# 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:** 13
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [June 4, 2003, 9:33am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/241 "2003-06-04T09:33:39Z")
> [@](#): > > \*Originally posted by Achernar \* > \*\*I humbly request a cite, preferably from a text on Analysis or some other math of a similar level of advancement. > > I am not saying that the notation 0.333… represents any finite number of 3’s. I am saying that the notation 0.333… represents the limit of a sequence whose Nth term is 0.333…333, where the number of 3’s is N. In your mind, this may be the same thing as an infinite number of 3’s, but in my mind, it’s not. Specifically, exactly what “an infinite number of 3’s” means mathematically is not well defined. \*\* Well in my mind infinite means, not-finite/unbounded, but I am not a mathematician by training and my university level maths training only extends to what is relevant to physics and not this sort of number theory. ---
**Author:** ![kabbes](https://avatars.discourse-cdn.com/v4/letter/k/e99b99/32.png) [@kabbes](https://boards.straightdope.com/u/kabbes)\ **Post date:** [June 4, 2003, 10:26am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/242 "2003-06-04T10:26:48Z")
Right, but 0.33333333333… where the number of threes is not finite or unbounded has no definition in the usual sense. We can only make sense of it by noting that: 0.1 = 10[sup]-1[/sup]; and 0.01 = 10[sup]-2[/sup] and so on, so that: 0.xyz = x.10[sup]-1[/sup] + y.10[sup]-2[/sup] + z.10[sup]-3[/sup]. It is then clear that 0.333… = 3×10[sup]-1[/sup] + 3×10[sup]-2[/sup] + 3×10[sup]-3[/sup] + … In other words, it’s not just that 0.333… can be represented as an infinite sum, it’s that the (most obvious) way that 0.333… is **defined** is as an infinite sum. Numbers are much trickier things than people give them credit for. There’s a reason that number theory generally offers the simplest sounding proposals with the most difficult proofs. pan ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [June 4, 2003, 10:31am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/243 "2003-06-04T10:31:35Z")
Well, I certainly do remebr doing series and thereare ways of represtening, defining and evalutaing the sum of an infinte series, where that sum is a real number. ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [June 4, 2003, 10:33am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/244 "2003-06-04T10:33:49Z")
Anyway, I suppose saying 0.333… has infinite 3’s is pretty meaningless, but saying it is the sum of an infinite series is meaningful. ---
**Author:** ![kabbes](https://avatars.discourse-cdn.com/v4/letter/k/e99b99/32.png) [@kabbes](https://boards.straightdope.com/u/kabbes)\ **Post date:** [June 4, 2003, 10:50am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/245 "2003-06-04T10:50:18Z")
Exactly. ---
**Author:** ![Tarantula](https://avatars.discourse-cdn.com/v4/letter/t/e274bd/32.png) [@Tarantula](https://boards.straightdope.com/u/Tarantula)\ **Post date:** [June 4, 2003, 12:25pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/246 "2003-06-04T12:25:44Z")
Just for all the idiots that don’t understand humour… QED in Mathematics stands for “Quod Erat Demonstratum” - “Because it was shown”. Quod Erat Demonstrandum “As it was to have been shown” is a latinist joke… tsk ! Sometimes you guys take all the fun out of being a latinist… Although \*\* Kabbes \*\* was funny - > [@](#): > > Believe me, mathematicians know what QED means - obviously, Kabbes, you’re not a mathematician then… Har har… ---
**Author:** ![RealityChuck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/realitychuck/32/195_2.png) [@RealityChuck](https://boards.straightdope.com/u/RealityChuck)\ **Post date:** [June 4, 2003, 12:26pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/247 "2003-06-04T12:26:22Z")
> [@](#): > > \*Originally posted by Phage \* > \*\*[(1/3) is represented in base-10 as (0.333…). No matter how far you go, the next digit will have to be a 3. The point is that you can never have enough 3s; you will never reach the infinite number of 3s you need. \*\* :rolleyes: How can anyone contradict himself so blatantly and not even notice? You say “no matter how far you go, the next digit will be three” and in the very next sentence say “you can never have enough 3s.” But, as you said, every digit, no matter how far you go, is a three. How, then could you “never have enough 3s?” You have an infinite number of threes, don’t you? If you’re short one, you just keep going; you can’t run out. So in two adjacent sentences you say, “you have an infinite number of threes” and “you never have enough.” Which is it? If you **don’t** have enough threes, it means that at least one digit **isn’t** three. But you’ve already stated “no matter how far you go, the next digit will be three.” Another contradiction. It’s pretty pathetic that you can’t even be consistent from sentence to sentence. ---
**Author:** ![Tarantula](https://avatars.discourse-cdn.com/v4/letter/t/e274bd/32.png) [@Tarantula](https://boards.straightdope.com/u/Tarantula)\ **Post date:** [June 4, 2003, 12:28pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/248 "2003-06-04T12:28:12Z")
Actually Chuck, that’s the same contradiction - you just mentioned it twice… ---
