# An infinite question: Why doesn't .999~ = 1? **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [September 24, 2016, 7:29am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711 "2016-09-24T07:29:45Z")\ **Posts on this page:** 20\ **Page:** 17
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [October 28, 2016, 7:06am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/321 "2016-10-28T07:06:00Z")
> [@DSYoungEsq](#): > > Folks, why is this “discussion” still ongoing? Eternal optimism. It’s also why we’re not a black clad army using fire to cleanse the world of immortal stupidity, so don’t take it away from us. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 28, 2016, 11:19am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/322 "2016-10-28T11:19:52Z")
> [@Xema](#): > > This is a somewhat puzzling response to the question. It could mean: > > 1. You believe there is no decimal expansion of 1/3 > 2. You suspect there may be one, but you are not knowledgeable enough to say what it is > 3. You wish to avoid taking a position on whether one exists > 4. Something else? > > Can you help by saying which of these is correct? > Also: Is 1/3 unique in this regard? Or are there other fractions whose decimal expansions you’re unable to see? 1. is correct. And yes, there are other fractions whose decimal expansions I am unable to see. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 28, 2016, 11:28am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/323 "2016-10-28T11:28:13Z")
> [@DSYoungEsq](#): > > Folks, why is this “discussion” still ongoing? All it does is prove his point (in a weird, not so very helpful way). > > He refuses to accept that the repetition of something “infinitely often” produces a result that is identical to the defined limit. So 0.999… != 1 because the first is a repeating sequence of digits that never “reaches infinity” in terms of how many '9’s there are. As long as he persists in this fundamentally different concept of how to treat something repeating infinitely often, no argument, discussion, reasoning, etc. is going to change his viewpoint. The same would be true of ANY repeating decimal expansion of a fraction (as well as, obviously, irrational numbers). So 1/7, 23/346, etc. are all going to have the same trouble. > > Indeed, if you will, we can understand his viewpoint best by considering irrationals. Take π. (3.1415926…). Since no matter how far we extend the decimal expansion, we can always extend it farther (infinitely far!), and since we cannot EVER know exactly how far off our finite expansion is, any decimal expansion of π is not equal to π. The assertion here is that ANY decimal expansion of 0.999… is not 1, because no matter how far you go along the infinitely long path, you always are just that little bit short. As **Exapno Mapcase** points out, if you view it that way, what use is that math? No. Even if you go through **ALL** the finitely indexed digits, you don’t reach 1. I have repeated it so many times: t = 0: I move my pen from point 0 to point 0.9 of the number line t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line … I am **NOT** stopping at any finite step! We “reached infinity” in terms of how many '9’s there are, and still don’t reach 1. ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [October 28, 2016, 11:31am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/324 "2016-10-28T11:31:04Z")
> [@netzweltler](#): > > No. Even if you go through **ALL** the finitely indexed digits, you don’t reach 1. I have repeated it so many times: > > t = 0: I move my pen from point 0 to point 0.9 of the number line > t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line > t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line > … > > I am **NOT** stopping at any finite step! We “reached infinity” in terms of how many '9’s there are, and still don’t reach 1. If you stop, you’re not at infinity. There’s no “still don’t reach 1”. ---
**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\ **Post date:** [October 28, 2016, 1:55pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/325 "2016-10-28T13:55:13Z")
> [@netzweltler](#): > > I am **NOT** stopping at any finite step! We “reached infinity” in terms of how many '9’s there are, and still don’t reach 1. Again, you’re trying to treat infinity as a really large finite quantity. This is a perfect example of why your question is not very well defined or useful. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [October 28, 2016, 3:16pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/326 "2016-10-28T15:16:37Z")
