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.999...=1?
Someone attempted to prove this...

 .999...=1? •StreetlightX Eppur si muove... Usergroup: Moderators Joined: Jun 12, 2010 Location: In medias res Total Topics: 42 Total Posts: 2691 #71 - Quote - Permalink 10 of 10 people found this post helpful Posted May 29, 2012 - 9:03 AM: I suspect that much of the misunderstanding regarding this issue is not necessarily one of misunderstanding the proofs, but rather stems from a rather subtle semantic (rather than merely mathematical) confusion regarding the notion of 'the same'. That is, when the doubter (let's call her) is presented with the proofs that .999.. is the same (or equal to) 1, doubt nonetheless remains because it's not altogether clear to the doubter how to make sense of the significance of the proof. That is, we've done all these fancy mathematical manipulations, but it isn't yet clear to the doubter the nature of what exactly has been proved.As such, I think that it's worth explicitly highlighting to anyone confused - as you've implicitly done all along Raugust - that when we say that .999... is "the same" as 1, what is meant that both numbers have the same mathematical value. That is to say, both 1 and .999... function in the exact same way when subjected to mathematical manipulation. That is the sense in which it is asserted that both numbers are "the same". So whenever we do calculations that both 1 and .999... are involved in, what remains 'the same' for both, what consigns to both their equivalent ('equal to') character, are the functions according to which both can be used, which in this case, are 'the same': whether we multiply .999 or 1 by 2, or raise either to the power of x, what we get 'in the end', are equivalent values, and this because both share, from the very beginning, the same values. This is the semantic significance, if I may put it that way, of what is demonstrated in the proofs so far offered by others in the thread.Philosophically speaking then, it seems perhaps that the confusion stems, above all, from a certain approach to thinking about the very notion of number as such. That is, from treating numbers in an essentialist way, as if the 'essence' of a number is somehow 'contained' within "it", as if there is a "Oneness" about 1, or a "Point NineNineNineRepeatingness" about .999 which cannot, despite the proofs, coincide. To dispel this, I think what is needed is to emphasize a shift in the very approach to thinking about number such that the identity of the number is given in reference to what it is that a number can do, rather than reifying numbers as entities unto themselves - a tendency which, despite it all, is quite understandable I think, and - if I may perhaps unfairly say - cultivated by the the slightly more "nominalist" approaches offered so far in this thread (".999... is 1, here's the proof...").**I don't say this as any sort of prolegomena to a theory of number - I'm far too unfamiliar with the philosophy of mathematics for that - but simply as a useful way of approaching this sort of problem which seems to be a bit of a stickler. Edited by StreetlightX on May 29, 2012 - 8:38 PM On May 29, 2012 - 9:28 AM, StreetlightX responded: Indeed, you know that I tend to approach identity in terms of function as a general rule - but something still bugs me, which is that what such a conception misses is that the significance of function cannot be said to be a 'natural' given. And given that identity is bound up with questions of significance, identity cannot be reduced to function, but must be temporally 'open', in the last instance, to ethical or political contestation. How to reconcile these thoughts with the question of numerical identity is beyond me though... Edited by StreetlightX on May 29, 2012 - 9:40 AM •Raugust Forum Veteran Usergroup: Sponsors Joined: May 21, 2005 Location: United States Total Topics: 33 Total Posts: 843 #72 - Quote - Permalink Posted May 29, 2012 - 10:19 AM: Indeed. Without the idea of equivalent value, it's not even clear what justifies us saying, e.g., that 1.0 = 1. I took this to be the point ViHart was making in her video in the first post:Victoria Hart wrote:So what does this statement mean? .999... is the same as 1? It looks pretty different. But it equals 1 in the same way that 1/2 equals .5 -- they have the same value. You can philosophize over whether, if 1 is the loneliest number, .78 + .22 is just as lonely, but there's no mathematical doubt that they have the same value. Just as 100 years of solitude is exactly as long as (10 + 40)2 years of solitude, or 99.999... years of solitude. •Agilious Newbie Usergroup: Members Joined: Jun 19, 2012 Total Topics: 0 Total Posts: 1 #73 - Quote - Permalink