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Thread: 0.999... again

  1. #1
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    0.999... again

    We aren't going to have a 1=.999999.... show up are we?
    Solfe

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    Quote Originally Posted by Solfe View Post
    We aren't going to have a 1=.999999.... show up are we?
    We do now...

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    Quote Originally Posted by Solfe View Post
    We aren't going to have a 1=.999999.... show up are we?
    I hope not, because it doesn't.

    1 = 1 and ".999999 to infinity" is a non-statement because "infinity" is a human construct with no representative in reality.

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    The first three posts were removed from this ATM thread:

    http://cosmoquest.org/forum/showthre...Second-Problem

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    All right mathletes, it wasn't my intention to start a 1 vs. .99999.... thread, but my question is, can you have a conjecture, theorem, or other math rule that can be negated by substituting .99999.... for one? It would seem to me that the answer is no, for the practical purpose that by whatever logic used to introduce the .99999.... value could be applied to every value in the conjecture, theorem, etc. and it would still have to be reasonably true.

    Is that correct or is my logic busted? Or is the reverse true, that you prove all sorts of things by creating a problem that requires the answer not to be 1 and injecting a .99999.... value to make it work.

    In the thread where this came from, it seemed to me that a requirement of "not 1" could be met by "is .99999...."
    Solfe

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    If you have something that's negated by 1=0.999... then it's not even false, it's simply not math, since 1 and 0.999... are two different decimal representations of the same member of the set of reals.
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    Vi Hart does some great math videos. For example: YouTube: Why Every Proof that .999... = 1 is Wrong

    (Note the date, before getting too wound up by it!)

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    Quote Originally Posted by HenrikOlsen View Post
    If you have something that's negated by 1=0.999... then it's not even false, it's simply not math, since 1 and 0.999... are two different decimal representations of the same member of the set of reals.
    The other thread was ATM...

    The set of reals is something that I have heard of, I think I need to read up on that and watch the video.

    I have this suspicion that people confuse the impossibility of writing out the value of .99999.... as a process instead of a representation a singular concept. I'd like to do it myself, but at least I understand that it is wrong.
    Solfe

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    Quote Originally Posted by Solfe View Post
    The set of reals is something that I have heard of, I think I need to read up on that and watch the video.
    make sure you watch her "other" video on the subject as well (or instead) otherwise you might end up very confused ...

    I have this suspicion that people confuse the impossibility of writing out the value of .99999.... as a process instead of a representation a singular concept.
    I think you are right. And once again, math trumps "common sense". (Sorry, Don!)

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    Quote Originally Posted by Solfe View Post
    All right mathletes, it wasn't my intention to start a 1 vs. .99999.... thread, but my question is, can you have a conjecture, theorem, or other math rule that can be negated by substituting .99999.... for one? It would seem to me that the answer is no, for the practical purpose that by whatever logic used to introduce the .99999.... value could be applied to every value in the conjecture, theorem, etc. and it would still have to be reasonably true.

    Is that correct or is my logic busted? Or is the reverse true, that you prove all sorts of things by creating a problem that requires the answer not to be 1 and injecting a .99999.... value to make it work.

    In the thread where this came from, it seemed to me that a requirement of "not 1" could be met by "is .99999...."
    You can't have a sensible discussion in mathematics unless you agree what you are talking about. In the case of whether 1=0.9999... you need to agree what kind of numbers you are talking about and what notations you use for those numbers. Once we agree we are talking about the real numbers, and agree that those are defined by a standard construction, eg Dedekind cuts, then that is a context in which 1=0.9999..., they are simply equivalent notations for the same thing within the real numbers. But if you refuse to agree on your terms of debate, your notation is ill-defined, and you could be talking about any kind of number system. It is straightforward to devise consistent definitions and notations where 1 and 0.9999... are different, but it won't be a number system of much applicability to any maths anyone is interested in.

    This basically is the sophistry that the contrarians employ: they refuse to agree to a defined number system. And obviously without defined terms and notations, anything goes.

    The number of cranks willing to deny long and strongly established mathematics is extraordinary. Prominent maths professors receive numerous unsolicited papers purporting to prove things contrary to long and established mathematics, where the proofs are in no doubt. I read recently a claim that the most popular subject for those cranks is Cantor's famous "diagonalisation proof" that the reals cannot be enumerated. Which is actually of the essence in the case of what we are talking about here, since it depends upon much the same thing.

