# 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:** 15
**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 24, 2016, 5:40pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/281 "2016-10-24T17:40:59Z")
> [@Exapno\_Mapcase](#): > > For the ten millionth time. Each of those points are finite. Just like the n you used earlier. You are trying to make a point about infinity by using a series of finite points. And you are failing. That should tell you that your tools are inadequate to the task. You should therefore immediately stop what you are doing and look for tools that are adequate to the task. The limit, e.g. > > Using the right tools is critical to all of math. You cannot use Euclid’s geometry to determine areas on the surface of the earth: it is the wrong tool for the job, no matter how good and useful it is for other tasks. > > You have been told repeatedly that you are going about the task in the wrong way. Arguing repeatedly in return that your method gives you a different answer is utterly futile. Of course it gives you the wrong answer. You are using the wrong tool. The discussion ends right there. **netzweltler** , please pay attention to this. If this were Ancient Greece, your “intuitive” approach to the apparent contradictions and paradoxes of infinities might be right up there on the leading edge of mathematics & philosophy. But many great minds have brought us a long way since then, among other things developing rigorous methods to deal with infinities in various contexts. By analogy, you’re arguing on about the same level as “if we evolved from monkeys, why are there still monkeys”. ---
**Author:** ![RitterSport](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rittersport/32/6326_2.png) [@RitterSport](https://boards.straightdope.com/u/RitterSport)\ **Post date:** [October 24, 2016, 6:08pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/282 "2016-10-24T18:08:20Z")
> [@netzweltler](#): > > Ok. But what kind of problem is this a solution for? The problem of notation. 1/3 really is 0.3333…, but the issue you may have is that 0.3333… seems to go on forever. However, 1/3 in base 3 is written as 0.1. No problems with notation. So, rather than 0.333…, 0.666…, 0.999… when writing 1/3, 2/3, 3/3 in base 10, let’s instead write it in base 3: 0.1, 0.2, 1. No problem with notation at all, and the “problem” with infinite sequences goes away. And, since we all agree (I hope) that 1/3 in base 3 is 0.1, then we all see that there’s no issue with 0.99999… = 1, since we all realize it’s just a notation problem, not a mathematical problem. What’s 0.333…+0.333…+0.333… equal to? Well, in base three, 0.1 + 0.1 + 0.1 = 1. The same number in different bases are still the same number, so 0.333…+0.333…+0.333… = 0.999… = 1. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 24, 2016, 6:46pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/283 "2016-10-24T18:46:24Z")
> [@netzweltler](#): > > The infinitely repeated answer is: > > For each of the infinitely many lines is valid, that I get closer to 1/3 but don’t reach 1/3. Tell me if you can find a line for which this does not hold true. Of course you reach 1/3 when we consider all the lines … you ask to pick just one line, but that’s not how infinity works … your “for each” is invalid, it should be “for each and every” … whichever line we find will only represent a very very large finite number. Well, since you’ve declared to hold to your flawed logic and will be repeating infinitely, I have no choice but to answer: Beetlejuice Beetlejuice Beetlejuice ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 24, 2016, 9:11pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/284 "2016-10-24T21:11:09Z")
> [@netzweltler](#): > > The infinitely repeated answer is: > > For each of the infinitely many lines is valid, that I get closer to 1/3 but don’t reach 1/3. Tell me if you can find a line for which this does not hold true. Can you tell us what other number than 1/3 can be expressed by 0.333~? If it isn’t 1/3, then what do you think it is? Can you construct this point? The problem is that, if you could, you would also be showing that 0.000~ is \> 0, and that’s nonsense. ---
**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 24, 2016, 10:57pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/285 "2016-10-24T22:57:02Z")
> [@RitterSport](#): > > The problem of notation. 1/3 really is 0.3333…, but the issue you may have is that 0.3333… seems to go on forever. However, 1/3 in base 3 is written as 0.1. No problems with notation. > > So, rather than 0.333…, 0.666…, 0.999… when writing 1/3, 2/3, 3/3 in base 10, let’s instead write it in base 3: > > 0.1, 0.2, 1. > > No problem with notation at all, and the “problem” with infinite sequences goes away. And, since we all agree (I hope) that 1/3 in base 3 is 0.1, then we all see that there’s no issue with 0.99999… = 1, since we all realize it’s just a notation problem, not a mathematical problem. > > What’s 0.333…+0.333…+0.333… equal to? Well, in base three, 0.1 + 0.1 + 0.1 = 1. The same number in different bases are still the same number, so 0.333…+0.333…+0.333… = 0.999… = 1. FTR, I found your argument cogent and utterly persuasive. ---