Posted Jun 19, 2012 - 4:53 PM: StreetlightX wrote: That is to say, both 1 and .999... function in the exact same way when subjected to mathematical manipulation. That is the sense in which it is asserted that both numbers are "the same". So whenever we do calculations that both 1 and .999... are involved in, what remains 'the same' for both, what consigns to both their equivalent ('equal to') character, are the functions according to which both can be used, which in this case, are 'the same': whether we multiply .999 or 1 by 2, or raise either to the power of x, what we get 'in the end', are equivalent values, and this because both share, from the very beginning, the same values. This is the semantic significance, if I may put it that way, of what is demonstrated in the proofs so far offered by others in the thread. Philosophically speaking then, it seems perhaps that the confusion stems, above all, from a certain approach to thinking about the very notion of number as such. That is, from treating numbers in an essentialist way, as if the 'essence' of a number is somehow 'contained' within "it", as if there is a "Oneness" about 1, or a "Point NineNineNineRepeatingness" about .999 which cannot, despite the proofs, coincide. To dispel this, I think what is needed is to emphasize a shift in the very approach to thinking about number such that the identity of the number is given in reference to what it is that a number can do, rather than reifying numbers as entities unto themselves I am new here and haven't gone through the entire thread yet. I think you are on the right track here, but I do have a question about our notion of numbers as such, and the notion of "the same" outlined above. I suspect that those who deny the proof in question will need to answer: what, then, exactly is the difference between 0.999... and 1, if they are not the same value? Surely one's best possible response can be only that there is an infinitessimal difference between the two different numbers. But, how does one represent such an amount; how does one represent an infinitely small numerical value? For, it seems impossible to write it in decimal form (ex. 0.000... 1, whereby there is mysteriously a 1 at the 'end' of an infinitely long decimal expansion of zeroes). Perhaps, then, the notion of the infinitessimal is best captured by the following: an infinitessimal is the limit of 1/X where X approaches the infinite. The limit is zero. The notation for problems like these make use of the '=' symbol. And yet, to say the limit is zero is still distinct from saying that [LIM: 1/X where X-->Infinite] is equivalent to zero. At least, they are not equivalent in the sense outlined above whereby 0 and [LIM: 1/X where X-->Infinite] can be replaced in any calculation without making a difference. That is, they do not server entirely the same function when subjected to mathematical manipulation. To support my claim, consider the following: [LIM: 1/X where X-->0], that is, the limit of 1/X where X approaches zero. There is no limit; the limit has no bound, or, very roughly, [LIM: 1/X where X-->0]=infinite. This is quite a trivial calculation, however, it indicates one important thing to me: there must be a difference in value between an infinitessimal amount and zero. This is because if there wasn't then [LIM: 1/X where X-->0] (where X is an infinitessimal amount) would be replacable with 1/0 (on the assumption that an infinitessimal amount and zero are replacable/serve entirely the same function), which we were all taught to be undefined. How can an infinitessimal be entirely equivalent to zero, yet usable as a denominator? I conclude, therefore, that either dividing by zero is a legitimate operation, or one can legitimately say there is an infinitessimal difference between 0.999... and 1. And yet, given the proofs demonstrating the equivalence of 0.999... and 1, this ultimately leads to a very non-essentialist concept of numbers, whereby one could say any number differs from itself by a non-zero infinitessimal value. Such a strange fluid notion of numbers as shifting processes is certainly peculiar and likely problematic. Fascinating, nonetheless... Cheers for reading, and let me know if I can elborate anywhere. •plikp1 Initiate Usergroup: Members Joined: Mar 11, 2012 Total Topics: 3 Total Posts: 54 #74 - Quote - Permalink Posted Jun 20, 2012 - 5:05 AM: SgtJava wrote:But you really cant prove this can you? .999... is not even a real number is it? Someone recently tried to prove this on youtube and it has really been bothering me. I feel like her logic was all off. Link to her explanation: http://www.youtube.com/watch?v=TINfzxSnnIE&feature=g-vrecFirst off she attempts to prove this algebraically. She starts off with this problem:x=0.999...