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    Quote Originally Posted by Ivan Viehoff View Post
    It is straightforward to devise consistent definitions and notations where 1 and 0.9999... are different, but it won't be a number system of much applicability to any maths anyone is interested in.
    The hyperreals are one such number system, and they have quite a significance in mathematics - they allow the construction of non-standard analysis. Just because it's not the real numbers doesn't mean that it hasn't much applicability to any maths anyone is interested in. In fact, if you ever used any calculus at all you could have used the hyperreals as a basis rather than the -formalism (see non-standard calculus).
    Last edited by caveman1917; 2013-Jun-04 at 01:22 PM.

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    Here is an interesting article on mathematical cranks, and how to respond to them: http://web.mst.edu/~lmhall/WhatToDoW...ectorComes.pdf

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    Quote Originally Posted by Strange View Post
    Here is an interesting article on mathematical cranks, and how to respond to them: http://web.mst.edu/~lmhall/WhatToDoW...ectorComes.pdf
    I find it entertaining that, since the paper was written before Wiles's proof of Fermat's Theorem, he lists proofs of Fermat's Conjecture as an example of the sort of thing he receives from cranks.

    Of course, I have no doubt that mathematicians do receive erroneous proofs of Fermat's Theorem, even today (and indeed, Wiles's proof is so fantastically complex, using mathematical techniques developed long after Fermat lived, that it seems almost certain that whatever Fermat himself had in mind as a proof was probably flawed). Even if something is true, it's still possible to have an incorrect proof of it.
    Conserve energy. Commute with the Hamiltonian.

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    Here is another article that points out that just occasionally it is worth reading the letter: http://www.nytimes.com/1999/02/09/sc...ath-crank.html

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    Quote Originally Posted by Strange View Post
    Here is another article that points out that just occasionally it is worth reading the letter: http://www.nytimes.com/1999/02/09/sc...ath-crank.html
    That's a great article. I had a college friend who had a method of trisecting an angle, which worked as nearly as we could measure it with a plastic protractor marked out in whole degrees. That, of course, is probably about as accurate as the method was. I wish I still remembered how to do it though.
    Cum catapultae proscriptae erunt tum soli proscript catapultas habebunt.

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    Last edited by grapes; 2018-Aug-10 at 09:54 AM. Reason: ETA:

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    Quote Originally Posted by Grey View Post
    I find it entertaining that, since the paper was written before Wiles's proof of Fermat's Theorem, he lists proofs of Fermat's Conjecture as an example of the sort of thing he receives from cranks.

    Of course, I have no doubt that mathematicians do receive erroneous proofs of Fermat's Theorem, even today (and indeed, Wiles's proof is so fantastically complex, using mathematical techniques developed long after Fermat lived, that it seems almost certain that whatever Fermat himself had in mind as a proof was probably flawed). Even if something is true, it's still possible to have an incorrect proof of it.
    I seriously love the story behind Fermat's Conjecture. I often wonder if he was expressing something completely non-informational such as his amazement and love of math instead of proposing a real proof. Wouldn't that be funny?
    Solfe

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    This is neat; "origami geometry" (or should that be origami kikagaku?) extends Euclid's axioms and means you can trisect an angle:
    http://www.sciencenews.org/view/gene...e_with_Origami

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    Sure you can trisect an angle, given the correct tools. But you can't trisect an angle with only a compass and an unmarked straightedge.

    Fred
    Hey, you! "It's" with an apostrophe means "it is" or "it has." "Its" without an apostrophe means "belongs to it."

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    Quote Originally Posted by Nowhere Man View Post
    Sure you can trisect an angle, given the correct tools. But you can't trisect an angle with only a compass and an unmarked straightedge.
    No, that's sufficient (see link above). There's one more restriction.

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    Quote Originally Posted by Nowhere Man View Post
    Sure you can trisect an angle, given the correct tools. But you can't trisect an angle with only a compass and an unmarked straightedge.
    And you have to follow all the rules. The method grapes uses only involves a compass and unmarked straightedge, but he (very subtly) breaks the rules for how to use the compass. It's a very nice example of how you can break the rules in a way that most people might not notice (many people might not be clear about exactly what the rules are).

    Edit to add: And grapes himself makes the same point, slightly faster than me.
    Last edited by Grey; 2013-Jun-04 at 10:51 PM.
    Conserve energy. Commute with the Hamiltonian.

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    Oops, I shoulda let you handle it, better.

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    There is a thread about constructions with only compasses and straight edges from about two years ago. I dropped in to mentioned that my drafting teacher had those the final exam.

