# .999 = 1? **URL:** \ **Category:** Factual Questions\ **Created:** [July 31, 2000, 6:28am UTC](https://boards.straightdope.com/t/999-1/27517 "2000-07-31T06:28:03Z")\ **Posts on this page:** 20\ **Page:** 55
**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:** [January 20, 2013, 9:11pm UTC](https://boards.straightdope.com/t/999-1/27517/1086 "2013-01-20T21:11:36Z")
> [@Freakenstein](#): > > This is what’s My problem with higher math. To Me math is numbers, not letters. Arithmetic is numbers. Math is letters. When you’re too young to understand the difference you get taught in school that math is numbers. A lot of people get stuck there. It’s similar to what happens in English [insert your local language] classes in which teachers have no choice but to give “rules” to make the complexity easier to swallow. People remember the “rules” and never quite grasp that those are stand-ins for nuances that take a lifetime to understand. ---
**Author:** ![zombywoof](https://avatars.discourse-cdn.com/v4/letter/z/9e8a1a/32.png) [@zombywoof](https://boards.straightdope.com/u/zombywoof)\ **Post date:** [January 20, 2013, 9:20pm UTC](https://boards.straightdope.com/t/999-1/27517/1087 "2013-01-20T21:20:35Z")
> [@Freakenstein](#): > > I know the meaning of every single word of this, but together - no idea. The number 4, for example, has two square roots, -2 and 2 (-2\*-2=4; 2\*2=4) - the positive square root is the positive one, 2. ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [January 20, 2013, 9:39pm UTC](https://boards.straightdope.com/t/999-1/27517/1088 "2013-01-20T21:39:50Z")
> [@Freakenstein](#): > > I know what all these words mean separately, I even checked them out to be sure, but I have absolutely no idea what they mean together here. “Unique decimal expansion” means “there is only one way to write this number down in decimal form.” Some numbers do have a unique decimal expansion, but the number 1 is not among them. (To be more explicit: Any number that you can write in decimal form with an infinite string of zeroes at the end, such as 1.000… or 0.5000…, can also be written in decimal form with an infinite string of nines at the end, such as 0.999… or 0.4999…, and both forms are the same number.) ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 20, 2013, 11:51pm UTC](https://boards.straightdope.com/t/999-1/27517/1089 "2013-01-20T23:51:46Z")
> [@Exapno\_Mapcase](#): > > And you show one of them. The difference between 9.999… and 1 is not infinitely small. It is 0. Exactly 0. They are indeed the same number expressed in two different ways. I hope this is obvious, but just in case anyone’s confused: I assume **Exapno** meant to say .999… here and not 9.999… ---
**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:** [January 21, 2013, 12:21am UTC](https://boards.straightdope.com/t/999-1/27517/1090 "2013-01-21T00:21:55Z")
> [@Thudlow\_Boink](#): > > I hope this is obvious, but just in case anyone’s confused: I assume **Exapno** meant to say .999… here and not 9.999… Yes. Sorry. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [June 8, 2014, 7:30am UTC](https://boards.straightdope.com/t/999-1/27517/1091 "2014-06-08T07:30:07Z")
Yes! It’s THIS thread again! :eek: Just thought I’d raise this zombie to point out this article in Slate yesterday: [Does 0.999… = 1? And Are Divergent Series the Invention of the Devil?](http://www.slate.com/blogs/how_not_to_be_wrong/2014/06/06/does_0_999_1_and_are_divergent_series_the_invention_of_the_devil.html) by Jordan Ellenberg. Attempts to splain it clearly. (My assessment: Probably only really clear to someone with reasonable mathematical understanding already. If you feel rather unclear on things like using letters for numbers, or understanding the whole, even when you understand the individual words, as **Freakenstein** admits several posts above, then I think this article won’t help much.) But this article also discusses several well-known divergent series, such as Grandi’s Series ( 1 - 1 + 1 - 1 + 1 - 1 + … ), showing why it might reasonably add up to 1/2, or the series ( 1 + 2 + 4 + 8 + 16 + … ) which might be considered to add up to -1. He also mentions ( 1 + 2 + 3 + 4 + … ) as adding up to -1/12 but doesn’t give any arguments why. I guess that one is left as an exercise for the reader. ---
