinal powers seems problematic.
But what if it's just a sum over the finite integers? For any infinitesimal h, all the finite partial sums are < 1-h; so it certainly shouldn't converge to 1; it can't converge to any ordinary real number < 1 because some of the partial sums are bigger; if it converges to 1-h for some infinitesimal h, why not 1-2h or 1-h/2?
And again, you have to deal with the geometric series proof. If .99999 is 1-h, wouldn't 9.99999 be both 10 - 10h (because you multiplied each term by 10) and 10 - h (because you added 9 to 1-h), so h = 0.
Well, if h is 1/omega then maybe 1-h is the *simplest* thing satisfying some convenient property.
Then the same simplicity would presumably give 9.99999.... = 10 - 1/omega, so you'd have to give up the "multiplication by a scalar" property in exchange for some as-yet-unspecified simplicity property. That could work. Andy
On Saturday 17 November 2012 15:50:47 Andy Latto wrote: [me, about surreal numbers:]
I think that if you reinterpret .9 + .09 + .009 + ... as a sum over *all the ordinals* then we can rightly say that it converges to 1.
I'm very skeptical that one could get a consistent definition for sums over classes that are that large. Even the first step of defining raising 10 to ordinal powers seems problematic.
Maybe you're right -- though there's a note on p38 of my edition of ONAG saying that Simon Norton invented a way of doing integration that lets you define logarithms and hence arbitrary (?) powers x^y. But it doesn't give any details).
But what if it's just a sum over the finite integers? For any infinitesimal h, all the finite partial sums are < 1-h; so it certainly shouldn't converge to 1; it can't converge to any ordinary real number < 1 because some of the partial sums are bigger; if it converges to 1-h for some infinitesimal h, why not 1-2h or 1-h/2?
And again, you have to deal with the geometric series proof. If .99999 is 1-h, wouldn't 9.99999 be both 10 - 10h (because you multiplied each term by 10) and 10 - h (because you added 9 to 1-h), so h = 0.
I repeat: it can't converge to 1 if "converge" means anything like what it means when dealing with ordinary real numbers, because there are lots of upper bounds for the series that are strictly less than 1.
Well, if h is 1/omega then maybe 1-h is the *simplest* thing satisfying some convenient property.
Then the same simplicity would presumably give 9.99999.... = 10 - 1/omega, so you'd have to give up the "multiplication by a scalar" property in exchange for some as-yet-unspecified simplicity property. That could work.
Right. (I thought I'd kinda said that, but on rereading what I wrote I see that I failed to. Sorry.) * On looking a bit further in ONAG I find the following proposal: write numbers as sums of the form sum over all Numbers y of r(y) omega^y where r(y) vanishes except for y in some descending sequence indexed by an ordinal. Then say that a series converges if (1) there's some single ordinal-indexed descending sequence in which all the nonzero exponents in the series lie, and (2) for each exponent, the sums of the corresponding r(y) converge. In the special case of series of real numbers, this is exactly, and rather boringly, the same as saying that the series converges in the usual sense in R. (Because then all your coefficients are zero except for those of omega^0.) This all seems rather unsatisfactory to me, but I haven't anything more convincing to offer. -- g
On 11/17/2012 6:25 AM, Gareth McCaughan wrote:
On Saturday 17 November 2012 05:10:23 Andy Latto wrote:
But I don't understand why you think the proof that .9999... = 1 requires continuity. [...] Then all we need to assume about such sums to prove that if it has a sum, that sum must be 1 is exactly the three properties you describe below:
If m in R and b_i = m a_i for all i in Z, then sum(b) = m sum(a). If c_i = a_i + b_i for all i in Z, then sum(c) = sum(a) + sum(b).
We would want sums to be preserved by shifting the sequence left or right,
e.g: If k in Z and b_i = a_{i+k} for all i in Z, then sum(b) = sum(a).
The second of those properties doesn't hold even in R without the proviso that all three sums converge; otherwise you'd have 1-1+1-1+1-1... being zero and similarly for -1+1-1+1-1+1..., and with those in hand plus the principle that a0 + sum{1..oo}ai = sum{0..oo}ai you can prove 0=1. So perhaps the right question is: is there actually any such thing as the value of .9 + .09 + .009 + ... in the surreal numbers? The above snippet was not in regard to surreal numbers, it was in regard to possibly assigning a consistent real value to double-ended infinite sums. I don't understand surreals well enough to remark intelligently on them, except to note that in that setting, it is perhaps possible or even reasonable to define 0.9999... in such a way that we could shoehorn an infinitesimal between 0.9999... and 1, thus providing a reasonable setting in which we might conclude 0.9999... =/= 1.