    It turns out that the final exam contained drawings of a trisecting an angel, doubling a cube and squaring a circle. The actual point was duplication of an image, correctly labelling and a few other items of interest in drafting. It had nothing to do with the actual construction with restricted tools and must have been a joke on the teacher's part. I didn't get it at all until I posted here.
    Solfe

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    Anything to keep this thread from becoming another long argument over whether or not 0.999... = 1.000...

    Fred
    Hey, you! "It's" with an apostrophe means "it is" or "it has." "Its" without an apostrophe means "belongs to it."

    "For shame, gentlemen, pack your evidence a little better against another time."
    -- John Dryden, "The Vindication of The Duke of Guise" 1684

    Earth's sole legacy will be a very slight increase (0.01%) of the solar metallicity.

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    I thought the previous thread was closed shortly after the claim was comprehensively proven. Something like:

    1 divided by 3 is 0.333...

    0.333... times 3 is 0.999...

    End of story, surely?

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    Quote Originally Posted by Solfe View Post
    It turns out that the final exam contained drawings of a trisecting an angel
    That sounds bloody.

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    Quote Originally Posted by Paul Beardsley View Post
    I thought the previous thread was closed shortly after the claim was comprehensively proven. Something like:

    1 divided by 3 is 0.333...

    0.333... times 3 is 0.999...

    End of story, surely?
    That only works if one accepts that 0.333... is actually 1/3.

    The better one is

    Let x = 0.999...
    multiply both sides by 10:
    10x = 9.999...
    so
    10x = 9 + 0.999...
    but since x=0.999...
    then 10x = 9 + x
    9x = 9
    x =1
    QED

    This is basically the way to convert repeating decimals into fractions.

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    Quote Originally Posted by NEOWatcher View Post
    That sounds bloody.
    And makes one wonder, how many thirds of an angel fit on a pinhead?

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    Quote Originally Posted by Solfe View Post
    All right mathletes, it wasn't my intention to start a 1 vs. .99999.... thread, but my question is, can you have a conjecture, theorem, or other math rule that can be negated by substituting .99999.... for one? It would seem to me that the answer is no, for the practical purpose that by whatever logic used to introduce the .99999.... value could be applied to every value in the conjecture, theorem, etc. and it would still have to be reasonably true.

    Is that correct or is my logic busted? Or is the reverse true, that you prove all sorts of things by creating a problem that requires the answer not to be 1 and injecting a .99999.... value to make it work.

    In the thread where this came from, it seemed to me that a requirement of "not 1" could be met by "is .99999...."
    I am a bit surprised that at a science-themed board, this would be controversial.

    There are some references to another thread - was that at this board? If so, where is it? (I went looking, without success, but I didn't really know a whole lot about what I was looking for.)

    I don't know what "reasonably true" in a maths setting is; we would usually talk about statements being true or false, not degrees of truth. But, if a statement about the number "1" is true, then it remains true if you replace "1" by "6/6" (provided the statement is about values, not about particular ways of representing the value). It will also remain true if "1" is replaced by "0.999...", since these are different ways of writing the same number.

    I find the complaint that 0.999... is a human construct to be particularly strange. Of course it is a human construct, like all numbers. Was the number "6" discovered by an archeologist in a cave somewhere? Numbers are human constructs. And since humans constructed the number 0.999... to have the same meaning as "one", it has all the same representations in reality that "one" has.

    Edited to add - I guess I should make it clear that the last paragraph is responding to someone else's post, not Solfe's.
    Last edited by ipsniffer; 2013-Jun-05 at 02:29 PM.

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    Quote Originally Posted by Solfe View Post
    It had nothing to do with the actual construction with restricted tools and must have been a joke on the teacher's part.
    That's awesome.


    Quote Originally Posted by Paul Beardsley View Post
    I thought the previous thread was closed shortly after the claim was comprehensively proven. Something like:

    1 divided by 3 is 0.333...

    0.333... times 3 is 0.999...

    End of story, surely?
    You'd think so, but there's this thread, and this thread, and this one, and this one. Notice how long some of those were. There may be others hiding somewhere.

    Quote Originally Posted by Nick Theodorakis View Post
    That only works if one accepts that 0.333... is actually 1/3.

    The better one is...
    Actually, form seeing lots of conversations about this, it appears that Paul's method is usually more convincing to people that don't initially accept the equality. I'm not sure why; maybe the method you're using (which is more commonly used, I think) seems to some people like there's some trickery in one of the steps somewhere.
    Conserve energy. Commute with the Hamiltonian.

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