# 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:** 12
**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 19, 2016, 5:09pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/221 "2016-10-19T17:09:41Z")
> [@netzweltler](#): > > Zeno’s dichotomy paradox has never been solved. Can you cite the working mathematician who claims this? And why? And if you do happen to know of one, why do the other million working mathematicians disagree? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 19, 2016, 5:30pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/222 "2016-10-19T17:30:47Z")
> [@Exapno\_Mapcase](#): > > Can you cite the working mathematician who claims this? And why? > > And if you do happen to know of one, why do the other million working mathematicians disagree? It’s obvious - for the same reason which we are discussing already. t = 0: I move my pen from point 0 to point 0.5 of the number line t = 0.5: I move my pen from point 0.5 to point 0.75 of the number line t = 0.75: I move my pen from point 0.75 to point 0.875 of the number line … None of the infinitely many lines tells us to move to point 1. If we reached 1 we have disobeyed the actions on the list. ---
**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 19, 2016, 5:33pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/223 "2016-10-19T17:33:35Z")
> [@netzweltler](#): > > The pen is not moving at constant speed - not as I have defined it. It is moving in discrete steps. There is no action defined for t = 1 on the list. > > What I want to make clear is, that moving the distance 0.999… does not get us to a defined point of the number line - not to point 1 and not to a point short of 1 either. Ah, I think I see. You want to persuade us that 0.999… \<\> 1 by having us first believe in the instantaneous movement of your pen? ---
**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 19, 2016, 5:43pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/224 "2016-10-19T17:43:48Z")
> [@netzweltler](#): > > None of the infinitely many lines tells us to move to point 1. If we reached 1 we have disobeyed the actions on the list. You can’t treat infinite as just a really big finite. ---
**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 19, 2016, 6:01pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/225 "2016-10-19T18:01:41Z")
There are some formulations of mathematics that can’t deal with Zeno’s Paradox. But why would one ever use such a formulation? Clearly, it doesn’t actually describe our world, since in our world, arrows can hit their targets. Nor does it lead to any particularly interesting results (at least, not ones that you couldn’t also get with other formulations that do deal with it). I mean, sure, you can use any mathematics you want, but why would you even want to use one like that? ---
**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 19, 2016, 7:50pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/226 "2016-10-19T19:50:01Z")
> [@netzweltler](#): > > Yes, with a bit of expansion. But the bit of expansion is not on the list. I meant expansion of the rules by the natural extension of your incomplete and uninteresting thought experiment. ---
**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 19, 2016, 8:17pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/227 "2016-10-19T20:17:16Z")
> [@netzweltler](#): > > It’s obvious - for the same reason which we are discussing already. Obviously it’s not obvious since obviously other people are disagreeing that it’s true. ---
**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 19, 2016, 8:54pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/228 "2016-10-19T20:54:05Z")
**Exapno** , I \*think \*what is being argued is that the LHS can never reach 1, therefore the RHS can never reach 1. Because the conclusion \*obviously \*follows from the antecedent, our hero thinks they have a knock-down argument. But – if I am right – then there has never been, nor could there ever be, a plainer example of begging the question. ---
**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 19, 2016, 10:56pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/229 "2016-10-19T22:56:46Z")
> [@The\_Great\_Unwashed](#): > > **Exapno** , I \*think \*what is being argued is that the LHS can never reach 1, therefore the RHS can never reach 1. Because the conclusion \*obviously \*follows from the antecedent, our hero thinks they have a knock-down argument. > > But – if I am right – then there has never been, nor could there ever be, a plainer example of begging the question. You’re the type of pedant who probably thinks that math has rules that everybody needs to agree on. 😉 ---
**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 19, 2016, 11:25pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/230 "2016-10-19T23:25:03Z")
> [@Exapno\_Mapcase](#): > > You’re the type of pedant who probably thinks that math has rules that everybody needs to agree on. 😉 No. I’m the other type of pedant. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 20, 2016, 7:49am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/231 "2016-10-20T07:49:12Z")
> [@The\_Great\_Unwashed](#): > > Ah, I think I see. You want to persuade us that 0.999… \<\> 1 by having us first believe in the instantaneous movement of your pen? I do not persuade anyone. The list is convincing. The pen is moving (jumping) in discrete steps - just like the addition 0.9 + 0.09 + 0.009 + … means discrete steps. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 20, 2016, 7:50am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/232 "2016-10-20T07:50:59Z")
