# .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:** 13
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 10:17am UTC](https://boards.straightdope.com/t/999-1/27517/241 "2012-08-06T10:17:18Z")
Alltrop, I am interested in how you got to point B though, very interested. Because I admit I don’t know, how did you get there? ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 6, 2012, 10:17am UTC](https://boards.straightdope.com/t/999-1/27517/242 "2012-08-06T10:17:58Z")
> [@erik150x](#): > > How do you know what distinciton I was trying to make? OK, here’s a very accessible article that explains it quite well. [http://www.sciencenews.org/view/generic/id/9269/title/Math\_Trek\_\_Small\_Infinity,\_Big\_Infinity](http://www.sciencenews.org/view/generic/id/9269/title/Math_Trek__Small_Infinity,_Big_Infinity) ---
**Author:** ![TATG](https://avatars.discourse-cdn.com/v4/letter/t/50afbb/32.png) [@TATG](https://boards.straightdope.com/u/TATG)\ **Post date:** [August 6, 2012, 10:20am UTC](https://boards.straightdope.com/t/999-1/27517/243 "2012-08-06T10:20:05Z")
> [@erik150x](#): > > I would argue this an artifically imposed conclusion/definition based on a real number system which has the notion of numbers as limits put on to them. It does not have to be that way. I don’t think I said anything about limits. What exactly are you rejecting in what I said? > [@erik150x](#): > > After all there are no natural numbers between 1 and 2, but they are not equal. And different sets of numbers have different properties; a fairly fundamental property of the reals is that “for any two real numbers there is one in-between.” I also don’t think we need to talk about limits to talk about this property. As others have said, you can have number systems different, or more expansive, than the reals. And I think that is where you will find your intuitions satiated. ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 6, 2012, 10:22am UTC](https://boards.straightdope.com/t/999-1/27517/244 "2012-08-06T10:22:21Z")
> [@erik150x](#): > > You made no definition or explination of infinity, you simply said what happens in your example fo going from point A ot B. I never siad you don’t reach point B. You did. Nor did you explain how you got there. I guess your explaination is it just is, it just happens. If that is satifactory to you, we have nothign to discuss really. > [@erik150x](#): > > Alltrop, I am interested in how you got to point B though, very interested. Because I admit I don’t know, how did you get there? Well,I guess at some point it’s a concept you either grasp or you don’t. Honestly, very few people DO get it to any meaningful degree. Just like there are some things I’ll never be able to wrap my head around. That doesn’t mean that I dismiss them as crap though. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 10:24am UTC](https://boards.straightdope.com/t/999-1/27517/245 "2012-08-06T10:24:10Z")
> [@TATG](#): > > I don’t think I said anything about limits. What exactly are you rejecting in what I said? > And different sets of numbers have different properties; a fairly fundamental property of the reals is that “for any two real numbers there is one in-between.” I also don’t think we need to talk about limits to talk about this property. > > As others have said, you can have number systems different, or more expansive, than the reals. And I think that is where you will find your intuitions satiated. I say this because we only say this due to the definiton of them uses limits (in the standard analysis). But if you would like to discuss some other number system I am happy to mine I see no reason to impose this restriciton that there must be another number between them. Perhaps they are right next to each other, no room for another number. Or maybe they are seperated by 1/infinity which we don’t really define, but we need to maybe to complete this number system. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 10:28am UTC](https://boards.straightdope.com/t/999-1/27517/246 "2012-08-06T10:28:32Z")
> [@allotrope](#): > > Well,I guess at some point it’s a concept you either grasp or you don’t. > > Honestly, very few people DO get it to any meaningful degree. Just like there are some things I’ll never be able to wrap my head around. That doesn’t mean that I dismiss them as crap though. Ture enough. I don’t fully understand Eisnteins Theorys, but I accpet them. This one though doesn’t ring true to me. I don’t see why at infinity that .999… becomes 1 and not less than 1 if only by 1/infinity. I totally understand where your going. Limits do solve the parodox of getting from point A to B if you include time by the way. But thats another topic. But limits don’t explain how this is true. They just say at Infinity they become equal. No how or why? That’s what I want. And that’s whats missing from Limits. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 10:31am UTC](https://boards.straightdope.com/t/999-1/27517/247 "2012-08-06T10:31:44Z")