**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:** [June 9, 2014, 3:04pm UTC](https://boards.straightdope.com/t/999-1/27517/1092 "2014-06-09T15:04:59Z")
I’ve been thinking about this for a while and I’ve come up with a different approach I’d like to try out. Start by looking at the problem in reverse. You start with 1 unit length and devise a procedure for cutting it into equal pieces. Into two pieces you get 1/2 + 1/2 = 1. Into four pieces you get 1/4 + 1/4+ 1/4 +1/4 = 1. The pieces must equal one because you started with 1 and you continue until nothing is left over or goes too far. Obviously, there are an infinite number of ways to do this. This is totally noncontroversial. Get a bit trickier, though. Use a new procedure: Take 1/2 and then 1/2 of the reminder and then 1/2 of the remainder and so on. You get 1/2 + 1/4 + 1/8 + 1/16 … More formally that’s: 1/2[sup]1[/sup] + 1/2[sup]2[/sup] + 1/2[sup]3[/sup] + 1/2[sup]4[/sup] + … Does this equal 1 and if not, why not? That question is Xeno’s paradox. He understood perfectly well that if you start with 1 and then say you’re cutting it into pieces, no matter what you do by definition those pieces must add up to 1. Not just a bit short of one, but 1. Complete and whole and always. His unease lay in his uncomfortableness with infinity. The sum MUST equal one but the procedure was not explainable in terms of what he knew about math. Math didn’t just happen. If you couldn’t explain math, then it wasn’t math. Could it really be that there was a bit left over? If so, what bit was that? If you can define that bit, then you resolve the paradox and have an explanation. You can never define that bit, though. Here’s why. Try a different but equivalent cutting procedure. Start with 2/3 and then take 2/3 of the remainder and then 2/3 of that… or 2/3 + 2/9 + 2/27 + … Formally that’s: 2/3[sup]1[/sup] + 2/3[sup]2[/sup] + 2/3[sup]3[/sup] + 2/3[sup]4[/sup] + … It also MUST add up to 1, but at any given point there will be a different bit left than in the earlier equation. How can you make a claim that x is the piece left over when we’ve shown that x has two different values? It’s gets worse. You can continue this indefinitely. 3/4[sup]1[/sup] + 3/4[sup]2[/sup] + 3/4[sup]3[/sup] + 3/4[sup]4[/sup] 4/5[sup]1[/sup] + 4/5[sup]2[/sup] + 4/5[sup]3[/sup] + 4/5[sup]4[/sup] 5/6[sup]1[/sup] + 5/6[sup]2[/sup] + 5/6[sup]3[/sup] + 5/6[sup]4[/sup] 6/7[sup]1[/sup] + 6/7[sup]2[/sup] + 6/7[sup]3[/sup] + 6/7[sup]4[/sup] 7/8[sup]1[/sup] + 7/8[sup]2[/sup] + 7/8[sup]3[/sup] + 7/8[sup]4[/sup] 8/9[sup]1[/sup] + 8/9[sup]2[/sup] + 8/9[sup]3[/sup] + 8/9[sup]4[/sup] 9/10[sup]1[/sup] + 9/10[sup]2[/sup] + 9/10[sup]3[/sup] + 9/10[sup]4[/sup] And there it is, our old friend 0.999999~. Yes, if you stop at any finite point you will have a tiny bit left out of the unit whole. But it’s a different tiny piece than any of the other procedures give you. That eliminates one of the main arguments that people used above. In addition, thinking about it as a procedure to cut 1 into pieces ensures that you are always looking at 1 as the answer, knowing that if you come up with any other answer you’re the one who must be doing something wrong. You can’t cut a perfect mathematical object into perfect mathematical objects and have a piece disappear into thin air. That’s a much bigger problem than mere infinite sums. One is difficult math; the other is magic. I know which one I reject. I know this isn’t rigorous math. I also know that in other forms of math you can indeed define infinitesimals and use them rigorously. I’m only concerned with this one problem in one familiar field, though: 0.999999~ = 1. Problems often vanish if you can look at them in a new way. I hope this works for some people who previously weren’t sure. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [June 9, 2014, 6:23pm UTC](https://boards.straightdope.com/t/999-1/27517/1093 "2014-06-09T18:23:59Z")
> [@Exapno\_Mapcase](#): > > That question is Xeno’s paradox. He understood perfectly well that if you start with 1 and then say you’re cutting it into pieces, no matter what you do by definition those pieces must add up to 1. Not just a bit short of one, but 1. Complete and whole and always. His unease lay in his uncomfortableness with infinity. The sum MUST equal one but the procedure was not explainable in terms of what he knew about math. Math didn’t just happen. If you couldn’t explain math, then it wasn’t math. Since by definition you can’t “reach” infinity, is that why they now speak of the limit of a series instead? ---
**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:** [June 9, 2014, 10:22pm UTC](https://boards.straightdope.com/t/999-1/27517/1094 "2014-06-09T22:22:01Z")
> [@Lumpy](#): > > Since by definition you can’t “reach” infinity, is that why they now speak of the limit of a series instead? Pretty much. ---
**Author:** ![Pedro](https://avatars.discourse-cdn.com/v4/letter/p/919ad9/32.png) [@Pedro](https://boards.straightdope.com/u/Pedro)\ **Post date:** [June 10, 2014, 7:06pm UTC](https://boards.straightdope.com/t/999-1/27517/1095 "2014-06-10T19:06:55Z")
> [@Exapno\_Mapcase](#): > > Does this equal 1 and if not, why not? That question is Xeno’s paradox. He understood perfectly well that if you start with 1 and then say you’re cutting it into pieces, no matter what you do by definition those pieces must add up to 1. Not just a bit short of one, but 1. Complete and whole and always. His unease lay in his uncomfortableness with infinity. The sum MUST equal one but the procedure was not explainable in terms of what he knew about math. Math didn’t just happen. If you couldn’t explain math, then it wasn’t math. That’s not the paradox, Zeno is deeper than that because it involves motion, i.e, traversing infinite points in a finite amount of time. ---
