# Why doesn't .9999~ = 1? **URL:** \ **Category:** Factual Questions\ **Created:** [June 1, 2003, 11:12pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019 "2003-06-01T23:12:58Z")\ **Posts on this page:** 20\ **Page:** 4
**Author:** ![dejahma](https://avatars.discourse-cdn.com/v4/letter/d/839c29/32.png) [@dejahma](https://boards.straightdope.com/u/dejahma)\ **Post date:** [June 2, 2003, 4:48am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/61 "2003-06-02T04:48:02Z")
ok x-ray here is the answer, in the words of our poet laureate Billy Collins: Of all the questions you might want to ask about angels, the only one you ever hear is how many can dance on the head of a pin. No curiosity about how they pass the eternal time besides circling the Throne chanting in Latin or delivering a crust of bread to a hermit on earth or guiding a boy and girl across a rickety wooden bridge. Do they fly through God’s body and come out singing? Do they swing like children from the hinges of the spirit world saying their names backwards and forwards? Do they sit alone in little gardens changing colors? What about their sleeping habits, the fabric of their robes, their diet of unfiltered divine light? What goes on inside their luminous heads? Is there a wall these tall presences can look over and see hell? If an angel fell off a cloud, would he leave a hole in a river and would the hole float along endlessly filled with the silent letters of every angelic word? If an angel delivered the mail, would he arrive in a blinding rush of wings or would he just assume the appearance of the regular mailman and whistle up the driveway reading the postcards? No, the medieval theologians control the court. The only question you ever hear is about the little dance floor on the head of a pin where halos are meant to converge and drift invisibly. It is designed to make us think in millions, billions, to make us run out of numbers and collapse into infinity, but perhaps the answer is simply one: one female angel dancing alone in her stocking feet, a small jazz combo working in the background. She sways like a branch in the wind, her beautiful eyes closed, and the tall thin bassist leans over to glance at his watch because she has been dancing forever, and now it is very late, even for musicians. and so the answer appears to be “one” I love this guy ---
**Author:** ![Phage](https://avatars.discourse-cdn.com/v4/letter/p/a4c791/32.png) [@Phage](https://boards.straightdope.com/u/Phage)\ **Post date:** [June 2, 2003, 4:49am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/62 "2003-06-02T04:49:26Z")
.99999… cannot be equal to one, as has been proven many times over in this thread. However, the attempt to prove .99999… = 1 that disturbs me the most is the argument that there is no number between .9999… and 1. Simply put, .9999… is the number that is closest to 1, yet still less than 1. This theoretical number can be considered abstractly; just because it does not conform to the properties of regular numbers does not disprove its existence. For every normal number there is always one larger, but there is no number larger than infinity. Does that mean that infinity cannot exist? For the record, 1/infinity DOES NOT equal zero. Think about it people, if you divide it an infinite number of times it will become infinitely small, but the mere fact that it is infinitely small requires that it exist. Therefore, it cannot equal zero. **Donut** offered probably the most reasonable reason for one to consider .999… equal to 1, in saying that the number is assumed to refer to the limit of a sequence. However, you should be aware that the concept of the limit is used so that a finite answer can be determined that while not being absolutely accurate, is close enough for any use. The sequence .999… never reaches 1, yet it is as close as anything can come. By considering .999… as 1 you can come up with a useful answer, in fact an answer that is the closest to the real answer as anything can come. **Exapno Mapcase** , limits and infinitesimals ARE NOT incompatible! The very IDEA of the limit was made to deal with the problems infinitesimals caused in calculations. Using the limit concept is merely a way of introducing acceptable error in order to make calculations possible. BTW, 1/3 does not really equal .3333…, but it is as close as we can get. 10 does not divide evenly by 3, so there is no way to completely accurately express such a number using base 10. .333… is infinitely close, so we can make-do with it. **Q.E.D.** I would suggest that you give a cursory examination of the concept of a limit, and then get out your hacksaw. ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [June 2, 2003, 6:55am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/63 "2003-06-02T06:55:27Z")