Also putting limits aside, it could very well be I think… that 1/2 + 1/4 + 1/8 + … + 1/infinity = 1 but 9/10 + 9/100 + 9/1000 + … + 9/infinity \< 1 ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [August 6, 2012, 10:33am UTC](https://boards.straightdope.com/t/999-1/27517/248 "2012-08-06T10:33:57Z")
> [@erik150x](#): > > … Look at the natural numbers. You don’t say because there are no numbers between 2 and 3, that they are the same number do you? … Right, between two natural numbers there may be no third natural number, because the natural numbers are not closed with respect to division, i.e., for some natural numbers n there may be no natural number n/2, n/3, etc. So, for 2 and 3 there is no natural number 5/2 or 8/3, which you would expect to lie between them. However, the rational numbers and the real number are closed with respect to division (except for division by zero). So between x and y there always is (x+y)/2. In fact, as a simple consequence of there being at least one number, there must be an infinite set of such numbers. This leads to an interesting question. If (as you are suggesting) .999… and 1 are different numbers, then the average ((1+.999…)/2) must be a different number again. In fact within this infinitesimal interval between .999… and 1 there ought to be an infinite number of different numbers. If there aren’t, where do you stop subdividing and finding new numbers? ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 6, 2012, 10:34am UTC](https://boards.straightdope.com/t/999-1/27517/249 "2012-08-06T10:34:56Z")
> [@erik150x](#): > > Ture enough. I don’t fully understand Eisnteins Theorys, but I accpet them. This one though doesn’t ring true to me. I don’t see why at infinity that .999… becomes 1 and not less than 1 if only by 1/infinity. > > I totally understand where your going. Limits do solve the parodox of getting from point A to B if you include time by the way. But thats another topic. > > But limits don’t explain how this is true. They just say at Infinity they become equal. No how or why? That’s what I want. And that’s whats missing from Limits. Your problem is getting hung up on the notion of a limit. I keep telling you, fuck limits. Limits are not relevant here. We are going to infinity and beyond. Any limit notation is necessary because we don’t have the tools to deal directly with infinities. But that’s what you’re proposing that we do with your infinite decimal. Accordingly, we have to take off the gloves and go mano-a-mano with your decimal representation and say what is it’s value AT infinity - NO LIMIT NOTATION. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 10:37am UTC](https://boards.straightdope.com/t/999-1/27517/250 "2012-08-06T10:37:43Z")
> [@TATG](#): > > Every real number can be represented by an infinite decimal expansion.\* For any two real numbers there is one in-between. If 1 and .999~ are real numbers, then either there is some real number that can be represented by an infinite decimal expansion which is between 1 and 0.999~; or 1 = .999~. > > - We can have any finite number of digits on the left hand side of the decimal, and denumerably many digits on the right had side. Sorry, but you can correct me if I wrong please, but please site the source if I am. This idea " For any two real numbers there is one in-between." is imposed by the definiton of real numbers which uses Limits. At least that is my understanding of things. Being that we define .999… as the limit of the series, then it is inevitbale that " For any two real numbers there is one in-between.". But other than the imposed limit definition, it perhaps does not have to be so. If you want to talk about uanother number system, ok. Can we just leave that definition out then? ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [August 6, 2012, 10:38am UTC](https://boards.straightdope.com/t/999-1/27517/251 "2012-08-06T10:38:16Z")
> [@erik150x](#): > > Also putting limits aside, it could very well be I think.. > > that 1/2 + 1/4 + 1/8 + … + 1/infinity = 1 > > but 9/10 + 9/100 + 9/1000 + … + 9/infinity \< 1 But every term in that first series is less than the corresponding term in the second: 1/2 \< 9/10 3/4 \< 99/100 7/8 \< 999/1000 15/16 \< 9999/10000 It would be very odd if the limit of the first were greater than the limit of the second, since it can never overtake it. (But it’s not odd that they have the same limit.) ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 10:39am UTC](https://boards.straightdope.com/t/999-1/27517/252 "2012-08-06T10:39:38Z")