**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:** [June 11, 2014, 12:12am UTC](https://boards.straightdope.com/t/999-1/27517/1096 "2014-06-11T00:12:35Z")
> [@Pedro](#): > > That’s not the paradox, Zeno is deeper than that because it involves motion, i.e, traversing infinite points in a finite amount of time. Technically, Zeno (brain fart with the X spelling) proposed a series of paradoxes though all of them come down to the same basic problem of dealing with an infinite series, whether it’s creating a sum or moving though an infinite number of points. The one you mention is known as the arrow paradox. The summing of infinite halves is known as the dichotomy paradox. Even that is slightly simplified. See [Zeno’s Paradoxes](http://en.wikipedia.org/wiki/Zeno's_paradoxes). ---
**Author:** ![Pedro](https://avatars.discourse-cdn.com/v4/letter/p/919ad9/32.png) [@Pedro](https://boards.straightdope.com/u/Pedro)\ **Post date:** [June 11, 2014, 2:15am UTC](https://boards.straightdope.com/t/999-1/27517/1097 "2014-06-11T02:15:21Z")
> [@Exapno\_Mapcase](#): > > Technically, Zeno (brain fart with the X spelling) proposed a series of paradoxes though all of them come down to the same basic problem of dealing with an infinite series, whether it’s creating a sum or moving though an infinite number of points. > > The one you mention is known as the arrow paradox. The summing of infinite halves is known as the dichotomy paradox. Even that is slightly simplified. See [Zeno’s Paradoxes](http://en.wikipedia.org/wiki/Zeno's_paradoxes). Maybe you should take a closer look at that (sub-standard) Wiki page. The title of “Paradoxes of Motion” should give you a big clue. [QUOTE=Wikipedia] Zeno’s arguments are often misrepresented in the popular literature. That is, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite.[36] However, none of the original ancient sources has Zeno discussing the sum of any infinite series. Simplicius has Zeno saying “it is impossible to traverse an infinite number of things in a finite time”. This presents Zeno’s problem not with finding the sum, but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a “last event”?[5][6][7][37] [/quote] ---
**Author:** ![j\_sum1](https://avatars.discourse-cdn.com/v4/letter/j/8baadc/32.png) [@j\_sum1](https://boards.straightdope.com/u/j_sum1)\ **Post date:** [June 11, 2014, 5:53am UTC](https://boards.straightdope.com/t/999-1/27517/1098 "2014-06-11T05:53:06Z")
I don’t get your problem here. Not that I have read them in the original Greek, but my understanding is that all of Zeno’s paradoxes deal with the mapping of an infinite number of definable discrete events onto a continuum of time or space or both. I personally like the arrow paradox and Archiles and the tortoise. The point is that the same paradox arises whenever one considers any infinite converging series. The 22 pages of this thread are testimony to the fact that the paradox, whatever context it appears in, is not easily resolvable in the average brain. However, the math geeks have worked out both a resolution and a language to describe it. That doesn’t necessarily make it easier for some people, but it is not an unanswered (or unanswerable) question. **Pedro** , it matters little which particular continuum one refers to – whether time or space or the number line. My limited knowledge of Zeno is that he was referring to both time and space in his various paradoxes. The context of this thread is any continuum. The same reasoning applies with the same result. Kudos to Zeno for creatively posing the question. I don’t think he was necessarily confused by the situations himself. But he was definitely exploring a language to describe such things. That is pretty sophisticated and right where we are at with this thread. ---
**Author:** ![Pedro](https://avatars.discourse-cdn.com/v4/letter/p/919ad9/32.png) [@Pedro](https://boards.straightdope.com/u/Pedro)\ **Post date:** [June 11, 2014, 6:28am UTC](https://boards.straightdope.com/t/999-1/27517/1099 "2014-06-11T06:28:04Z")
> [@j\_sum1](#): > > I don’t get your problem here. My problem is the difference between summing infinitesimal magnitudes, which the greeks had no problem with, and integrating velocity, which the Greeks did not understand and gave rise to Zeno’s paradox, although Aristotles had given a satisfactory answer already. I was just remarking that the convergence of the reciprocal sum of powers of two is not “Zeno’s paradox”. By the way this thread convienced me that .999… = 3/3. I look forward to the sister threads, the polemic “.666… = 2/3?” and the strangely vituperative “.333… = 1/3?”. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [June 11, 2014, 6:49am UTC](https://boards.straightdope.com/t/999-1/27517/1100 "2014-06-11T06:49:20Z")