**Phage** , May I suggest that you take another look at the concept of limit. The number 0.999… is defined to be a limit and that limit is equal to 1.0. Limits allow us to determine what happens when we add an infinite number of terms. The concept is exact and does not disregard an error. If you think an error is disregarded, please express the error. This is the same as Zeno’s paradox (if you expressed the question in base two). Would you say you cannot move to a new location because you must travel 1/2 the distance and then 1/4 the distance and then 1/8 the distance and then . . . ? * * * Virtually yours, **DrMatrix** - If i’ve told you once, I’ve told you 0.999… times. ---
**Author:** ![3\_14159265358979323846](https://avatars.discourse-cdn.com/v4/letter/3/eada6e/32.png) [@3\_14159265358979323846](https://boards.straightdope.com/u/3_14159265358979323846)\ **Post date:** [June 2, 2003, 6:59am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/64 "2003-06-02T06:59:35Z")
> [@](#): > > \*Originally posted by x-ray vision \* > > > [@](#): > > > > _Originally posted by 3\_14159265358979323846_ > > No, you can’t. That’s just it. You’ll die, eventually. > > C’mon now. In your example, you stated that you could “go on [adding 9s] forever”, and not get zero. But that isn’t true. You _can’t_ go on forever. You’ll eventually assume that your case is proven and quit, as you did, or you’ll expire from sheer exhaustion. In either case, you haven’t proven that .999… does not equal 1. You’ve proven that .9 does not equal 1, and you’ve proven that .99 does not equal 1. You’ve proven that .999 doesn’t not equal 1. You might even prove that .9999999999999999999999999999999999999999999999999999999999999999999 does not equal 1. But that doesn’t prove that .999… (infinitely repeating) does not equal zero. ---
**Author:** ![something\_clever](https://avatars.discourse-cdn.com/v4/letter/s/3bc359/32.png) [@something\_clever](https://boards.straightdope.com/u/something_clever)\ **Post date:** [June 2, 2003, 8:18am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/65 "2003-06-02T08:18:00Z")
> [@](#): > > \*Originally posted by 3\_14159265358979323846 \* > \*\*No, you _can’t_. That’s just it. You’ll die, eventually. > \*\* Isnt this why we invented computers… ? 😃 ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [June 2, 2003, 11:55am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/66 "2003-06-02T11:55:45Z")
> [@](#): > > _Originally posted by x-ray vision_ > **That’s a contradiction I just can’t grasp.** Hmmm…let me try rephrasing it a bit. If 0.999… and 1 _aren’t_ equal, then 0.999… must be less than 1. That means that 1-0.999… must be a positive number. Call this number y. Then 1/y is also a positive number. In particular it’s finite. So 1/y must be less than 1000…000 for some number of zeroes. Let’s say for sake of argument that 1/y is less than 10000. Then y is greater than 0.0001. But that’s impossible, because 1-0.999… \< 1-0.9999 = 0.0001. Okay, maybe 1/y is less than 100000 instead. Then y is greater than 0.00001. But that’s impossible too, because 1-0.999… \< 1-0.99999 = 0.00001. See how it works? No matter how many zeroes I use, 1/y _can’t_ be less than 1000…000 because y is demonstrably less than 0.000…001. But if y is positive than 1/y _must_ be less than 1000…000 for some number of zeroes. That’s a contradiction, and that’s why 0.999… and 1 must be the same number. Now, on to **Phage** … > [@](#): > > \*Originally posted by Phage \* > **.99999… cannot be equal to one, as has been proven many times over in this thread.** What thread have you been reading? Seriously. I was pretty sure _this_ thread consisted mostly of people trying to explain to **x-ray vision** why 0.9999… _did_ equal 1. Where are these mystery proofs that they’re unequal? > [@](#): > > \*\* > However, the attempt to prove .99999… = 1 that disturbs me the most is the argument that there is no number between .9999… and 1. Simply put, .9999… is the number that is closest to 1, yet still less than 1. \*\* Okay, then. Suppose 0.999…=x. Answer me this: what is (1+x)/2? Is (1+x)/2 bigger than one? Less than one? Equal to one? Is (1+x)/2 bigger than x? Less than x? Equal to x? I can guarantee you that anything you choose either leads to the conclusion that x=1, or else leads to a contradiction. But then we get to this gem… > [@](#): > > \*\* > This theoretical number can be considered abstractly; just because it does not conform to the properties of regular numbers does not disprove its existence. \*\* So it doesn’t matter to you