# 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:** 11
**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 17, 2016, 4:42pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/201 "2016-10-17T16:42:15Z")
> [@Chronos](#): > > It didn’t even take Cantor to explain how. Newton had already done everything that needed to be done centuries earlier, and Archimedes had done most of what needed done milennia before that. The calculus does deal with infinities but doesn’t place them in a rigorous framework. I’m not sure that the 19th century knew what Archimedes had done. The [Archimedes palimpsest](http://archimedespalimpsest.org/about/scholarship/method-infinity.php) wasn’t found until 1906 and a critical passage in it not until 2001. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [October 17, 2016, 5:57pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/202 "2016-10-17T17:57:01Z")
> [@netzweltler](#): > > To make it even more obvious: > > 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 0.99 to point 0.999 of the number line > t = 0.99: I move my pen from point 0.999 to point 0.9999 of the number line > … > > t \>= 1: which point of the number line my pen is pointing to? Yes, but what does it prove? To me it just says if you could write an infinite string of numbers (which of course you can’t do), you could write an infinite string of 9s preceded by “0.” Is it true? Either way it doesn’t have that much to do with math(s) to me as maths is about abstract notions. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 17, 2016, 7:21pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/203 "2016-10-17T19:21:05Z")
> [@moes\_lotion](#): > > If you are hung up on needing to know for which value of n the sequence 1 - 10^(-n) becomes equal to 1, then it’s not clear that you understand the meaning of the limit as n approaches infinity. The limit clearly is 1. It is the lowest value which is higher than any element of the sequence (0.9, 0.99, 0.999, …). I still don’t think that this is hard to understand. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 17, 2016, 7:24pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/204 "2016-10-17T19:24:26Z")
> [@Asympotically\_fat](#): > > > [@netzweltler](#): > > > > To make it even more obvious: > > > > 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 0.99 to point 0.999 of the number line > > t = 0.99: I move my pen from point 0.999 to point 0.9999 of the number line > > … > > > > t \>= 1: which point of the number line my pen is pointing to? > > Yes, but what does it prove? To me it just says if you could write an infinite string of numbers (which of course you can’t do), you could write an infinite string of 9s preceded by “0.” > > Is it true? Either way it doesn’t have that much to do with math(s) to me as maths is about abstract notions. Did you quote the right post? ---
**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 17, 2016, 8:00pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/205 "2016-10-17T20:00:12Z")
> [@](#): > > Quoth **Exapno** : > > The calculus does deal with infinities but doesn’t place them in a rigorous framework. But we don’t need a rigorous framework for infinities. We only need a rigorous framework for limits. Which is exactly what calculus is. ---
**Author:** ![Quercus](https://avatars.discourse-cdn.com/v4/letter/q/7ab992/32.png) [@Quercus](https://boards.straightdope.com/u/Quercus)\ **Post date:** [October 18, 2016, 8:02pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/206 "2016-10-18T20:02:19Z")
> [@netzweltler](#): > > The limit clearly is 1. It is the lowest value which is higher than any element of the sequence (0.9, 0.99, 0.999, …). I still don’t think that this is hard to understand. For the record, the limit is clearly 1, but the reasoning (‘lowest value which is higher than any element of the sequence’) isn’t really how limits are defined. You could have a sequence [such as (1/n)\*(-1)^n, for n=1, 2, 3, 4,…] that has a limit, but converges on the limit from both above and below. ---
**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, 12:45am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/207 "2016-10-19T00:45:53Z")
> [@Quercus](#): > > For the record, the limit is clearly 1, but the reasoning (‘lowest value which is higher than any element of the sequence’) isn’t really how limits are defined. You could have a sequence [such as (1/n)\*(-1)^n, for n=1, 2, 3, 4,…] that has a limit, but converges on the limit from both above and below. The question to solve is whether “reaching the limit” is one of the actions given on this list or not: 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 0.99 to point 0.999 of the number line t = 0.99: I move my pen from point 0.999 to point 0.9999 of the number line … Which point of the number line my pen is pointing to at t \>= 1? ---
**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, 1:37am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/208 "2016-10-19T01:37:52Z")
> [@netzweltler](#): > > The question to solve is whether “reaching the limit” is one of the actions given on this list or not: > > 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 0.99 to point 0.999 of the number line > t = 0.99: I move my pen from point 0.999 to point 0.9999 of the number line > … > > Which point of the number line my pen is pointing to at t \>= 1? You have not defined what happens when t is greater than 1. Which is why definitions are so important. Or do you mean when t = 1? Which is why agreed upon symbolism is so important. ---