> [@Telemark](#): > > You can’t treat infinite as just a really big finite. Do I? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 20, 2016, 7:54am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/233 "2016-10-20T07:54:15Z")
> [@Chronos](#): > > There are some formulations of mathematics that can’t deal with Zeno’s Paradox. But why would one ever use such a formulation? Clearly, it doesn’t actually describe our world, since in our world, arrows can hit their targets. Nor does it lead to any particularly interesting results (at least, not ones that you couldn’t also get with other formulations that do deal with it). I mean, sure, you can use any mathematics you want, but why would you even want to use one like that? I am not saying that the arrow doesn’t hit the target. I am saying that 0.5 + 0.25 + 0.125 + … is not the correct description of _how_ the arrow hits the target. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 20, 2016, 7:57am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/234 "2016-10-20T07:57:35Z")
> [@Exapno\_Mapcase](#): > > Obviously it’s not obvious since obviously other people are disagreeing that it’s true. How can it be _not_ obvious that there is simply no line on the list that shows how we reach point 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 20, 2016, 8:04am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/235 "2016-10-20T08:04:41Z")
> [@netzweltler](#): > > I am not saying that the arrow doesn’t hit the target. I am saying that 0.5 + 0.25 + 0.125 + … is not the correct description of _how_ the arrow hits the target. Why should we care? Your description is not relevant to the discussion at hand. We already know dealing with infinities in this way leads to Zeno’s paradoxes and that the solution is to not deal with infinities in this way. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 20, 2016, 8:12am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/236 "2016-10-20T08:12:25Z")
> [@naita](#): > > Why should we care? Your description is not relevant to the discussion at hand. We already know dealing with infinities in this way leads to Zeno’s paradoxes and that the solution is to not deal with infinities in this way. But 0.999… is nothing else than 0.9 + 0.09 + 0.009 + … There is nothing else to it! ---
**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 20, 2016, 8:22am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/237 "2016-10-20T08:22:47Z")
> [@netzweltler](#): > > But 0.999… is nothing else than 0.9 + 0.09 + 0.009 + … > There is nothing else to it! But you’re making it into something else. You’re taking the mathematical concept of infinite nines after the decimal points and turning it into a process of adding nines. Now Zeno’s paradoxes show that using this approach of applying certain infinite processes to real world situations give results that don’t match reality. The mathematical way to deal with it is acknowledge that Zeno’s rules are inapproriate and to look at the limit of the sequence, which works quite sensibly to resolve the reality-contradicting paradoxes. I’m still not sure what you’re trying to say though. Do you disagree with the mathematical definition that defines the infinite series 0.9 + 0.09 + 0.009 + … to be equal to its limit? Do you disagree that the limit is 1? Or are you just quibbling with describing it as a process, which to everyone here seems to be something of your own making? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 20, 2016, 8:49am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/238 "2016-10-20T08:49:29Z")
> [@naita](#): > > But you’re making it into something else. You’re taking the mathematical concept of infinite nines after the decimal points and turning it into a process of adding nines. Now Zeno’s paradoxes show that using this approach of applying certain infinite processes to real world situations give results that don’t match reality. The mathematical way to deal with it is acknowledge that Zeno’s rules are inapproriate and to look at the limit of the sequence, which works quite sensibly to resolve the reality-contradicting paradoxes. > > I’m still not sure what you’re trying to say though. > > Do you disagree with the mathematical definition that defines the infinite series 0.9 + 0.09 + 0.009 + … to be equal to its limit? > Do you disagree that the limit is 1? > Or are you just quibbling with describing it as a process, which to everyone here seems to be something of your own making? The limit is 1. It is a defined point on the number line. The limit is not 0.999…, because it’s position on the number line is NOT defined. Its neither at position 1 nor is it short of 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 20, 2016, 9:01am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/239 "2016-10-20T09:01:28Z")
> [@netzweltler](#): > > The limit is 1. It is a defined point on the number line. > The limit is not 0.999…, because it’s position on the number line is NOT defined. Its neither at position 1 nor is it short of 1. That’s only if you use your obviously flawed approach, shown over 2000 years ago not to work when dealing with infinities. Do you think the square root of two has a defined position on the number line? And if so, where is that? ---
**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 20, 2016, 10:12am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/240 "2016-10-20T10:12:07Z")
> [@netzweltler](#): > > Do I? Yes, that’s exactly what you are doing by treating .999… as a process of adding '9’s. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=11) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=13)