that I can prove, and have proved, that _there is no real number between x and 1_? You’ll just claim that there’s some new kind of Phage-number there instead? > [@](#): > > \*\* > For every normal number there is always one larger, but there is no number larger than infinity. Does that mean that infinity cannot exist? \*\* It means that infinity is not a “normal number”. Which it isn’t. Infinity is not an integer, or a rational number, or a real number. It’s something else. But it’s not a real number, because as you’ve just proved it doesn’t satisfy one of the properties of a real number. Now x=0.999… and 1, on the other hand, _are_ real numbers. So they have to satisfy the properties of real numbers. In particular, _if_ they’re different then there has to be a third number between them because _that’s a property of the real numbers_. But (and yes I’m shouting) _ **there is no real number between 0.999… and 1** _. I have offered a proof of this fact. You have not disproved it. … I’m getting testy again, so it’s probably a good thing that I’m going to be busy for the next two days. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 2, 2003, 1:45pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/67 "2003-06-02T13:45:18Z")
The real numbers are dense, meaning that if x and y are distinct real numbers with x \< y, there’s a real number z such that x \< z \< y. (x + y)/2 is a good choice. Did you know that a bounded mononotically increasing sequence converges to its least upper bound? ---
**Author:** ![kabbes](https://avatars.discourse-cdn.com/v4/letter/k/e99b99/32.png) [@kabbes](https://boards.straightdope.com/u/kabbes)\ **Post date:** [June 2, 2003, 2:21pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/68 "2003-06-02T14:21:16Z")
You know, I once came across somebody who refused to accept the existence of irrational numbers too. None of the standard proofs I knew would phase him, because the belief that everything was rational just ran too deep. It happens. These threads rumble on and on long after everyone who can be convinced by logical step-by-step proofs is convinced. There are always just one or two folks who insist on ignoring every single post that shows them to be wrong and repeating the same old many-refuted mantra again and again. Whatcha gonna do? Still, for completeness’ sake, here’s another method that helps some people with the intuitive feel for _why_ 0.999… = 1: Don’t get so hung up on base 10. Consider base 12 for a second. 0.4[sub]12[/sub] + 0.4[sub]12[/sub] + 0.4[sub]12[/sub] = 1. And yet 0.4[sub]12[/sub] is 0.333… in base-10. Now, since the basis is irrelevant to the calculation, it follows by replacing each number with its base-10 equivalent that 0.333… + 0.333… + 0.333… = 1. But in any basis greater than base-10, 0.333… + 0.333… + 0.333… = 0.999… so by equating the two RH sides, 0.999… in base 10 = 1. No infinities, no calculation that may be viewed as suspicious, just a simple shift of perspective from one basis to another. The specific problem of 0.999… is purely an artefact of the decimal basis. It’s artificial. If you claim that there is something between 0.999… and 1, you have to consider what that something would be once you switch to a basis that removes the repeating digit. pan ---
**Author:** ![InLikeFlynn](https://avatars.discourse-cdn.com/v4/letter/i/f19dbf/32.png) [@InLikeFlynn](https://boards.straightdope.com/u/InLikeFlynn)\ **Post date:** [June 2, 2003, 3:07pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/69 "2003-06-02T15:07:55Z")
> [@](#): > > \*Originally posted by Q.E.D. \* > \*\*Well, I tell you what. If you can find ONE source with correct proof that .9999… != 1, i’ll eat my computer. \*\* I thought you were trying to argue the point that .999 = 1! I **correctly** say that it doesn’t!! Was that a typo or are you on my side now? 😕 There is no such thing as an angel!! How many “trollers” does it take to kill this board?? only one named **dejahma** ---
**Author:** ![robertliguori](https://avatars.discourse-cdn.com/v4/letter/r/46a35a/32.png) [@robertliguori](https://boards.straightdope.com/u/robertliguori)\ **Post date:** [June 2, 2003, 3:25pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/70 "2003-06-02T15:25:27Z")
Two angels. Because only Satan and Azraephale know how to dance, and neither of them are constrained by physics. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [June 2, 2003, 3:37pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/71 "2003-06-02T15:37:00Z")