With regard to the double-ended sums, I was hoping to treat them as a formalism, and avoid issues of convergence (which end do you converge toward?). Thus we would assign a reasonable value to some class of basic expressions (e.g, ... + 0 + 0 + 0 + 9 + 0 + 0 + 0 + ... = 9), and apply some formal arithmetic rules (value preserved by shifting, term-by-term sum, term-by-term scalar multiplication) and see where that leads us (I expect to a contradiction). But supposing a consistent theory of formal double-ended sums can be developed this way, by starting with base class sums (the values we decree for certain simple double-ended sums) and geometric double-ended sums (which always sum to 0 according to the arithmetic formalisms), we might construct consistent sums for a more general class of double-ended sums. Then by embedding standard right-infinite sums by left zero padding, we might arrive at a way to consistently evaluate left-infinite sums and perhaps even some divergent sums (e.g, 1 + 2 + 4 + 8 + ... = -1), which certainly cannot be done on the basis of convergence.
Well, what's the right way to understand infinite sums generally, or infinite decimals in particular, in the surreal numbers? In R, you can say e.g. that an infinite sum of nonnegative terms is the l.u.b. of the finite partial sums, but you can't do that in the surreals because you don't have lubs. You could handle infinite decimals specially, as per Scott's suggestion that .abcd... means {.a, .ab, .abc, .abcd, ... | .A, .aB, .abC, .abcD, ... } where I hope my notation is obvious, and indeed that will get you plausible answers for infinite decimals, but all that's doing is defining your notation so that infinite decimals denote (something like) the same real numbers as they usually do; it doesn't seem to shine much light on infinite series of surreal numbers in general.
I think that if you reinterpret .9 + .09 + .009 + ... as a sum over *all the ordinals* then we can rightly say that it converges to 1. But what if it's just a sum over the finite integers? For any infinitesimal h, all the finite partial sums are < 1-h; so it certainly shouldn't converge to 1; it can't converge to any ordinary real number < 1 because some of the partial sums are bigger; if it converges to 1-h for some infinitesimal h, why not 1-2h or 1-h/2?
Well, if h is 1/omega then maybe 1-h is the *simplest* thing satisfying some convenient property. As Scott says, that's what you get from his interpretation of infinite decimals in this case. (On the other hand, one can tweak Scott's definition a little -- e.g., add 2 rather than 1 to each final digit -- so that all infinite decimals, even terminating ones, "converge" to their conventional values, and that might be a better convention.)
I'm not sure how you'd make that work for infinite series generally, but I bet it would end up violating the first of those properties: you'd get cases where two series converge to a-1/omega and b-1/omega and their termwise sum converges to a+b-1/omega instead of a+b-2/omega.
On 11/17/2012 12:10 AM, Andy Latto wrote:
On Fri, Nov 16, 2012 at 11:57 PM, David Wilson <davidwwilson@comcast.net> wrote:
It occurred to me that the standard proof of 0.9999... = 1 depends on the continuity of the reals numbers, which I believe the surreals lack. I don't know what you mean by "the continuity of the reals"; do you mean completeness? I know what it means for a function to be continuous, but not a set. My terminology is not always correct. It is difficult to look up the terminology by mathemetical concept, even online, and I haven't done any analysis to speak of 30 years.
But yes, I think I meant completeness in the sense that limits of sequences of set elements are in the set.
But I don't understand why you think the proof that .9999... = 1 requires continuity. It seems to me that if we write this as a two-way-infinite series,
... + 0 + 0 + 0 + .9 + .09 + .009 + ..., with a_1 = .9
Then all we need to assume about such sums to prove that if it has a sum, that sum must be 1 is exactly the three properties you describe below:
If m in R and b_i = m a_i for all i in Z, then sum(b) = m sum(a). If c_i = a_i + b_i for all i in Z, then sum(c) = sum(a) + sum(b).
We would want sums to be preserved by shifting the sequence left or right, e.g:
If k in Z and b_i = a_{i+k} for all i in Z, then sum(b) = sum(a).