(multiplies both sides by 10)10x=9.999...(subtracts .999...)10x-.999...=9(because x=.999..., she simplifies)9x=9x=1I found many things wrong with her logic here. First of all she multiplied .999.. by 10. If you multiply .999... repeating by 10, it actually = 9.999...-0.000...9. Essentially, by multiplying a repeating number is is no longer repeating. She failed to recognize this and from this moment on her logic was flawed. When she later subtracted .999... from both sides, the right side did not actually equal 9. It equaled a number very close to 9, but not nine as it was multiplied by 10 and no longer held the same repeating value.Anyways, whose logic is correct? Am I stupid? Or is she using flawed logic? Thanks for all feedback!**keep in mind I'm a freshman in high school, there are many areas in math that I have not studied. Most of this stuff is instinct and self taught on my part.You're right that her math is wrong, but not for the reason you said. It's fine to say that 9.99...*10=9.99..., because if an infinite series of 9s are shifted left, they don't extend and less far: infinity-1=infinity. Her mistake is when she says that 10-.999... =9, because that is presupposing that .999...=1.Here's a valid proof (someone has probably already said this):x=0.999...10x=9.999...x+9=9.999...x+9=10x9=9x1=x1=0.999... •Raugust Forum Veteran Usergroup: Sponsors Joined: May 21, 2005 Location: United States Total Topics: 33 Total Posts: 843 #75 - Quote - Permalink 1 of 1 people found this post helpful Posted Jun 20, 2012 - 6:52 AM: plikp1 wrote:You're right that her math is wrong, but not for the reason you said. [...] Her mistake is when she says that 10-.999... =9, because that is presupposing that .999...=1.Nowhere in her proof does she subtract .999... from 10 to get 9. (The number 10 never even explicitly occurs in her proof, though you could incidentally use her proof to show that 9.999... was 'secretly' 10 all along.) All she assumes is that x is equal to 0.999.... Thus she can subtract x from one side and .999... from the other, since she defined them as equal. So her proof goes:x = 0.999...(premise)10x = 9.999...(multiply by 10)9x = 9.000...(subtract x from 10x, subtract 0.999... from 9.999...)x = 1.000...(divide by 9)0.999... = 1.000...(by definition of "x")A very elegant and unimpeachably correct proof. Your proof is the same as hers, except you start from "x+9 = 10x" and then subtract x from both sides, whereas ViHart starts from "9.999... = 10x" and then subtracts 0.999 or x from each side. (Lines 2 and 3 of your proof, after the definition, are not doing any work and can be eliminated.) The only advantage of your version is that you aren't forced to subtract by different forms of the same number; the only advantage of ViHart's version is that she has a more straightforward way to move from the first line of her proof to the second, without any sudden 'leaps.' But they both get the job done. •Kelvin ==== Usergroup: Sponsors Joined: Jul 06, 2011 Total Topics: 84 Total Posts: 1916 #76 - Quote - Permalink Posted Jun 20, 2012 - 8:03 AM: Raugust wrote: A very elegant and unimpeachably correct proof. Yea, it works. The main difference between .999... and 1 is that if you ask the baker for .999.... creme puff, you'll never get your creme puff because you'll be standing there forever saying "9." •plikp1 Initiate Usergroup: Members Joined: Mar 11, 2012 Total Topics: 3 Total Posts: 54 #77 - Quote - Permalink Posted Jun 20, 2012 - 8:00 PM: Raugust wrote:Nowhere in her proof does she subtract .999... from 10 to get 9. (The number 10 never even explicitly occurs in her proof, though you could incidentally use her proof to show that 9.999... was 'secretly' 10 all along.) All she assumes is that x is equal to 0.999.... Thus she can subtract x from one side and .999... from the other, since she defined them as equal. So her proof goes:x = 0.999...(premise)10x = 9.999...(multiply by 10)9x = 9.000...(subtract x from 10x, subtract 0.999... from 9.999...)x = 1.000...(divide by 9)0.999... = 1.000...(by definition of "x")A very elegant and unimpeachably correct proof. Your proof is the same as hers, except you start from "x+9 = 10x" and then subtract x from both sides, whereas ViHart starts from "9.999... = 10x" and then subtracts 0.999 or x from each side. (Lines 2 and 3 of your proof, after the definition, are not doing any work and can be eliminated.) The only advantage of your version is that you aren't forced to subtract by different forms of the same number; the only advantage of ViHart's version is that she has a more straightforward way to move from the first line of her proof to the second, without any sudden 'leaps.' But they both get the job done.OK, you're right. I see now. •Raugust Forum Veteran Usergroup: Sponsors Joined: May 21, 2005 Location: United States Total Topics: 33 Total Posts: 843 #78 - Quote - Permalink Posted Jun 20, 2012 - 8:53 PM: We noted that to deny the fractional proof, infinitesimalists had to deny that 1/3 = 0.333..., in effect asserting that 0.333... 