**Author:** ![RitterSport](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rittersport/32/6326_2.png) [@RitterSport](https://boards.straightdope.com/u/RitterSport)\ **Post date:** [October 24, 2016, 11:14pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/286 "2016-10-24T23:14:11Z")
> [@The\_Great\_Unwashed](#): > > FTR, I found your argument cogent and utterly persuasive. Well, we were both born, SDMB-wise, in March of 2002, so, as twins, it only makes sense that we’d find each other persuasive. ---
**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 24, 2016, 11:58pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/287 "2016-10-24T23:58:45Z")
> [@RitterSport](#): > > Well, we were both born, SDMB-wise, in March of 2002, so, as twins, it only makes sense that we’d find each other persuasive. Now you’re just being an idiot. (Don’t make me post a smiley.) ---
**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 25, 2016, 12:05am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/288 "2016-10-25T00:05:01Z")
My join date is also shown as March 2002. But I really joined several months earlier. All my early posts and my join date were lost in the winter of missed content. So I was born again in March 2002. Maybe you guys were as well. Sometimes facts aren’t facts. ---
**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 25, 2016, 12:15am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/289 "2016-10-25T00:15:40Z")
> [@Exapno\_Mapcase](#): > > My join date is also shown as March 2002. But I really joined several months earlier. All my early posts and my join date were lost in the winter of missed content. So I was born again in March 2002. Maybe you guys were as well. Sometimes facts aren’t facts. My point entirely! Before the great virtual death of 2002 I had been very witty and universally liked. It’s had having to drag your zombied corpse back from oblivion. ---
**Author:** ![RitterSport](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rittersport/32/6326_2.png) [@RitterSport](https://boards.straightdope.com/u/RitterSport)\ **Post date:** [October 25, 2016, 12:19am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/290 "2016-10-25T00:19:02Z")
> [@The\_Great\_Unwashed](#): > > Now you’re just being an idiot. > > (Don’t make me post a smiley.) Hey, I resemble that remark! This thread has taken a weird turn. ---
**Author:** ![TroutMan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/troutman/32/6721_2.png) [@TroutMan](https://boards.straightdope.com/u/TroutMan)\ **Post date:** [October 25, 2016, 12:38am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/291 "2016-10-25T00:38:14Z")
If you start moving your pencil on the calendar, you’ll reach my join date at t=6 years. ---
**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 25, 2016, 12:47am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/292 "2016-10-25T00:47:13Z")
Ahem! 5.999… years! ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [October 25, 2016, 3:31am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/293 "2016-10-25T03:31:46Z")
> [@](#): > > Join Date: Mar 2002 That’s actually kinda creepy … ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 25, 2016, 8:02am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/294 "2016-10-25T08:02:19Z")
> [@RitterSport](#): > > The problem of notation. 1/3 really is 0.3333…, but the issue you may have is that 0.3333… seems to go on forever. However, 1/3 in base 3 is written as 0.1. No problems with notation. > > So, rather than 0.333…, 0.666…, 0.999… when writing 1/3, 2/3, 3/3 in base 10, let’s instead write it in base 3: > > 0.1, 0.2, 1. > > No problem with notation at all, and the “problem” with infinite sequences goes away. And, since we all agree (I hope) that 1/3 in base 3 is 0.1, then we all see that there’s no issue with 0.99999… = 1, since we all realize it’s just a notation problem, not a mathematical problem. > > What’s 0.333…+0.333…+0.333… equal to? Well, in base three, 0.1 + 0.1 + 0.1 = 1. The same number in different bases are still the same number, so 0.333…+0.333…+0.333… = 0.999… = 1. If you avoid using infinite sequences then the problem is gone. I agree. This doesn’t mean that the “problem” with infinite sequences goes away. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 25, 2016, 8:16am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/295 "2016-10-25T08:16:49Z")