> [@Pedro](#): > > By the way this thread convienced me that .999… = 3/3. > > I look forward to the sister threads, the polemic “.666… = 2/3?” and the strangely vituperative “.333… = 1/3?”. As **j\_sum1** noted, this thread spent 22 pages proving that 0.999… = 1. Any 7 to 8 pages of it should suffice to prove that 0.333… = 1/3, and any 14 to 15 pages of it should suffice to prove that 0.666… = 2/3. ---
**Author:** ![j\_sum1](https://avatars.discourse-cdn.com/v4/letter/j/8baadc/32.png) [@j\_sum1](https://boards.straightdope.com/u/j_sum1)\ **Post date:** [June 11, 2014, 8:53am UTC](https://boards.straightdope.com/t/999-1/27517/1101 "2014-06-11T08:53:08Z")
Well differentiating distance wrt time gives velocity and integrating velocity gives distance. (Although the Greeks never understood it in those terms. Calculus was still many centuries away.) I am pretty confident that they understood the relationship between the two. I still don’t understand your problem. Are you criticising Zeno for not framing the question in modern terms? His paradoxes relate directly to infinite series which have been thoroughly discussed here. ---
**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:** [June 11, 2014, 1:12pm UTC](https://boards.straightdope.com/t/999-1/27517/1102 "2014-06-11T13:12:37Z")
I hope the [Stanford Encyclopedia of Philosophy](http://plato.stanford.edu/entries/paradox-zeno/) can be considered a rigorous site. ---
**Author:** ![Pedro](https://avatars.discourse-cdn.com/v4/letter/p/919ad9/32.png) [@Pedro](https://boards.straightdope.com/u/Pedro)\ **Post date:** [June 11, 2014, 7:59pm UTC](https://boards.straightdope.com/t/999-1/27517/1103 "2014-06-11T19:59:38Z")
> [@Exapno\_Mapcase](#): > > I hope the [Stanford Encyclopedia of Philosophy](http://plato.stanford.edu/entries/paradox-zeno/) can be considered a rigorous site. I don’t know why you are being so obtuse about this. My comment about the Wiki link was just to say that it is below the generally high standard of Wikipedia, not that it’s not rigorous. Do you not understand that Zeno’s argument centering around motion requires “reaching” the least upper bound of 1/2 + 1/4 + 1/8 + 1/16 + … in a finite amount of time? And how that fundamentally differs from your representation of the paradox of dichotomy? This is the crux of the issue: > [@Exapno Mapcase](#): > > Could it really be that there was a bit left over? Zeno argues (correctly) that there is always a bit left over, because each step in the sequence is assumed to take a finite non-infinitesimal amount of time to complete. A more interesting question would be reconciling this quote with the Banach-Tarski paradox: > [@Exapno Mapcase](#): > > In addition, thinking about it as a procedure to cut 1 into pieces ensures that you are always looking at 1 as the answer, knowing that if you come up with any other answer you’re the one who must be doing something wrong. You can’t cut a perfect mathematical object into perfect mathematical objects and have a piece disappear into thin air. That’s a much bigger problem than mere infinite sums. One is difficult math; the other is magic. I know which one I reject. ---
**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:** [June 11, 2014, 10:07pm UTC](https://boards.straightdope.com/t/999-1/27517/1104 "2014-06-11T22:07:40Z")
> [@Pedro](#): > > I don’t know why you are being so obtuse about this. Right back atcha. I am not arguing what Zeno did or did not say. Have you read any this thread? Anyone who has would understand what I am arguing and who with. Zeno is an irrelevant footnote that I mentioned as an aside. People who argue with the last two centuries of math are the problem, not Zeno. ---
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [June 12, 2014, 2:40am UTC](https://boards.straightdope.com/t/999-1/27517/1105 "2014-06-12T02:40:04Z")
> [@Pedro](#): > > I don’t know why you are being so obtuse about this. . . > > Do you not understand . . . Even if **Exapno** _was_ being obtuse (which he was most certainly not), _you_ are the one who doesn’t understand. You don’t understand that **Exapno** has contributed substantially to this thread and, I might add, done so in his typical patient and perspicuous manner. On the other hand you barge in - a Johnny Come Lately - and contribute four posts tainted, if not made unreadable, by their manifest snark. In other words, what _you_ don’t understand is that people will be turned off by your attitude before they’ve even had a chance to consider whether any of the insights you purport to have might be of interest to them. [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=54) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=56)