# .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:** 6
**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:** [August 4, 2012, 4:33pm UTC](https://boards.straightdope.com/t/999-1/27517/101 "2012-08-04T16:33:20Z")
What’s 9/9? You could say 1; or you could divide it like this: ```auto 0.99999 9|9.00000 81 90 81 90 81 90 81 90 81 ``` etc. ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [August 4, 2012, 4:35pm UTC](https://boards.straightdope.com/t/999-1/27517/102 "2012-08-04T16:35:48Z")
Remember: It’s turtles all the way down. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 4, 2012, 4:39pm UTC](https://boards.straightdope.com/t/999-1/27517/103 "2012-08-04T16:39:35Z")
> [@Exapno\_Mapcase](#): > > Once again, this is wrong. Utterly, completely, factually wrong. No mathematicians would ever describe this as a trick. It is a proof. You need to read more. You say that, but don’t really know it. You can find lots of debate on it. Sure if your in the camp that says .999… = 1 you probably accept it, but I have seen some that even though they believe .999.. = 1 , don’t accept that as a rigorous proof. ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [August 4, 2012, 4:44pm UTC](https://boards.straightdope.com/t/999-1/27517/104 "2012-08-04T16:44:49Z")
> [@erik150x](#): > > Your presupposing that 10 x .999… = 9.999… can you prove that? **Great Antibob** did just that in post 95. ---
**Author:** ![The\_Second\_Stone](https://avatars.discourse-cdn.com/v4/letter/t/bbe5ce/32.png) [@The\_Second\_Stone](https://boards.straightdope.com/u/The_Second_Stone)\ **Post date:** [August 4, 2012, 4:44pm UTC](https://boards.straightdope.com/t/999-1/27517/105 "2012-08-04T16:44:56Z")
At the risk of starting something new, is: .999… \< 1 ? ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 4, 2012, 4:49pm UTC](https://boards.straightdope.com/t/999-1/27517/106 "2012-08-04T16:49:36Z")
> [@Exapno\_Mapcase](#): > > It’s an infinite number of 9s because I’m using the definition of what an infinite number is. That you don’t know the definition of what an infinite number is pretty much sums up what everybody has been trying to explain to you. There is no such thing as an infinite number. Infinity is an abstract concept, yet you claim .999… is an infinite number? I’d like to see the text book that says that. It has an infinite number of decimal places. What happens when you shift an infinite number of decimal places? Do you actually learn that somewhere? Or someone showed you a proof involving 10 x .999… and you accepted it. Which set has a greater number of elements: Natural Numbers or Rational Numbers? and also Rational Numbers or Real Numbers? ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [August 4, 2012, 4:50pm UTC](https://boards.straightdope.com/t/999-1/27517/107 "2012-08-04T16:50:21Z")
> [@The\_Second\_Stone](#): > > At the risk of starting something new, is: > > .999… \< 1 ? The irony is that one of the biggest puzzles in early mathematics was how an infinite series could sum to any finite value at all. The intuition was that it should clearly be itself infinite in value, being made from an infinite number of non-zero terms. But there it is, finite. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 4, 2012, 4:50pm UTC](https://boards.straightdope.com/t/999-1/27517/108 "2012-08-04T16:50:47Z")
> [@The\_Second\_Stone](#): > > At the risk of starting something new, is: > > .999… \< 1 ? yes. if you ask me. 😉 ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 4, 2012, 4:55pm UTC](https://boards.straightdope.com/t/999-1/27517/109 "2012-08-04T16:55:58Z")
> [@Francis\_Vaughan](#): > > The irony is that one of the biggest puzzles in early mathematics was how an infinite series could sum to any finite value at all. The intuition was that it should clearly be itself infinite in value, being made from an infinite number of non-zero terms. But there it is, finite. I am well aware of the summation of an infinite series which converges. It relies on the concept of limits for it answer, which again for the the umpteenth time simply asks you to accept that if you can prove a function approaches with arbitrary precision some number that it is actual = the number, but never proved. It is a fundamental principle of calculus, and indeed very practical, but none the less not really a proof in and of it self that .999… = 1. If you don’t understand that then you don’t understand the definition of a limit. ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [August 4, 2012, 4:59pm UTC](https://boards.straightdope.com/t/999-1/27517/110 "2012-08-04T16:59:09Z")