'stops' somehow, and therefore never 'makes it' to being a complete one-third -- i.e., since 0.999... is missing a value of .000...1 that would be needed to 'push it over' to 1.000..., 0.333... must likewise be missing 1/3 of this .000...1. Of course, we can't denote this difference between 1/3 and 0.333... by .000...333... since this would fall victim to the same problem of not equaling a third of 1. So if the difference between .999... and 1 is .000...1, the difference between .333... and 1/3 must likewise be 0.000...1/3. In other words, there simply is no real number corresponding to 1/3; it's impossible to express in any decimal notation.Are there any similarly interesting consequences of denying the algebraic proof above? The only step that the skeptic seems able to conceivably deny is the leap from "x = 0.999..." to "10x = 9.999...". Clearly 10 * x = 10x, so the difficulty will have to arise on the numerical side.0.999 * 10 = 9.999...How could one reject this? One could assert that 0.999... is the same as 0.999...9, which is why you can 'add 1' to the end of this infinity to make it equal 1 -- in other words, this is one way of notating the idea that 0.999... is infintesimally close to, but not the same as, 1. In standard algebra, 1.000... - 0.999... equals 0.000..., hence 1 is the same as 0.999.... But if 1.000... - 0.999... instead equals 0.000...1, then the two aren't equal, and we can say that the numbers are more properly expressed: 1.000...0 and 0.999...9.Now, what happens when I multiply 10.000...0 by 0.999...9? Do we end up with 9.999...9, as ViHart's proof claims? No! Because we've lost a 9: All the 9s got moved up a decimal place, so that what we really end up with is 9.999...0, or perhaps 9.999...∅. If we then continue the proof from here, its new form is:x = 0.999...9(premise)10x = 9.999...0(multiply by 10)9x = 8.999...1(subtract x from 10x, subtract 0.999...9 from 9.999...0)x = 0.999...9(divide by 9)So the proof is trivial. We learn nothing about 1.000...0 from 0.999...9, without adding the missing 0.000...1.Notice that at the second step, things have already started going wrong, because we have 9.999...0 where the 1=0.999... dogmatist claims we should have 9.999...9. The problem becomes more apparent when we try to subtract 0.999...9 (i.e., x) from both sides, and end up with 8.999...1 instead of the 9.000...0 the dogmatist desires. The 0.000...9 that we lost when we carried all the 9s over during multiplication makes all the difference, literally -- if you re-added that 0.000...9 to 8.999...1 you'd get 9.000...0, and the dogmatist's proof would work. So, if the fallacy in the fractional proof was to assume that 0.333...3 (i.e., a third of 0.999...9) is the same as 0.333...1/3 (i.e., a third of 1), the Cantorian fallacy in this algebraic proof was to assume that 0.999...9 doesn't 'lose a digit' when you multiply it by 10, that 0.999...9 is not meaningfully distinct from 0.999...90 or for that matter 0.999...99 -- i.e., the fallacy was to assume that infinity is infinite, that there is no 'last digit' of the sort I've been talking about. •StreetlightX Eppur si muove... Usergroup: Moderators Joined: Jun 12, 2010 Location: In medias res Total Topics: 42 Total Posts: 2691 #79 - Quote - Permalink Posted Jun 20, 2012 - 11:42 PM: Agilious wrote:I am new here and haven't gone through the entire thread yet. I think you are on the right track here, but I do have a question about our notion of numbers as such, and the notion of "the same" outlined above. I suspect that those who deny the proof in question will need to answer: what, then, exactly is the difference between 0.999... and 1, if they are not the same value? Surely one's best possible response can be only that there is an infinitessimal difference between the two different numbers. But, how does one represent such an amount; how does one represent an infinitely small numerical value? For, it seems impossible to write it in decimal form (ex. 0.000... 