**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:** [June 4, 2003, 12:45pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/249 "2003-06-04T12:45:24Z")
> [@](#): > > \*Originally posted by Tarantula \* > \*\*Just for all the idiots that don’t understand humour… > > QED in Mathematics stands for “Quod Erat Demonstratum” - “Because it was shown”. > > Quod Erat Demonstrandum “As it was to have been shown” is a latinist joke… > > tsk ! > > Sometimes you guys take all the fun out of being a latinist… > > Although \*\* Kabbes \*\* was funny - - obviously, Kabbes, you’re not a mathematician then… > > Har har… \*\* I don’t fall into your the category addressed (I doubt you’ll find too many that do on these boards, maybe you should try somewhere else), but I’ll answer you anyway: I have always understood **Q.E.D** to stand for _quod erat demonstrandum_ and meaning _which was to be demonstrated_. ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [June 4, 2003, 12:48pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/250 "2003-06-04T12:48:30Z")
Look, if f(x) = 1/x, then the limit of f(x) as x approaches infinity is zero. HOWEVER, at no point does f(x) actually equal zero. And it’s misleading to argue that f(infinity) = 0 because infinity is a concept, not a number. ---
**Author:** ![RealityChuck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/realitychuck/32/195_2.png) [@RealityChuck](https://boards.straightdope.com/u/RealityChuck)\ **Post date:** [June 4, 2003, 12:51pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/251 "2003-06-04T12:51:43Z")
> [@](#): > > \*Originally posted by kabbes \* > \*\*(As an aside, **RealityChuck** , therein also lies the problem with the “10x - x = 9” proof. Because 9.999… is an infinite sum, you have to prove that it converges before you’re allowed to do things like multiply and subtract it… but once you’ve proved it converges _you’ve already shown that it equals its limit and don’t need to do the multiplication-and-sutraction thing anyway_! Therefore whilst the “proof” is definitely useful for giving an intuitive feel for the reason the solution works, it is actually begging the question. > > Sadly, it isn’t as simple as saying “multiplication just moves the decimal place” because multiplciation on the reals is a field operation that doesn’t work on infinite sums unless they converge, leading you back once again to the same starting place) \*\* No, you don’t have to prove anything about it being a limit. Remember, for 10 \* 0.9999… != 9.99999… to be correct, there needs to be at least one digit in 10 \* 0.9999… that isn’t equal to 9. If we discovered, that the result was, say 9.99…9899…, it would prove the multiplication false. Yet, limit or not, the effect of multiplying the number by ten _does not change any digit to a number other than nine_ (assuming the rules of multiplication do not change). Whether 0.999… expresses a limit or not is irrelevant to the proof; all I am assuming is that if you multiply by 10, all digits remain nine, and this assumption is firmly based on the rules of multiplication. (And I know only a little about limits, but are you saying that if you have a divergent limit, you can’t multiply each of its elements by a number? And that the rules of multiplication are different from divergent limits? Doesn’t seem likely to me.) But you get a lot of credit from me for being one of the few on the board to use “begging the question” correctly. 🙂 ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [June 4, 2003, 12:54pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/252 "2003-06-04T12:54:05Z")
Look, if f(x) = 1/x, then the limit of f(x) as x approaches infinity is zero. HOWEVER, at no point does f(x) actually equal zero. And it’s misleading to argue that f(infinity) = 0 because infinity is a concept, not a number. ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [June 4, 2003, 12:57pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/253 "2003-06-04T12:57:10Z")
> [@](#): > > \*Originally posted by RealityChuck \* > \*\*No, you don’t have to prove anything about it being a limit. > > Remember, for 10 \* 0.9999… != 9.99999… to be correct, there needs to be at least one digit in 10 \* 0.9999… that isn’t equal to 9. \*\* It seems to me that this is not the fatal flaw in your proof. The problem is that if A = B, then it does not necessarily follow that A-X = B-X. (When you are dealing with infinite quantities). ---
**Author:** ![Tarantula](https://avatars.discourse-cdn.com/v4/letter/t/e274bd/32.png) [@Tarantula](https://boards.straightdope.com/u/Tarantula)\ **Post date:** [June 4, 2003, 12:59pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/254 "2003-06-04T12:59:07Z")
**Unwashed** - no, just another of the perils of not being a latinist. Consider yourself enlightened. Latinists may be nearly ALL unemployed, but we know the unless things… ---
**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 4, 2003, 1:16pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/255 "2003-06-04T13:16:14Z")
I can’t believe this thread has spilled over to a sixth page because people are debating a _fact_. It’s as if the OP had asked “What color is the sky”, and the first reply had been “Blue” then ten people come in to say “It cannot be blue and the reason is…” Gah! ---