If the sum is S, the first property tells us that the sum of ... + 0 + 0 + 0 + 9 + .9 + .09 +... with a_1 = 9, is 10 * S
The third property says that the sum of this is still 10 * S when we shift it over by 1, so that a_0 = 9 and a_1 = .9.
Now the second property (subtracting rather than adding, but we can use the first property to multiply the first sequence by -1) says that the sum of the sequence
...0 + 0 + 0 + 9 + 0 + 0 + 0 + ...
So any notion of summability that assigns the sum 9 to this series, and satisfies your three properties, must assign the sum 0 to the sum 1 you mean sequence ... 0 + 0 + 0 + .9 + .09 + .009 + ....., with no assumption of "continuity" needed.
Andy What I said was: the *standard* proof that 0.9999... = 1 does require completeness.
The standard proof does not conclude that "if 0.9999... has a value, it is 1", it proves "0.9999... = 1". It does so first establishing a recursive definition for finite decimals which does not involve a limit. It then establishes a value for infinite decimals as a limit of finite decimals, the latter definition requiring completeness to ensure that this limit is indeed a real number in all cases. But in this case, I could allow you are right. You can show that the particular decimal 0.9999... has real value 1 without having to show that all infinite decimals have real limit values (which I think would require at least a weakened version of completeness). However, in our double-ended formalism above, I have not actually defined a sum value for any of the double-ended sums. I have only said things like "if this double-ended sum has value a, then that double ended sum have value b". From these sorts of statements, the best you can prove is "if this double-ended sum has a value, it is a". Clearly we want sum({0}) = 0, but even in conjunction with the other formal properties, I don't that good enough to establish ... + 0 + 0 + 0 + 9 + 0 + 0 + 0 + ... = 9. At any rate, I expect this whole house of cards to collapse, which is to say, I expect that the sum() function cannot be consistently defined so as to give the intuitive values in the simple cases and also conform to the sum, scalar multiplication, and shifting rules.
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On Fri, Nov 16, 2012 at 6:54 PM, Allan Wechsler <acwacw@gmail.com> wrote:
People often don't like to talk about n-adic numbers when n isn't prime, because the resulting structure fails to be a field. I think that's the problem, anyway -- I don't remember too clearly.
It's worse than that; there's no good way to define multiplication in the n-adic numbers for n composite. You can define a metric on the rationals by | a - b | = 1/n^k, where k is the largest power of n dividing a - b. The n-adics are then formed by taking the topological completion of this metric space. If multiplication is continuous, you can then extend multiplication to the n-adics by continuity. But continuity of multiplication depends on the fact that | a | * | b | = | (a * b) | which only holds for n prime. Andy
On Fri, Nov 16, 2012 at 6:48 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm not very familiar with the n-adic numbers, but I think that
...999.0... = -1
is true in the 10-adic numbers, where the reasoning Marc uses can be justified rigorously.
(Alternatively, just add 1 to ...999.0... to get 0.)
--Dan
On 2012-11-16, at 6:32 AM, Andy Latto wrote:
On Fri, Nov 16, 2012 at 12:52 AM, Marc LeBrun <mlb@well.com> wrote:
Since this was originally an historical question perhaps as an arithmetical post-modernist punk I should hold my piece, but perhaps those that hold that
...0.999... = 1
will also equably agree that
...999.0... = -1?
After all, are we not told (presented with precisely the same corroboration as given for 0.999...=1), that the geometrical series with term ratio r=10:
9 10^0 + 9 10^1 + 9 10^2 + ...
must be equal to 9/(1-r) = 9/(1-10) = 9/-9 = -1?
This isn't post-modern, it's pre-classical. This is exactly the way Euler reasoned, and he came up with the same sort of conclusions. But the reasons I, and a modern mathematician, say that .99999.... = 1 not just because of the proof of the formula for the sum of a geometric series (which, if you examine it closely, shows only that *if* the series has a limit, *then* the limit is 1), but because of the epsilon-definition of a limit, and a proof I can provide, using that definition, that this series indeed does sum to 1. If you try to provide a similar epsilon-delta proof that the series ...9999.0 sums to -1, you will fail, and I can provide an epsilon-delta proof that this series does *not* have a limit.
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* Dan Asimov <dasimov@earthlink.net> [Nov 18. 2012 08:36]:
I'm not very familiar with the n-adic numbers, but I think that
...999.0... = -1
is true in the 10-adic numbers, where the reasoning Marc uses can be justified rigorously.