> [@netzweltler](#): > > I am **NOT** stopping at any finite step! We “reached infinity” in terms of how many '9’s there are, and still don’t reach 1. You cannot reach infinity. Thinking about it in that way is dead wrong. You said earlier “There is an n-th digit for every n ∈ ℕ.” That means every n is a finite number. Although mathematicians dislike the simplistic phrase that infinity means endless, it is of help here in showing how this thinking cannot be applied to infinity. You cannot have “endless” and “reach” in the same concept. Now take a break and answer my questions from before. If you reject that 0.999~ = 1, what math can you do? And if you can’t do any math, why aren’t you changing your thinking to a different technique that is useful? Your “math” is broken. Why not fix it? ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 28, 2016, 7:26pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/327 "2016-10-28T19:26:13Z")
> [@netzweltler](#): > > No. Even if you go through **ALL** the finitely indexed digits, you don’t reach 1. I have repeated it so many times: > > t = 0: I move my pen from point 0 to point 0.9 of the number line > t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line > t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line > … > > I am **NOT** stopping at any finite step! We “reached infinity” in terms of how many '9’s there are, and still don’t reach 1. If there are n iterations, and n ∈ ℕ, then indeed you’ve stopped after a finite number of iterations. If you’re trying to consider _all_ n iterations, then this doesn’t make sense at all; for n = 500 and for n = 1,000 will give profoundly different answers. I think you’re restricting yourself by requiring n ∈ ℕ, there’s no reason why it should be in all cases. So in those cases where n !∈ ℕ, you’ll be coming up with bad results. ---
**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\ **Post date:** [October 28, 2016, 10:20pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/328 "2016-10-28T22:20:40Z")
> [@netzweltler](#): > > 1. is correct. And yes, there are other fractions whose decimal expansions I am unable to see. Do you know a bright 8-year-old who you can ask to explain long division to you? [If only there were a distributed network of useful information that one could access at the touch of a few keys.](http://www.coolmath.com/prealgebra/02-decimals/13-decimals-converting-fraction-to-decimal-part2-01) Honestly and sincerely, biting my tongue so hard that it hurts, until you have read and understood the method for dividing one by three _or_ until you are prepared to coherently and directly explain why the method is flawed you should absent yourself from this thread because, until then, you have zero, zilch and zip (what’s left after you take 0.999… away from 1) to say of the least interest. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [October 29, 2016, 12:22am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/329 "2016-10-29T00:22:25Z")
Hey, netzweltler, want to see the ultimate result of your thinking? [Look here](http://i.stack.imgur.com/znQDV.png). Repeat to infinity! … would make a good sig. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 29, 2016, 7:47am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/330 "2016-10-29T07:47:24Z")
> [@Exapno\_Mapcase](#): > > You cannot reach infinity. Thinking about it in that way is dead wrong. You said earlier “There is an n-th digit for every n ∈ ℕ.” That means every n is a finite number. Although mathematicians dislike the simplistic phrase that infinity means endless, it is of help here in showing how this thinking cannot be applied to infinity. You cannot have “endless” and “reach” in the same concept. Nevertheless, it is a mathematical truth that I have reached all the digits (every n-th digit for every n ∈ ℕ) by applying this process t = 0: I move my pen from point 0 to point 0.9 of the number line t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line … And this means reaching infinitely many digits, and it means completing 0.999… > [@Exapno\_Mapcase](#): > > Now take a break and answer my questions from before. If you reject that 0.999~ = 1, what math can you do? And if you can’t do any math, why aren’t you changing your thinking to a different technique that is useful? Your “math” is broken. Why not fix it? I don’t care what math can be done or not. I don’t think it would change math no matter if I accept 0.999~ = 1 or not. I just want to clarify what is true about points on the number line, and I want to clarify if this point is well-defined or not. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 29, 2016, 7:52am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/331 "2016-10-29T07:52:12Z")
> [@watchwolf49](#): > > If there are n iterations, and n ∈ ℕ, then indeed you’ve stopped after a finite number of iterations. If you’re trying to consider _all_ n iterations, then this doesn’t make sense at all; for n = 500 and for n = 1,000 will give profoundly different answers. I think you’re restricting yourself by requiring n ∈ ℕ, there’s no reason why it should be in all cases. So in those cases where n !∈ ℕ, you’ll be coming up with bad results. I am considering n for all n ∈ ℕ. What’s wrong about that? That’s standard math. What do you mean by n ∉ ℕ? ---
