# An infinite question: Why doesn't .999~ = 1? **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [September 24, 2016, 7:29am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711 "2016-09-24T07:29:45Z")\ **Posts on this page:** 20\ **Page:** 32
**Author:** ![Itself](https://avatars.discourse-cdn.com/v4/letter/i/d07c76/32.png) [@Itself](https://boards.straightdope.com/u/Itself)\ **Post date:** [November 19, 2016, 8:41pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/621 "2016-11-19T20:41:09Z")
Nope, not doing this again. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 19, 2016, 9:17pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/622 "2016-11-19T21:17:57Z")
> [@watchwolf49](#): > > I understand you answered a question that wasn’t asked, so to be more specific would be to ask you to answer the questions that _are_ being asked … which step in the equation’s derivation do you think is in error? That question is open to anybody, I make no claim that this is a perfect proof … … Step #1 is in error already. The limit of (0.9, 0.99, 0.999, …) is 1. 0.999… is not the limit since it is not a point on the number line. 0.999… is greater than every element in { 0.9, 0.99, 0.999, … }, but you cannot find a segment in 0.999… which is not already a segment of one of the numbers 0.9, 0.99, 0.999, … ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [November 19, 2016, 11:51pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/623 "2016-11-19T23:51:54Z")
So, after all this time, the key point is that you refuse to accept 0.9999 as an alternate notation for 1.0000. It might have saved some time if you’d made that point explicit earlier. by the same logic, do you not accept 0.3333 an an alternate notation for 1/3 ? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 20, 2016, 8:16am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/624 "2016-11-20T08:16:34Z")
> [@septimus](#): > > So, after all this time, the key point is that you refuse to accept 0.9999 as an alternate notation for 1.0000. It might have saved some time if you’d made that point explicit earlier. > > by the same logic, do you not accept 0.3333 an an alternate notation for 1/3 ? The limit of (0.3, 0.33, 0.333, …) is ⅓. 0.333… is not the limit since it is not a point on the number line. 0.333… is greater than every element in { 0.3, 0.33, 0.333, … }, but you cannot find a segment in 0.333… which is not already a segment of one of the numbers 0.3, 0.33, 0.333, … ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [November 20, 2016, 9:01am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/625 "2016-11-20T09:01:16Z")
So “number”, I guess, means real number. But only a tiny subset of the rationals (those with only 2 and 5 as prime factors in the denominator) are “points on the number line.” _ **Got it** _. Now that that’s settled, can you point us to the scientific fields where this new “insight” is useful? String theory? Computational geometry? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 20, 2016, 9:17am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/626 "2016-11-20T09:17:06Z")
We are dealing with simple logic. That’s it. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 20, 2016, 9:26am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/627 "2016-11-20T09:26:33Z")
> [@netzweltler](#): > > We are dealing with simple logic. That’s it. 0.9 0.99 0.999 … can be written as [0, 0.9] [0, 0.9]∪[0.9, 0,99] [0, 0.9]∪[0.9, 0,99]∪[0.99, 0,999] … 0.999… can be written as [0, 0.9]∪[0.9, 0,99]∪[0.99, 0,999]∪… There is no segment in 0.999… which is not already a segment of one of the finite numbers in the list. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 20, 2016, 9:48am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/628 "2016-11-20T09:48:19Z")
> [@netzweltler](#): > > There is no segment in 0.999… which is not already a segment of one of the finite numbers in the list. Usually we are saying that a number is greater than a set of smaller numbers, if it contains at least one segment which is not a segment of one of the smaller numbers. ---
**Author:** ![Budget\_Player\_Cadet](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/budget_player_cadet/32/205_2.png) [@Budget\_Player\_Cadet](https://boards.straightdope.com/u/Budget_Player_Cadet)\ **Post date:** [November 20, 2016, 12:24pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/629 "2016-11-20T12:24:13Z")
> [@netzweltler](#): > > We are dealing with simple logic. That’s it. Yes, the simple logic of a 4-year-old who can’t figure out that the sun is remaining in the same place relative to the earth, and merely appears to be moving because the earth is spinning. “But we haven’t moved, _it’s_ moving!” ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [November 20, 2016, 1:27pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/630 "2016-11-20T13:27:16Z")