(Alternatively, just add 1 to ...999.0... to get 0.)
--Dan
[...]
For me the left side of ...999999.0 = -1.0 is just ten's complement, a neat way to write negative numbers without the minus sign. In general, to negate a (non-negative) number, first complement the digits ( d --> 9-d ), then add 1. Example: 123 --> ...9999876 --> ...9999877 == -123 Check (negate again): ...9999877 --> ...0000122 --> 123 Omit the "add 1" step to get nine's complement (with two zeros, 0 and ...9999999 ). Engineers learn that for base 2, and about every computer in existence uses two's complement for integers.
Along this same vein of limits, I have never quite understood the staircase paradox, that a sequence of staircases from (0,0) to (1,1) with decreasing step size pointwise approaches a diagonal line of length sqrt(2), but the staircases themselves always have length 2. Anyway, I was wondering, does 0.99999... = 1 in the surreal numbers?
David Wilson asks:
Anyway, I was wondering, does 0.99999... = 1 in the surreal numbers?
That depends on how you map infinite decimals into surreals. For sets of numbers A and B with a<b whenever a in A and b in B, surreal "A | B" is the "simplest" number between A and B. The surreals that have finite sets of reals for A and B are dyadic rationals (including integers). If .9999... is mapped to {.9, .99, .999, .9999, ...} | {} then it is 1 in the surreals. But with this mapping, .3333... becomes {.3, .33, .333, .3333, ...} | {} which is 1/2 in the surreals. A better (IMO) mapping of infinite decimals to surreals is .3333... = {.3, .33, .333, .3333, ...} | {.3+.1, .33+.01, .333+.001, ...} which is 1/3 in the surreals. This mapping gives .9999... = {.9, .99, .999, .9999, ...} | (1,1,1,1,...} which is 1 - 1/omega in the surreals. In general, for numeral D an infinite decimal converging to real r, this interpretation in surreals yields D = r - 1/omega if r is a dyadic rational D = r if r is not a dyadic rational
Minor correction: I said {.3, .33, .333, .3333, ...} | {} is 1/2 in the surreals. Actually, it is 1. Still reinforcing that {.3, .33, .333, .3333, ...} | {.3+.1, .33+.01, .333+.001, ...} = 1/3 Is the better way to map infinite decimals into surreals.
I missed the first post, but: I thought surreals were constructed with a surreal on the left and a surreal on the right. So what does it mean to have a sequence on the left and/or right? I might be able to guess, but would rather know how (and where) this is defined. Thanks, Dan On 2012-11-15, at 9:39 PM, Huddleston, Scott wrote:
Minor correction: I said {.3, .33, .333, .3333, ...} | {} is 1/2 in the surreals. Actually, it is 1. Still reinforcing that
{.3, .33, .333, .3333, ...} | {.3+.1, .33+.01, .333+.001, ...} = 1/3 Is the better way to map infinite decimals into surreals.
Each surreal number is constructed with a set of surreals to the left and right, so each such construction has a left and right set, not merely a left and right number. So, in the equation { 0 | 1 } = 1/2, the expression on the left side means a surreal whose left set is {0} and whose right set is {1}. The left and right sets can be infinite, otherwise we would never be able to construct infinitesimals, or even 1/3, for that matter. On Fri, Nov 16, 2012 at 2:12 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I missed the first post, but: I thought surreals were constructed with a surreal on the left and a surreal on the right.
So what does it mean to have a sequence on the left and/or right? I might be able to guess, but would rather know how (and where) this is defined.
Thanks,
Dan
On 2012-11-15, at 9:39 PM, Huddleston, Scott wrote:
Minor correction: I said {.3, .33, .333, .3333, ...} | {} is 1/2 in the surreals. Actually, it is 1. Still reinforcing that
{.3, .33, .333, .3333, ...} | {.3+.1, .33+.01, .333+.001, ...} = 1/3 Is the better way to map infinite decimals into surreals.
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participants (16)
-
Allan Wechsler -
Andy Latto -
Charles Greathouse -
Dan Asimov -
David Wilson -
Gareth McCaughan -
Gary Antonick -
Huddleston, Scott -
j -
James Buddenhagen -
Joerg Arndt -
Marc LeBrun -
meekerdb -
Mike Speciner -
Thane Plambeck -
W. Edwin Clark