> [@](#): > > \*Originally posted by InLikeFlynn \* > \*\* I thought you were trying to argue the point that .999 = 1! > > I **correctly** say that it doesn’t!!\*\* I was arguing no such point. .999 does _not_ equal one. but .999… _does_. Also, I do _strongly_ suggest you refrain from using the word “troll” in this forum. Accusations of trolling can get you banned. please see the sticky concerning this topic at the top of the thread list. ---
**Author:** ![Big-Ole-Steve](https://avatars.discourse-cdn.com/v4/letter/b/4da419/32.png) [@Big-Ole-Steve](https://boards.straightdope.com/u/Big-Ole-Steve)\ **Post date:** [June 2, 2003, 3:43pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/72 "2003-06-02T15:43:47Z")
**Q.E.D.** I think **InLikeFlynn** was confused by your use of “!=” to represent “does not equal.” Steve ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [June 2, 2003, 4:02pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/73 "2003-06-02T16:02:09Z")
> [@](#): > > \*Originally posted by Big-Ole-Steve \* > **Q.E.D. I think InLikeFlynn was confused by your use of “!=” to represent "does not equal."** Possibly. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [June 2, 2003, 4:44pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/74 "2003-06-02T16:44:06Z")
My fellow mathematicians, let me point something out. The concept of infinity _is not necessary_ for addressing this problem. You do not need infinity to define the natural numbers, or the real numbers, or a sequence, or a series, or a limit, or the ellipsis notation. It’s very tempting to throw it in, but you do not need it. Furthermore, I strongly believe that if you use it, you do more harm than good; infinity is such a loaded, ambiguous, counter-intuitive concept that statements cannot help but come out wrong. So saying things like: 0.999… is “0.” with infinity 9’s after it. 1/infinity = 0. After an infinite amount of time… Don’t really help explain things. Maybe it’s just me, but I don’t trust people to understand how to work with infinity without a lot of practice, and I think it would be better if it were avoided altogether. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [June 2, 2003, 4:55pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/75 "2003-06-02T16:55:02Z")
I think **RealityChuck** did that quite nicely, **Achernar** , but that didn’t seem to help either. I give up. I can’t explain it any better than I and everyone else already has. ---
**Author:** ![DarrenS](https://avatars.discourse-cdn.com/v4/letter/d/977dab/32.png) [@DarrenS](https://boards.straightdope.com/u/DarrenS)\ **Post date:** [June 2, 2003, 5:06pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/76 "2003-06-02T17:06:46Z")
Kudos to **Achernar** , **QED** , **ultrafilter** et al for being patient enough to explain (and re-explain) something that has been done to death on this board, and a million other places on the web (even in the sci.math FAQ I believe). It seems to me the problem here is the “non-believers” are tacitly insisting that mathematics conform to their intuition. “But…but…how can an infinite series of 9s after the point equal 1, when **in practice** I’m used to any finite amount of 9s being less than 1?” It reminds me of an amusing quote I once read - “a googol is already larger than most people’s concept of infinity.” The analogy to Zeno’s paradoxes is a good one - they still fascinate people even today because they show how flawed our intuition really is when considering the infinite. ---
**Author:** ![Terminus\_Est](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/terminus_est/32/3087_2.png) [@Terminus\_Est](https://boards.straightdope.com/u/Terminus_Est)\ **Post date:** [June 2, 2003, 5:25pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/77 "2003-06-02T17:25:26Z")
This question is beginning to outrank “the third -gry word”, “echoing quack”, “missing dollar”, and “Monty Hall” in terms of sheer tiresomeness. Every nitwit who thinks they understand math wants to argue the point. It’s just an accident of the decimal system. If you switch to binary, then 0.111… = 1. In octal, 0.777… = 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:** [June 2, 2003, 5:34pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/78 "2003-06-02T17:34:04Z")