> [@allotrope](#): > > Your problem is getting hung up on the notion of a limit. I keep telling you, fuck limits. Limits are not relevant here. We are going to infinity and beyond. > > Any limit notation is necessary because we don’t have the tools to deal directly with infinities. But that’s what you’re proposing that we do with your infinite decimal. Accordingly, we have to take off the gloves and go mano-a-mano with your decimal representation and say what is it’s value AT infinity - NO LIMIT NOTATION. I love you man. Most especially for this: “Any limit notation is necessary because we don’t have the tools to deal directly with infinities” ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 6, 2012, 10:43am UTC](https://boards.straightdope.com/t/999-1/27517/253 "2012-08-06T10:43:43Z")
That’s very kind, but I’ll be happier if this all seems to be a little more real a little less like some sleazy math trick to you. Yes? No? ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 10:45am UTC](https://boards.straightdope.com/t/999-1/27517/254 "2012-08-06T10:45:53Z")
The value of .999… at infinity is .999… or if you prefer 1 - 1/infinity. Yes I realize 1/infinity is undefined, and no I can;t precisely definie it. But someone shoudl get busy on it 😉 ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [August 6, 2012, 10:46am UTC](https://boards.straightdope.com/t/999-1/27517/255 "2012-08-06T10:46:37Z")
> [@erik150x](#): > > This idea " For any two real numbers there is one in-between." is imposed by the definiton of real numbers which uses Limits. At least that is my understanding of things. No, it’s imposed by two things: (1) the ordinary rules of arithmetic, and (2) the fact that the sum of two real numbers and half of a real number are real numbers themselves. Limits are quite irrelevant. From that, if x\ **Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 10:47am UTC](https://boards.straightdope.com/t/999-1/27517/256 "2012-08-06T10:47:29Z") > [@allotrope](#): > > That’s very kind, but I’ll be happier if this all seems to be a little more real a little less like some sleazy math trick to you. > > Yes? No? Oh yes yes please! 😃 I mean that if you can show me how .999… = 1 without limits I will be very happy! ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 10:49am UTC](https://boards.straightdope.com/t/999-1/27517/257 "2012-08-06T10:49:43Z")
> [@Giles](#): > > No, it’s imposed by two things: > (1) the ordinary rules of arithmetic, and > (2) the fact that the sum of two real numbers and half of a real number are real numbers themselves. > Limits are quite irrelevant. > > From that, if x\ > It’s a bit harder to prove that between any two real numbers there is a rational number, but that theorem isn’t really relevant here. Can you explain to me why 1 and 2 must be true? what ordinary rules of arithmetic? I am not sure how 2) applys here? ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 6, 2012, 10:53am UTC](https://boards.straightdope.com/t/999-1/27517/258 "2012-08-06T10:53:25Z")
> [@erik150x](#): > > Oh yes yes please! 😃 > > I mean that if you can show me how .999… = 1 without limits I will be very happy! OK, well, that was what all of the “at infinity” stuff was about. But I’m guessing your response will be something along the lines of even at infinity there will still be an infinitely small difference. Except, at infinity, no, even that infinitely small difference has been gobbled up by all of those repeating 9’s. You have gone out to infinity such that there is no end to the decimal where you could find that infinitesimal difference. It’s gone, buried in infinity. Sorry, but that’s about the best I can do by way of an explanation. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 10:58am UTC](https://boards.straightdope.com/t/999-1/27517/259 "2012-08-06T10:58:13Z")
> [@Giles](#): > > But every term in that first series is less than the corresponding term in the second: > > 1/2 \< 9/10 > 3/4 \< 99/100 > 7/8 \< 999/1000 > 15/16 \< 9999/10000 > > It would be very odd if the limit of the first were greater than the limit of the second, since it can never overtake it. (But it’s not odd that they have the same limit.) Your right. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 11:01am UTC](https://boards.straightdope.com/t/999-1/27517/260 "2012-08-06T11:01:55Z")
> [@allotrope](#): > > OK, well, that was what all of the “at infinity” stuff was about. > > But I’m guessing your response will be something along the lines of even at infinity there will still be an infinitely small difference. > > Except, at infinity, no, even that infinitely small difference has been gobbled up by all of those repeating 9’s. > > You have gone out to infinity such that there is no end to the decimal where you could find that infinitesimal difference. It’s gone, buried in infinity. > > Sorry, but that’s about the best I can do by way of an explanation. Well, thanks for trying. Sorry I just don’t see it. All I can see is something infinitly close to 1, but not 1. [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=12) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=14)