> [@erik150x](#): > > I am well aware of the summation of an infinite series which converges. If you don’t understand that then you don’t understand the definition of a limit. I wasn’t replying to any of your posts. ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [August 4, 2012, 5:03pm UTC](https://boards.straightdope.com/t/999-1/27517/111 "2012-08-04T17:03:30Z")
> [@](#): > > There is no such thing as an infinite number. Infinity is an abstract concept, So are the reals and the imaginary numbers. Possibly the negative integers and some might argue zero. If you claim there is no such thing as an infinite number, would you care to proffer a definition for an infinite series? One that we might all agree on? ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 4, 2012, 5:09pm UTC](https://boards.straightdope.com/t/999-1/27517/112 "2012-08-04T17:09:26Z")
> [@Francis\_Vaughan](#): > > **Great Antibob** did just that in post 95. sum(i = 1 to “infinity”) [9/10^(i-1)] Written in a more ‘standard’ form, this is 9.99999… So now you are summing not to infinity but (i -1), I hardly see how that is 9.999…, or at least a given. Why not just start out with 10^(i-1), or (i-2) or (i-3) or (i-4)… It would seem your division by 10 is almost meaningless? But I would contest that in the process you adding [9/10^infinity] to your answer - which you may call zero, but I don’t. I’ll give you the best attempt yet. 😉 ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 4, 2012, 5:15pm UTC](https://boards.straightdope.com/t/999-1/27517/113 "2012-08-04T17:15:11Z")
> [@Francis\_Vaughan](#): > > So are the reals and the imaginary numbers. Possibly the negative integers and some might argue zero. > > If you claim there is no such thing as an infinite number, would you care to proffer a definition for an infinite series? One that we might all agree on? What I am trying to point out here is there is room for uncertainty. The set of Natural numbers is countably infinite. So is the set of Rationals, in fact exactly the same ordinal for that matter. See Cantor’s diagonal. But the set of Reals is uncountably infinite and thus larger. What kind of sense does that make…? I don’t really know, but Cantor proved it. So when you go start talking about how to manipulate an infinite number of 9s multiplication-wise or other. You are in very uncertain territory. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 4, 2012, 5:17pm UTC](https://boards.straightdope.com/t/999-1/27517/114 "2012-08-04T17:17:41Z")
> [@Francis\_Vaughan](#): > > I wasn’t replying to any of your posts. Sorry, I am a little on the defensive here, as you might be able to gauge. My apologizes. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 4, 2012, 5:21pm UTC](https://boards.straightdope.com/t/999-1/27517/115 "2012-08-04T17:21:18Z")
> [@erik150x](#): > > What I am trying to point out here is there is room for uncertainty. > > The set of Natural numbers is countably infinite. So is the set of Rationals, in fact exactly the same ordinal for that matter. See Cantor’s diagonal. But the set of Reals is uncountably infinite and thus larger. What kind of sense does that make…? I don’t really know, but Cantor proved it. So when you go start talking about how to manipulate an infinite number of 9s multiplication-wise or other. You are in very uncertain territory. “So is the set of Rationals, in fact exactly the same ordinal for that matter.” I meant …the same cardinality for that matter. ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [August 4, 2012, 5:23pm UTC](https://boards.straightdope.com/t/999-1/27517/116 "2012-08-04T17:23:59Z")
> [@](#): > > So when you go start talking about how to manipulate an infinite number of 9s multiplication-wise or other. You are in very uncertain territory. That is trivial. Colloquially we mean Aleph Null when we say infinity. That is all. Go back in all the arguments and substitute Aleph Null where needed. We do not ever need any other transfinite numbers for this discussion. Having done that, answer my question - define an infinite series. > [@](#): > > So now you are summing not to infinity but (i -1), Which by definition is infinity. That is the critical point. Infinity + 1 is infinity. Infinity - 1 is infinity. Infinity \* infinity is infinity. It simply doesn’t change the meaning of the expression to use the “- 1” If you don;t accept that infinity behaves like this, say so, but then offer your own explanation of what these expressions mean. (And again, note that by infinity I mean Aleph Null.) ---