**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\ **Post date:** [October 29, 2016, 11:23am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/332 "2016-10-29T11:23:35Z")
> [@netzweltler](#): > > And this means reaching infinitely many digits, and it means completing 0.999… This sentence clearly encapsulates everything wrong about your approach. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [October 29, 2016, 12:03pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/333 "2016-10-29T12:03:07Z")
And again, 0.99… is not a process. You don’t complete it. It’s the result of a process, and it’s by definition already complete. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [October 29, 2016, 2:53pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/334 "2016-10-29T14:53:43Z")
> [@netzweltler](#): > > I don’t care what math can be done or not. I don’t think it would change math no matter if I accept 0.999~ = 1 or not. I just want to clarify what is true about points on the number line, and I want to clarify if this point is well-defined or not. Well, you’re spending an incredible amount of time on this so if you want to achieve clarity you should be clear that you are wrong and all the mathematicians in the world are right. It is clear that the point is well-defined. It is clear that you are using finite points to make a statement about infinity. It is clear you can’t change other peoples’ minds because they clearly understand your mistake. Clearly, reaching “infinitely many digits” has null content because it is impossible and every thinker from Zeno on understood that it was impossible and that the problem clearly couldn’t be approached that way. You’re a flat earther, in essence. Come on over to the round side. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 29, 2016, 5:19pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/335 "2016-10-29T17:19:38Z")
> [@netzweltler](#): > > I am considering n for all n ∈ ℕ. What’s wrong about that? That’s standard math. > What do you mean by n ∉ ℕ? Obviously I misunderstand was you mean by ℕ … I thought you meant natural numbers … however my argument still stands whichever number system ℕ represents … In calculus, we have a thing called a differential, and my understanding is that every differential _dx_ ∉ ℕ … and differential equations are fairly ubiquitous in standard math … perhaps you mean standard arithmetic? One of the many useful features of the calculus is it’s ability to handle infinity in a rigid way … Give it a try, I think you’ll be impressed … Of course (0.999…) is well defined on the the number line … as well defined as 16/32’s … I think you mistake not finding these decimal numbers with them not existing … you’re just looking in the wrong place is all … can you at least try and stand over here and see what we’re seeing? ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [October 29, 2016, 6:12pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/336 "2016-10-29T18:12:39Z")
> [@watchwolf49](#): > > One of the many useful features of the calculus is it’s ability to handle infinity in a rigid way… Is rigid calculus something to do with limited-slip differentials? ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [October 29, 2016, 6:17pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/337 "2016-10-29T18:17:23Z")
> [@Riemann](#): > > Is rigid calculus something to do with limited-slip differentials? Don’t you think that math is a turn-on? ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 29, 2016, 9:53pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/338 "2016-10-29T21:53:44Z")
> [@Riemann](#): > > Is rigid calculus something to do with limited-slip differentials? Ha ha, very funny, if you’re slipping your own bounds you won’t stay within your own area … there’s limits here ya know … ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 30, 2016, 1:54am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/339 "2016-10-30T01:54:58Z")
1/4 is not equal to .25000~ either, because (by the reasoning presented) you have to keep adding zeroes. No matter how many zeroes you add, you can always add one more. The task is never completed, and thus the two numbers aren’t equal. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 30, 2016, 7:50am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/340 "2016-10-30T07:50:15Z")
> [@Trinopus](#): > > 1/4 is not equal to .25000~ either, because (by the reasoning presented) you have to keep adding zeroes. No matter how many zeroes you add, you can always add one more. The task is never completed, and thus the two numbers aren’t equal. That’s ridiculous. Apply the process suggested and you will see what really happens. t = 0: I move my pen from point 0 to point 0.25 of the number line t = 0.9: I move my pen from point 0.25 to point 0.250 of the number line t = 0.99: I move my pen from point 0.250 to point 0.2500 of the number line … I have completed the task adding infinitely many zeros by t = 1. The position of the pen hasn’t changed and is equal to 1/4 from t = 0 to t = 1. So, “by the reasoning presented” 1/4 **is** equal to 0.25000~. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=16) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=18)