1, whereby there is mysteriously a 1 at the 'end' of an infinitely long decimal expansion of zeroes). Perhaps, then, the notion of the infinitessimal is best captured by the following: an infinitessimal is the limit of 1/X where X approaches the infinite. The limit is zero. The notation for problems like these make use of the '=' symbol. And yet, to say the limit is zero is still distinct from saying that [LIM: 1/X where X-->Infinite] is equivalent to zero. At least, they are not equivalent in the sense outlined above whereby 0 and [LIM: 1/X where X-->Infinite] can be replaced in any calculation without making a difference. That is, they do not server entirely the same function when subjected to mathematical manipulation. To support my claim, consider the following: [LIM: 1/X where X-->0], that is, the limit of 1/X where X approaches zero. There is no limit; the limit has no bound, or, very roughly, [LIM: 1/X where X-->0]=infinite. This is quite a trivial calculation, however, it indicates one important thing to me: there must be a difference in value between an infinitessimal amount and zero. This is because if there wasn't then [LIM: 1/X where X-->0] (where X is an infinitessimal amount) would be replacable with 1/0 (on the assumption that an infinitessimal amount and zero are replacable/serve entirely the same function), which we were all taught to be undefined. How can an infinitessimal be entirely equivalent to zero, yet usable as a denominator? I conclude, therefore, that either dividing by zero is a legitimate operation, or one can legitimately say there is an infinitessimal difference between 0.999... and 1. And yet, given the proofs demonstrating the equivalence of 0.999... and 1, this ultimately leads to a very non-essentialist concept of numbers, whereby one could say any number differs from itself by a non-zero infinitessimal value. Such a strange fluid notion of numbers as shifting processes is certainly peculiar and likely problematic. Fascinating, nonetheless... Cheers for reading, and let me know if I can elborate anywhere.Great first post! However, there is I think a problem here, which is that while you're right that there is a difference between an infinitesimal and zero, the difference is not a difference in degree, but a difference in kind. More specifically, the limit of a function is not the value of that function. While the limit names a value, it is not the value of the function. It simply designates the value that it approaches but never is. As such, we can't substitute [LIM: 1/X where X -> 0] with 1/0. At X=0, 1/X remains undefined, because the limit does not define the value of the function. Edited by StreetlightX on Jun 21, 2012 - 9:04 AM •ssu PF Addon Usergroup: Sponsors Joined: Jun 02, 2007 Location: north Total Topics: 25 Total Posts: 2197 #80 - Quote - Permalink Posted Jun 30, 2012 - 1:25 PM: One question: What is the problem that we have with infinitesimals, really?People are OK with infinitesimals if one assumes non-standard analysis and the hyperreals of Abraham Robinson. Yet otherwise, if we don't assume the work of Robinson, it seems that infinitesimals are banned and the notion of limit is used as a replacement. Well, for an amateur like me the definition given of a limit isn't the most simple and the most beautiful, if I'm allowed to say so. To me it's more like "here's a way to say it like a lawyer would say in order that we can bypass the obvious problems that arise and go on". Now I'm not in any way saying here that limits wouldn't work, no, the math we had does work, yet the philosophical understanding seems to be lagging behind. Because this question pops out again and again and again...Yes, I admit that for me mathematics isn't this optional game where one choose one set of axioms over another from a multitude axioms where everything in the end goes. There isn't a mathematical multiverse, even if there can be various geometries at use. I believe that in math there is beauty and that this beauty is simple and in the end it is easy to understand. As I said earlier, that this question (1=0,999...) comes over and over again just shows that it isn't that we do not have yet the most simple theorem or law about this. For example, nobody is questioning the existence of irrational numbers here. There hasn't been a thread that "All numbers are rational" or "Irrational numbers do not exist". That there are irrational numbers is obvious to us. Well, it wasn't so during early ancient times. People really had this idea that all numbers are rational because obviously math was so harmonious. And so I hope someday would be with infinitesimals: that there would be this obvious proof (even if many will likely say that with limits the case is closed) that will explain to the vast majority why 1=0.999... or what does it mean. In my view, it would all have to start from a very simply an obvious proof that there exist logical limitations to arithmetic (addition and division and so on) and once you have infinitesimals at play it is simply illogical to assume that you can do basic arithmetic with them and they would behave like other numbers.

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