**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:** [August 4, 2012, 5:35pm UTC](https://boards.straightdope.com/t/999-1/27517/117 "2012-08-04T17:35:27Z")
> [@erik150x](#): > > What I am trying to point out here is there is room for uncertainty. > > The set of Natural numbers is countably infinite. So is the set of Rationals, in fact exactly the same ordinal for that matter. See Cantor’s diagonal. But the set of Reals is uncountably infinite and thus larger. What kind of sense does that make…? I don’t really know, but Cantor proved it. So when you go start talking about how to manipulate an infinite number of 9s multiplication-wise or other. You are in very uncertain territory. Let me get this straight. You’re willing to accept Cantor’s proof, even though you don’t seem to know what it is, while an infinitely more simple proof of .99999… = 1 can’t be right? Let’s face it, you can throw terms around, but you really are shaky on your knowledge of math. Also, the number of digits in .99999… is aleph null, since the number of digit is always a whole number. Bringing in C is just a smokescreen without meaning. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 4, 2012, 5:54pm UTC](https://boards.straightdope.com/t/999-1/27517/118 "2012-08-04T17:54:48Z")
> [@erik150x](#): > > Many thanks to Indistinguishable and Senegoid for well thought out responses. > > Re: “Similarly, when a mathematician says “0.9999…”, what they mean, by that same definition, is “The number which is \>= 0, and also \>= 0.9, and also \>= 0.99, and also \>= 0.999, and so on, AND also \<= 1, and also \<= 1.0, and also \<= 1.00, and also \<= 1.000, and so on.” What number satisfies all these properties? 1 satisfies all these properties. Thus, when a mathematician says “0.9999…”, what they mean, by definition, is 1” > > Pretty stupid notation, why not just say “1”. 😉 Seriously is the some Mathematical Authority which has proclaimed this definition? What if I just want to talk about the geometric series 9/(10^x ) for x = 1 to Infinity and I do not want any assumptions that say well intuitively if we get arbitrarily close to some number we’ll just call it that number. I’m not to say that this is not reasonable to make such assumptions, but they are assumptions not proofs. > > What is the smallest number less than 1? If I say it is .999… as defined by the geometric series geometric series 9/(10^x ) for x = 1, what proof can be offered I am wrong with out the assumptions made by a limit process? > > I don’t know why any one would find that mathematical trickery like saying 10 x .999… is 9.999… where one must add this mysterious 9/infinity to the end is comforting. I would argue it is the opposite, a trick is something to fool you into believing something other than the actual reality. > > I will repeat once more that when you use the Limit process you are making an assumption that x is close enough to n, that we will just call it n. Perhaps it is n, but it is not a proof is it? > > I don’t need a proof that 1 -1 = 0. I can accept that a given. But to me it seems many have taken the result of a limits as a mathematical practicality to the conclusion that they are in fact some undeniable truth. > > Okay so you can say by the axioms of the real number system, .999… must be 1. Fine but the real number system is not perfect is it? Does it describe (-1)^(1/2)? Does it describe 1/infinity? > > I can accept that by the rules proclaimed by the axiom of Real Numbers or whatever .999… must be 1. But can some one tell me what 1/infinity is or why it is not logical to assume 1 - .999… would be 1/infinity? I said I would leave having said my piece. However, I want to point out one more thing: Nothing you’ve said in this post is really objectionable. We CAN make up mathematical systems which have infinitesimals, and they ARE useful for some purposes. We could even decide to have some interpretation of non-terminating decimal notation into those systems on which 0.999… = 1. For example, one simple system is like so: let’s say a hyperrational is a non-terminating sequence of rationals; for example, \<0, 0, 0, …\> or \<2, 3, 4, 5, …\> or \<3, 3.1, 3.14, 3.141, …\>. All operations you can think of will be done component-wise, so, for example, \<2, 3, 4, 5, …\> + \<3, 3.1, 3.14, 3.141, …\> = \<5, 6.1, 7.14, 8.141, …\>, and max(\<2, 3, 4, 5, …\>, \<3, 3.1, 3.14, 3.141, …\>) = \<3, 3.1, 4, 