> [@netzweltler](#): > > You don’t give up your notion that 9(⅒) + 9(⅒)² + 9(⅒)³ + … is a continuous process, and I don’t give up the notion that 9(⅒) + 9(⅒)² + 9(⅒)³ + … is a step-by-step-approach. I think we stuck in a loop. That’s not what I described at all. I guess you’re just not able to understand what we’re saying. No point in continuing. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 20, 2016, 2:25pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/631 "2016-11-20T14:25:51Z")
> [@netzweltler](#): > > Step #1 is in error already. The limit of (0.9, 0.99, 0.999, …) is 1. 0.999… is not the limit since it is not a point on the number line. 0.999… is greater than every element in { 0.9, 0.99, 0.999, … }, but you cannot find a segment in 0.999… which is not already a segment of one of the numbers 0.9, 0.99, 0.999, … Please, don’t randomly change the equation as given … I asked about the limit of the summation … and I’m assuming here you agree that the limit of the summation as n –\> ∞ is in fact 1 … **Thank You** … So, it looks like the issue here is that you don’t believe that for most functions, f(x) … the limit as x –\> a of f(x) is equal to f(a) … For example if g(x) = 2x + 3 … then the limit as x –\> 4 of g(x) is equal to 11 … sure as shit, g(4) is also equal to 11 … there are exceptions to this … for h(x) = 1/x (where x ≠ 0) … the limit as x –\> 0 is equal to infinity … however h(0) is undefined, since x ≠ 0 … but that’s not the case in the summation we’re working with here … all the terms are clearly defined … So let me ask another question … why is it you don’t believe us when we say “the limit as x –\> a of f(x) is equal to f(a) in this case”? We can prove this to you, but you’re going to need two years college calculus under your belt … no way can we teach you calculus here on these boards … no way can we develop the theorems needed which takes a full 24 units of college credit … At some point here … you’re going to have to call us liars … millions upon millions of mathematicians all conspiring just to deceive you … and only you … and all seven billion of us in on the deception … just to deceive you … and only you … James Maxwell, Albert Einstein, Stephen Hawking … their entire body of work all designed to deceive you … and only you … [sigh] > [@netzweltler](#): > > We are dealing with simple logic. That’s it. You’re ending your logic too soon … giving you an incomplete conclusion … only a half truth … one that fails in the real world … ---
**Author:** ![Budget\_Player\_Cadet](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/budget_player_cadet/32/205_2.png) [@Budget\_Player\_Cadet](https://boards.straightdope.com/u/Budget_Player_Cadet)\ **Post date:** [November 20, 2016, 2:29pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/632 "2016-11-20T14:29:28Z")
> [@netzweltler](#): > > 0.9 > 0.99 > 0.999 > … > > can be written as > > [0, 0.9] > [0, 0.9]∪[0.9, 0,99] > [0, 0.9]∪[0.9, 0,99]∪[0.99, 0,999] > … > > 0.999… can be written as [0, 0.9]∪[0.9, 0,99]∪[0.99, 0,999]∪… > > There is no segment in 0.999… which is not already a segment of one of the finite numbers in the list. And yet, as a whole, 0.99… goes beyond those finite numbers. Next thing, you’re going to be claiming that Gabriel’s Horn couldn’t possibly have finite volume, because it has infinite area. You don’t have the slightest clue what you’re talking about and your “logic” was discarded as useless and/or meaningless more than 2000 years ago. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [November 20, 2016, 7:17pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/633 "2016-11-20T19:17:24Z")
> [@netzweltler](#): > > We are dealing with simple logic. That’s it. Trouble with logic is that if your premises are false, your conclusion cannot be trusted. All horses are female. ‘Dobbin’ is a horse. Dobbin is female. The logic is valid. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [November 20, 2016, 9:15pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/634 "2016-11-20T21:15:43Z")
OK, let’s review basic logic. We’ll start by proving the well-known fact that all horses have an infinite number of legs. > [@](#): > > Lemma: All horses are the same color. > Proof (by induction): > Case n = 1: In a set with only one horse, it is obvious that all horses in that set are the same color. > Case n = k: Suppose you have a set of k+1 horses. Pull one of these horses out of the set, so that you have k horses. We know by the inductive hypothesis that all of these horses are the same color. Now put back the horse that you took out, and pull out a different one. Again, all of the k horses now in the set are the same color. Then the set of k+1 horses are all the same color. We have k true =\> k+1 true; therefore all horses are the same color. > > Theorem: All horses have an infinite number of legs. > Proof (by intimidation): > Everyone would agree that all horses have an even number of legs. It is also well-known that horses have forelegs in front and two legs in back. 4 + 2 = 6 legs, which is certainly an odd number of legs for a horse to have! Now the only number that is both even and odd is infinity; therefore all horses have an infinite number of legs. However, suppose that there is a horse somewhere that does not have an infinite number of legs. Well, that would be a horse of a different color; and by the Lemma, it doesn’t exist. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 20, 2016, 9:40pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/635 "2016-11-20T21:40:18Z")