> [@watchwolf49](#): > > Of course you reach 1/3 when we consider all the lines … you ask to pick just one line, but that’s not how infinity works … your “for each” is invalid, it should be “for each and every” … whichever line we find will only represent a very very large finite number. How do you distinguish “for each” and “for each and every”? By the way, whichever digit we find in 0.999… will only represent a finitely indexed digit - even if there are infinitely many of them. > [@watchwolf49](#): > > Well, since you’ve declared to hold to your flawed logic and will be repeating infinitely, I have no choice but to answer: > > Beetlejuice > Beetlejuice > Beetlejuice ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 25, 2016, 8:21am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/296 "2016-10-25T08:21:09Z")
> [@Trinopus](#): > > Can you tell us what other number than 1/3 can be expressed by 0.333~? > > If it isn’t 1/3, then what do you think it is? Can you construct this point? > > The problem is that, if you could, you would also be showing that 0.000~ is \> 0, and that’s nonsense. As soon as we can define a point 0.333~ on the number line we are able to define what number can be expressed by 0.333~. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 25, 2016, 8:51am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/297 "2016-10-25T08:51:19Z")
> [@watchwolf49](#): > > Of course you reach 1/3 when we consider all the lines … […] It is true that we complete 0.333… if we consider all the lines. This doesn’t mean that we reached 1/3, because none of the infinitely many lines shows how we get there. ---
**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 25, 2016, 9:15am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/298 "2016-10-25T09:15:50Z")
**netzweltler** , If I understand your argument correctly (one cannot place 0.999… on the number line because it isn’t equal to any of {0.9, 0.99, 0.999, etc}) then similarly one cannot place any other repeating decimal nor any irrational number. But as shown by **RitterSport** if we consider bases other than 10, a given repeating decimal can be transformed into a non-repeating decimal and easily placed on the number line. Numbers cannot pop into and out of existence according to how many fingers we have. π, e, √2 and at least a couple more irrational numbers are perfectly well defined despite your supposed limitation. PS: Did you duck the question: “What is the decimal expansion of 1/3?” ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [October 25, 2016, 9:39am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/299 "2016-10-25T09:39:46Z")
> [@The\_Great\_Unwashed](#): > > **netzweltler** , If I understand your argument correctly (one cannot place 0.999… on the number line because it isn’t equal to any of {0.9, 0.99, 0.999, etc}) then similarly one cannot place any other repeating decimal nor any irrational number. > > But as shown by **RitterSport** if we consider bases other than 10, a given repeating decimal can be transformed into a non-repeating decimal and easily placed on the number line. Numbers cannot pop into and out of existence according to how many fingers we have. > > π, e, √2 and at least a couple more irrational numbers are perfectly well defined despite your supposed limitation. > PS: Did you duck the question: “What is the decimal expansion of 1/3?” I don’t think so. See my answer to **Trinopus’** question. All I have seen what **RitterSport** has shown is that 1/3 can be written 0.1 in base 3. So what? Both notations are well-defined points on their number lines. ---
**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 25, 2016, 10:21am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/300 "2016-10-25T10:21:13Z")
> [@netzweltler](#): > > I don’t think so. See my answer to **Trinopus’** question. [Post #296?](http://boards.straightdope.com/sdmb/showpost.php?p=19726376&postcount=296) – with all due respect, you ducked the question there too. I think I see why… > [@netzweltler](#): > > All I have seen what **RitterSport** has shown is that 1/3 can be written 0.1 in base 3. So what? Both notations are well-defined points on their number lines. The so what is : 1/3 = 0.1 (base 3) – a point easily placed on the number line and 1/3 = 0.333… (base 10) – a point you claim is impossible to place on the number line So, now it is quite pivotal that you state in plain and direct language what you think the decimal expansion of 1/3 is, because clearly you reject the second of these two assertions _or_ you wish to claim that numbers flip-flop in-and-out of existence according to which base we choose to name them in. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=14) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=16)