5, …\>). And in the same way we can talk about hyperintegers and hyperbooleans (Yes or No values) and hyper-anything else you like, and operations between them. Finally, we’ll consider two hyper-whatevers to be equal so long as their components are equal _from some point on_. Thus, \<2, 3, 4, 5, …\> = \<3, 3.1, 4, 5, …\>. Put another way, we’ll consider a hyper-boolean to be straight-up Yes just in case its components are all Yes from some point on; thus, the question “Is \<2, 3, 4, 5, …\> greater than \<4, 4, 4, 4, …\>?” has the hyperboolean answer “\”, which amounts to straight-up “Yes”. This system acts a lot like ordinary arithmetic. But it has infinite and infinitesimal values. For example, \<2, 3, 4, 5, ..\> is infinite, in the sense that it is larger than 0, larger than 1, larger than 2, larger than 3, etc. It is larger than any standard integer. And its reciprocal \<1/2, 1/3, 1/4, 1/5, …\>, conversely, is infinitesimal; positive but smaller than 1/n for any standard integer n. And there’s a natural way to interpret non-terminating decimal notation into this system: interpret a.bcd… as \. So 0.999… becomes \<0, 0.9, 0.99, 0.999, …\>, and 1.000… becomes, of course, \<1, 1, 1, 1, …\>. And are these equal? \. The difference between them is the infinitesimal value \<1, 0.1, 0.01, 0.0001, …\>; that is, 1/10^infinity, where “infinity” is the canonical infinite value \<0, 1, 2, 3, 4, …\>. This system probably captures very closely the intuitions you yourself are trying to express. For example, it has a value halfway between 0.999… and 1: \<0.5, 0.95, 0.995, 0.9995, …\>. This value, alas, has no representation in ordinary decimal notation, but we couldn’t have expected it to. Still, it’s there and acts exactly like you’d want it to. And this system IS useful, and used to do nontrivial mathematics. If names matter, it provides the underpinnings of “(Robinson-style) nonstandard analysis”. So your line of thought is not useless, and not fundamentally broken. HOWEVER: For many purposes, people don’t care to discuss infinite values, and don’t care to distinguish between values that are infinitesimally close. If we restrict ourselves to the finite hyperrationals, and stop distinguishing between hyperrationals that are infinitesimally close (so, for example, \<1/2, 1/3, 1/4, 1/5, …\> would be treated as equal to \<0, 0, 0, 0, …\>), then we get… the standard system of “real numbers”, in all its Archimedean glory. And, in particular, 0.999… and 1 become equal, because they are infinitesimally close. So that’s why “real numbers” are useful: they model reasoning at any level where you aren’t actually concerned with drawing fine distinctions between values which are infinitesimally close. And at that level of coarseness, 0.999… and 1 are going to be equal. If you want to draw finer distinctions, you can, and you can make up rules for doing so. For example, the above. These rules will have their drawbacks so far as interfacing with decimal notation goes (no longer will every value have a decimal representation; no longer will it be possible to multiply values by 10 simply by shifting their decimal representation [this will only work for values with finite decimal representations]), but that can be alright. In mathematics, it’s up to you what you want to model and how you want to model it. Always. The only thing is that you need to know and understand others’ conventions when talking to them. And other people very often are talking about real numbers and very rarely are talking about hyperrationals. ---
**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:** [August 4, 2012, 6:00pm UTC](https://boards.straightdope.com/t/999-1/27517/119 "2012-08-04T18:00:19Z")
To **erik150x** : Forgetting about all the proofs and definitions and “arguments”, do you honestly believe that you have discovered something that’s been missed by every mathematician who’s ever lived? I am very interested to hear your answer. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 4, 2012, 6:03pm UTC](https://boards.straightdope.com/t/999-1/27517/120 "2012-08-04T18:03:49Z")
> [@Indistinguishable](#): > > We CAN make up mathematical systems which have infinitesimals, and they ARE useful for some purposes. We could even decide to have some interpretation of non-terminating decimal notation into those systems **on which 0.999… = 1.** Er, I of course meant “on which 0.999… _doesn’t equal_ 1”. [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=5) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=7)