Casinos ban winning gambling systems … casinos ban throwing the dice as hard as one can … therefore, throwing the dice as hard as one can is a winning gambling system … I’ve actually tested this to some degree, usually the dealers don’t say anything until someone gets hurt … but every time I get the dice to bounce off _both_ end walls … they come up a winner … ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 21, 2016, 8:56am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/636 "2016-11-21T08:56:01Z")
> [@Budget\_Player\_Cadet](#): > > > [@netzweltler](#): > > > > 0.9 > > 0.99 > > 0.999 > > … > > > > can be written as > > > > [0, 0.9] > > [0, 0.9]∪[0.9, 0,99] > > [0, 0.9]∪[0.9, 0,99]∪[0.99, 0,999] > > … > > > > 0.999… can be written as [0, 0.9]∪[0.9, 0,99]∪[0.99, 0,999]∪… > > > > There is no segment in 0.999… which is not already a segment of one of the finite numbers in the list. > > And yet, as a whole, 0.99… goes beyond those finite numbers. [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… **means** 0.999… as a whole. I bet you cannot name a segment in 0.999… which is not a segment of one of the numbers of the list. And yet, you are claiming that there is. > [@Budget\_Player\_Cadet](#): > > Next thing, you’re going to be claiming that Gabriel’s Horn couldn’t possibly have finite volume, because it has infinite area. Not my claim. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 21, 2016, 8:59am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/637 "2016-11-21T08:59:26Z")
> [@watchwolf49](#): > > So let me ask another question … why is it you don’t believe us when we say “the limit as x –\> a of f(x) is equal to f(a) in this case”? No one has shown yet how we reach the limit point in my previous examples. ---
**Author:** ![Budget\_Player\_Cadet](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/budget_player_cadet/32/205_2.png) [@Budget\_Player\_Cadet](https://boards.straightdope.com/u/Budget_Player_Cadet)\ **Post date:** [November 21, 2016, 9:00am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/638 "2016-11-21T09:00:39Z")
> [@netzweltler](#): > > [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… **means** 0.999… as a whole. > I bet you cannot name a segment in 0.999… which is not a segment of one of the numbers of the list. And yet, you are claiming that there is. [0.9, 1], because 0.99… = 1, because there is no distance you can point to between 0.99… and 1. Done. That was easy. > [@](#): > > Not my claim. No, but it’s exactly the same kind of intuitive, ultimately faulty logic. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 21, 2016, 12:10pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/639 "2016-11-21T12:10:21Z")
> [@netzweltler](#): > > No one has shown yet how we reach the limit point in my previous examples. Your previous examples required we treat infinity as a very very large finite number … which is untrue … and the limits of your education don’t allow for you to understand what you have been shown … please check with your mom or dad, see if either still have their old calculus textbook you can borrow … we can try to explain a simplistic epsilon/delta proof (again). ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 22, 2016, 7:25am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/640 "2016-11-22T07:25:20Z")
> [@Budget\_Player\_Cadet](#): > > > [@netzweltler](#): > > > > [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… **means** 0.999… as a whole. > > I bet you cannot name a segment in 0.999… which is not a segment of one of the numbers of the list. And yet, you are claiming that there is. > > [0.9, 1], because 0.99… = 1, because there is no distance you can point to between 0.99… and 1. > > Done. > > That was easy. [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… doesn’t include 1 as an endpoint. [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1). Try again. > [@Budget\_Player\_Cadet](#): > > > [@netzweltler](#): > > > > Not my claim. > > No, but it’s exactly the same kind of intuitive, ultimately faulty logic. Even in my “copy and paste” examples I am dealing with infinity which is completed in a finite amount of time. So, I definitely agree that Gabriel’s Horn has finite volume although it has infinite area. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=31) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=33)