Andy Latto writes: "But I don't think anyone has ever proposed any useful or interesting definition under which .9999.... is not equal to 1". If one is willing to live without subtraction, then there is a consistent mathematical structure in which .9999... is different from 1; see my preprint "Carrying On with Infinite Decimals" ( http://jamespropp.org/carrying.pdf), which my students presented at a recent Gathering for Gardner, and which I will eventually get around to publishing in one of the G4G books. (If any of the editors of the series are reading this, please feel free to gently hassle me about this, now or at some future time.) This number system is certainly not useful, and in and of itself it is not very interesting, but it does hold some interest as a special case of a more general concept in the theory of abelian sandpiles. Jim Propp On Friday, November 16, 2012, Andy Latto wrote:
On Fri, Nov 16, 2012 at 5:20 PM, Tom Duff <td@pixar.com> wrote:
On Fri, 16 Nov 2012, math-fun-request@mailman.xmission.com wrote:
Yes. Looking more closely (by a factor of 1+epsilon) though, the result you really have is that if the epsilon-delta definition is the correct definition of what it means to be a limit, then there is no limit. So either it's not the limit, or the definition is bogus. What we're really exploring here is whether the second option is the case.
There are other ways to define a limit, that assign a value to some series that the e-d definition calls divergent, like Caesero summability. But for any meaning of the sum of a series in which .999999... isn't 1, either .999.... * 10 isn't equal to 9.99999.... (that is, you can't multiply a series term-by-term by a constant, and expect the sum to be multiplied by that same constant), or 9.999999... - .999.... isn't equal to 9 (that is, you can't do simple manipulations like shifting a series over by one without changing the sum, or you can't subtract the terms of two convergent series term-by-term and get a series that sums to the difference of the original series.
It's hard for me to imagine a notion of infinite sums that doesn't have one of these two properties that is in any way useful or interesting.
Yes, there are other interesting ways to define the limit of a series.
In some of them, ....666.0 = -1 (using base 7) In some of them, I suspect, 1 - 2 + 3 - 4 + ... = -1/4 (Euler)
But I don't think anyone has ever proposed any useful or interesting definition under which .9999.... is not equal to 1, because any such definition would have to sacrifice one of the properties that makes this infinite sum feel in any way like a sum.
Yes, sometimes a false but appealing statement can be re-interpreted in a way that makes it true, and interesting mathematics can result. But some false statements are just false statements, and there's no interesting way to make them true.
" They laughed at Columbus, they laughed at Fulton, they laughed at the Wright brothers. But they also laughed at Bozo the Clown." --Carl Sagan.
Andy
historical question is how we came to agree that e-d is the correct definition. There are multiple ways that Euler's intuitive approach to limits could have been formalized, and somehow we settled on this particular one, which fails to capture important parts of his intuition.
This is historically interesting because in other cases, like the definition of the definite integral, there is still no consensus but we're mostly OK with that. We are taught the Riemann definition in introductory calc, but eventually we may want to work in more complicated spaces than Rn or handle functions that Riemann fails on where some other definition will win out.
-- Tom Duff. Linux? Is that an OS, like Pentium?
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