**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, 4:48am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/209 "2016-10-19T04:48:00Z")
> [@Exapno\_Mapcase](#): > > > [@netzweltler](#): > > > > The question to solve is whether “reaching the limit” is one of the actions given on this list or not: > > > > 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 0.99 to point 0.999 of the number line > > t = 0.99: I move my pen from point 0.999 to point 0.9999 of the number line > > … > > > > Which point of the number line my pen is pointing to at t \>= 1? > > You have not defined what happens when t is greater than 1. Which is why definitions are so important. > > Or do you mean when t = 1? Which is why agreed upon symbolism is so important. On the list you do not find definitions for t = 1 and t \> 1 either. All I want to know is which point the pen is pointing to after completing the actions given 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, 5:13am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/210 "2016-10-19T05:13:14Z")
> [@netzweltler](#): > > All I want to know is which point the pen is pointing to after completing the actions given on the list! 1. ONE. It is pointing at 1. IT’S ONE. ---
**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, 5:42am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/211 "2016-10-19T05:42:16Z")
> [@The\_Great\_Unwashed](#): > > 1. > > ONE. > > It is pointing at 1. > > IT’S ONE. The actions on the list don’t allow me to reach point 1. Each of the infinitely many lines of the list shows how I get closer to 1 but not reach 1. So, what does make you thinking it’s pointing to 1? ---
**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, 6:00am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/212 "2016-10-19T06:00:24Z")
> [@The\_Great\_Unwashed](#): > > IT’S ONE. Even if you tell me 1000 times that it’s 1: The list tells me infinitely many times that it’s not 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 19, 2016, 9:36am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/213 "2016-10-19T09:36:21Z")
> [@netzweltler](#): > > Even if you tell me 1000 times that it’s 1: The list tells me infinitely many times that it’s not 1. The list tells you where it’s pointing when t\<1. With a bit of expansion you can show that the limit is 1 and the limit is 0.999… so 0.999… = 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 19, 2016, 10:04am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/214 "2016-10-19T10:04:10Z")
> [@netzweltler](#): > > Even if you tell me 1000 times that it’s 1: The list tells me infinitely many times that it’s not 1. You don’t understand what the infinite list is saying. ---
**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, 10:50am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/215 "2016-10-19T10:50:59Z")
> [@netzweltler](#): > > The actions on the list are taking place between t = 0 and t = 1. t \>= 1 is after the actions. > [@](#): > > between t = 0 and t = 0.9: I move my pen from point 0 to point 0.9 of the number line > between t = 0.9 and t = 0.99: I move my pen from point 0.9 to point 0.99 of the number line > between t = 0.99 and t = 0.999: I move my pen from point 0.99 to point 0.999 of the number line > … > > t \>= 1: which point of the number line my pen is pointing to? (My corrections) As far as I can tell, your pen is moving with constant speed. When t is 1 your pen is at 1. Why don’t you just tell us what you want us to say? ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [October 19, 2016, 2:53pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/216 "2016-10-19T14:53:25Z")
Apparently, **netzweltler** is confused by thinking that Zeno’s paradox still is unsolved. The poor arrow never reaches the tree because it’s confused by thinking it’s in some infinitely long series of actions. ☹ ---
**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, 4:48pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/217 "2016-10-19T16:48:02Z")
> [@naita](#): > > The list tells you where it’s pointing when t\<1. With a bit of expansion you can show that the limit is 1 and the limit is 0.999… so 0.999… = 1. Yes, with a bit of expansion. But the bit of expansion is not on the list. ---
**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, 4:49pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/218 "2016-10-19T16:49:05Z")
> [@Telemark](#): > > You don’t understand what the infinite list is saying. O yes. It says, don’t reach point 1. ---
**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, 4:54pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/219 "2016-10-19T16:54:24Z")
> [@The\_Great\_Unwashed](#): > > As far as I can tell, your pen is moving with constant speed. When t is 1 your pen is at 1. > > Why don’t you just tell us what you want us to say? 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. ---
**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, 4:56pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/220 "2016-10-19T16:56:28Z")
> [@DSYoungEsq](#): > > Apparently, **netzweltler** is confused by thinking that Zeno’s paradox still is unsolved. The poor arrow never reaches the tree because it’s confused by thinking it’s in some infinitely long series of actions. ☹ Zeno’s dichotomy paradox has never been solved. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=10) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=12)