**Author:** ![kabbes](https://avatars.discourse-cdn.com/v4/letter/k/e99b99/32.png) [@kabbes](https://boards.straightdope.com/u/kabbes)\ **Post date:** [June 4, 2003, 1:22pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/256 "2003-06-04T13:22:54Z")
I think the **Great Unwashed** is right actually. As a non-mathematician, what makes you so sure what mathematicians mean by QED? Every mathematician I’ve ever spoken to (and contrary to your suggestion, that does include me) has used the “As it was to be shown” definition of QED. I can assure you that every time I wrote “QED”, I had the phrase “As it was to be shown” in my head. Why the hell would we put “Because it was shown” at the end of a proof? It’s only three letters after all. It means what the author wants it to mean. I am still none the wise, however, why after “reading the whole thread” you are so sure that 0.999… != 1. Explain that, oh humour-enlightened one. **RealityChuck** : it’s not that you can’t multiply each element per se, it’s more that having done so the answer you get doesn’t necessarily mean anything. After all, if the sum is divergent then it goes to infinity. What does 10 time infinity mean? It sounds simple, but what you have to show is that the sum of 0.9 + 0.09 + 0.009 + … isn’t infinite. Until you show that, you can’t multiply it by ten and subtract what you first thought of (else you are doing no more than 10 times infinity minus infinity). It’s not good enough to point at it and assert that it isn’t infinite - you have to _prove_ it. How does one prove it? Why - by finding the limit of course! And the limit of 0.9 + 0.09 + 0.009 + … is one. Now, having established this we can do 10×1 - 1 and complete your proof. But since we’ve already shown that the limit is 1, there isn’t much point! Do you see where the objection is now? [ul][li]x = 0.999… can only really be defined as an infinite sum; it’s shorthand for x = 0.9 + 0.09 + 0.009 + … [/li][li]Therefore you can’t do that 10x - x thing until you prove that x isn’t infinite…[/li][li]… but the proof that x isn’t infinite shows that x = 1 all by itself![/ul]This apparently trivial problem with the process is just one of the many little annoying things that appear in higher mathematics. I remember well the day that I had to prove that 1 + 1 = 2.[/li] pan ---
**Author:** ![Tarantula](https://avatars.discourse-cdn.com/v4/letter/t/e274bd/32.png) [@Tarantula](https://boards.straightdope.com/u/Tarantula)\ **Post date:** [June 4, 2003, 1:29pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/257 "2003-06-04T13:29:43Z")
Kabbes - “As it was to be shown” is simply re-stating the question. That is not what ANY mathematician means when he / she writes QED. QED is used as a terminator, - this is finished BECAUSE of what I have just written - “because it was shown”. You could use “As it was to be shown” if you had made a mistake and you were refering back to the error you made in a previous piece of work… So did you make a mistake ? If not, you meant “quod erat demonstratum” ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [June 4, 2003, 1:37pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/258 "2003-06-04T13:37:15Z")
QED stands for _Quod Erat Demonstrandum_, “which was to be proved”. It was added to the end of the proofs in Euclid ( though obviously not by Euclid himself, as he wrote in Greek). Similarly, demonstrations of constructions were ended with QEF, _Quod Erat Faciendum_, “Which was to be done”. ---
**Author:** ![SCSimmons](https://avatars.discourse-cdn.com/v4/letter/s/e495f1/32.png) [@SCSimmons](https://boards.straightdope.com/u/SCSimmons)\ **Post date:** [June 4, 2003, 1:37pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/259 "2003-06-04T13:37:48Z")
> [@](#): > > \*Originally posted by lucwarm \* > \*\*Look, if f(x) = 1/x, then the limit of f(x) as x approaches infinity is zero. HOWEVER, at no point does f(x) actually equal zero. And it’s misleading to argue that f(infinity) = 0 because infinity is a concept, not a number. \*\* Nobody is arguing that, lucwarm. The point is that: a) 0.999… means the sum as n goes from 1 to infinity of 9\*(10^-n) b) the value of an infinite series is the limit of the values of the partial sums c) the partial sum of the series identified in a) after k terms is 1 - 10^-k d) the limit of 1 - 10^-k as k goes to infinity is 1 e) therefore, 0.999… = 1. It doesn’t matter that 10^-infinity isn’t well-defined. This is irrelevant to the claim, which relies only on the convergence of the infinite sum … ---
**Author:** ![23skidoo](https://avatars.discourse-cdn.com/v4/letter/2/f04885/32.png) [@23skidoo](https://boards.straightdope.com/u/23skidoo)\ **Post date:** [June 4, 2003, 1:41pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/260 "2003-06-04T13:41:36Z")
> [@](#): > > \*Originally posted by kabbes \* > \*\*(As an aside, **RealityChuck** , therein also lies the problem with the “10x - x = 9” proof. Because 9.999… is an infinite sum, you have to prove that it converges before you’re allowed to do things like multiply and subtract it… but once you’ve proved it converges _you’ve already shown that it equals its limit and don’t need to do the multiplication-and-sutraction thing anyway_! Therefore whilst the “proof” is definitely useful for giving an intuitive feel for the reason the solution works, it is actually begging the question. > > Sadly, it isn’t as simple as saying “multiplication just moves the decimal place” because multiplciation on the reals is a field operation that doesn’t work on infinite sums unless they converge, leading you back once again to the same starting place)\*\* Is there any repeating decimal notation of an infinite sum that doesn’t converge, where the “10x - x =9” proof wouldn’t work? Or, do all repeating decimals converge? [Previous page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=12) [Next page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=14)