I’m seriously interested in the psychology of those who are arguing that .9999… does not equal zero. Arguments can be made in some fields, but not in math. In math to say something is wrong you must either provide a proof or a counter-example. And in this case neither can be done. This equality is a fact. It is subject to mathematical proof as easily as **RealityChuck** showed. Every mathematician in the world accepts and understands this result. It can be a difficult fact to grasp. Despite what **Achernar** says it most certainly involves grasping the concept of infinity (even if it not technically necessary to the proof). If you don’t understand this you come up with absurd statements like **Phage’s** “Simply put, .9999… is the number that is closest to 1, yet still less than 1.” But the very definition of the real number line states that between any two numbers, no matter how close together, you can place an infinity of other numbers. That’s the difference between finite numbers and the infinity of numbers on the real number line. (The irrationals are a higher-order infinity than the rationals.) There is also the matter of the definition of a limit, which most emphatically does not mean coming closer and closer to but never reaching but means exactly equal to.) Again, this is where the answer to Zeno’s paradox comes in.) You can’t grasp this without making the mental leap to infinity as unending rather than growing larger and larger. So what you wind up with is a non-mathematician attempting to argue, on the basis of misunderstood math, that a mathematical proof is incorrect. And that’s the point I can’t grasp. I can see asking for help understanding this subtlety. I can’t wrap my head around your standing up and saying that every mathematician in the world is wrong just because you don’t understand something. That’s as alien to me as .999999… = 0 seems to be to those of you arguing against it. So I’m seriously asking for help in comprehending a subtlety. Why are you making these arguments instead of asking for help in understanding? ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [June 2, 2003, 5:47pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/79 "2003-06-02T17:47:37Z")
> [@](#): > > \*Originally posted by Q.E.D. \* > I think **RealityChuck** did that quite nicely, **Achernar** , but that didn’t seem to help either. Nothing personal, but I don’t like the proof that **RealityChuck** gave, and here’s why. As **Jabba** correctly pointed out, although there is no overt Analysis going on, the concept of a limit is implicit in the ellipsis notation. Instead of performing arithmetic operations on normal, real numbers, you’re performing them on infinite series. This is justified _only if_ you have proven that these series converge. This is fine, and the proof that they converge is simple, but in doing that proof, you’ve basically done exactly what it is you’re trying to prove anyway. That is, when you prove that the series associated with 0.999… converges, you’re one step away from proving that its limit is 1. There’s then no need to go into the arithmetic of that classic proof. **Exapno Mapcase** , I disagree with you on one point, ie, the point you disagree with me on. The concept of infinity is not necessary to grasp in order to understand a limit. I would be interested in hearing an argument to the contrary, though. ---
**Author:** ![SCSimmons](https://avatars.discourse-cdn.com/v4/letter/s/e495f1/32.png) [@SCSimmons](https://boards.straightdope.com/u/SCSimmons)\ **Post date:** [June 2, 2003, 5:50pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/80 "2003-06-02T17:50:52Z")
> [@](#): > > \*Originally posted by Phage \* > \*\*.99999… cannot be equal to one, as has been proven many times over in this thread. However, the attempt to prove .99999… = 1 that disturbs me the most is the argument that there is no number between .9999… and 1. Simply put, .9999… is the number that is closest to 1, yet still less than 1. This theoretical number can be considered abstractly; just because it does not conform to the properties of regular numbers does not disprove its existence. \*\* Yeah, but when it doesn’t have properties like being able to be added/subtracted/multiplied/divided with other numbers, it’s real hard to call it a number. Assumption 1: .999… \< 1 Assumption 2: There is no number n such that .999… \< n and n \< 1 Suppose: 1 + .999… \<2. Then (1 + .999…)/2 \<1 Therefore, by Assumption 2, (1 + .999…)/2 \<= .999… (Otherwise, (1+.999…)/2 is a number between .999… and 1, which does not exist.) However, if (1 + .999…)/2 = .999…, then 2\*(1 + .999…)/2 = 2 \* .999… Which gives 1 + .999… = .999… + .999… 1 = .999… Which contradicts Assumption 1. Therefore, (1 + .999…)/2 \< .999… Doubling both sides again, 1 + .999… \< .999… + .999… 1 \< .999… Which again contradicts assumption 1. The only conclusion I can see to draw is that .999… cannot be meaningfully combined in arithmetical operations with other numbers. Great, but I think I’ll stick with modern limit mathematics, which both permits these types of numbers and is internally consistent. [Previous page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=3) [Next page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=5)