# .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:** 17
**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, 3:35pm UTC](https://boards.straightdope.com/t/999-1/27517/321 "2012-08-06T15:35:09Z")
> [@Francis\_Vaughan](#): > > No, I gave you the solution. It is you that denies the solution by denying the convergence of the distance intervals. You allow time to reach the limit, but not the distance. So you are happy to allow limits in time, but not in distance. This is why the proof fails. You either let the sequence converge to the limit or not - but you can’t have it both ways - which you are trying to do. I can move happily, mostly because believe that the sum of 1/2 + 1/4 + 1/8 = 1, due to this I can move past the time when I reach the end of the line. You are not sure why you can move past the end of the line, just that you can. I denied no convergence? I said nothing about moving past the end of the line? 1 hour total time, 1 mile total 1/2 mile mark - 30 minuts next 1/4 mile 15 minutes next 1/8 mile 7min 30 sec .. let both time and distance converge 1 mile, 1 hour I have reached my infinit’th point…right? ## I can note that a 0 goes on top and 9 on the bottom start my subtraction: 1.000…0 -0.999…9 0.000…1 There is no need to actually do the subtraction is there, once we found the end of the 0s and 9s? We know the answer to the problem. ---
**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, 3:36pm UTC](https://boards.straightdope.com/t/999-1/27517/322 "2012-08-06T15:36:32Z")
> [@Frylock](#): > > Unfortunately, I don’t have the expertise. And anyway, it’s not so much that there’s a flaw in your proof–rather it’s that no one can know if there’s a flaw in your proof until you’ve laid out all of your assumptions and all of your logical steps. That’s not something I can do for you because I don’t know what your assumptions are. If I had a background in math, I could help draw them out, but I don’t, so I can’t. Well, thanks. I appreciate you rinput. As I said. I am sure it’s flawed. But I don’t see it. ---
**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 6, 2012, 3:39pm UTC](https://boards.straightdope.com/t/999-1/27517/323 "2012-08-06T15:39:57Z")
> [@](#): > > f wh know all the points up to and including the infinite’th one? You continue to somehow believe that the infinite’th point is a discrete point. It isn’t. There exists a infinite + 1 ordinal point and an infinite + infinite ordinal point. They all occupy the same location on the line. Which is why you can’t start at the end of the line and pick one out. You are assuming that you can locate a discrete point **for which there are no further points with a greater ordinal**. If you can locate this point you have contradicted the definition of infinity, or you have a finite ordinal point. Infinite simply doesn’t work this way. It doesn’t obey the rules that apply to normal numbers. Continuing to try to force it to does get you anywhere. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 6, 2012, 3:41pm UTC](https://boards.straightdope.com/t/999-1/27517/324 "2012-08-06T15:41:11Z")
> [@erik150x](#): > > I denied no convergence? > > I said nothing about moving past the end of the line? > > 1 hour total time, 1 mile total > > 1/2 mile mark - 30 minuts > next 1/4 mile 15 minutes > next 1/8 mile 7min 30 sec > .. > let both time and distance converge > 1 mile, 1 hour > > I have reached my infinit’th point…right? > > ## I can note that a 0 goes on top and 9 on the bottom start my subtraction: 1.000…0 -0.999…9 > > 0.000…1 > > There is no need to actually do the subtraction is there, once we found the end of the 0s and 9s? We know the answer to the problem. Since, in “normal math,” there is no such thing as the last digit in either of those two decimal expansions, it’s illegitimate to use a “normal math” subtraction algorithm which requires that you start from the last digit of a decimal expansion. ---
**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, 4:02pm UTC](https://boards.straightdope.com/t/999-1/27517/325 "2012-08-06T16:02:27Z")
> [@Frylock](#): > > Since, in “normal math,” there is no such thing as the last digit in either of those two decimal expansions, it’s illegitimate to use a “normal math” subtraction algorithm which requires that you start from the last digit of a decimal expansion. Hmmm, I don’t know. I would potenitally argue there is a last digit at the infinite’th place. It is common to think of infinity as never ending, I know. But a line segment can be divided into an infinite number of points. Why Can I not relate those to holding places for my numerals in the decimal expansion of .999… and 1.000…? Why normal math of subtraction may not apply… perhaps you are right? But I do not see why not here. And one of the morels of Zeno’s Pardox seems to be we can indeed reach the end of an infinite sequence, so… that seems to be not true that an infinite sequence has no end. Is it somehow there are 2 different infinites here? ---
**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, 4:05pm UTC](https://boards.straightdope.com/t/999-1/27517/326 "2012-08-06T16:05:30Z")
> [@Francis\_Vaughan](#): > > You continue to somehow believe that the infinite’th point is a discrete point. It isn’t. There exists a infinite + 1 ordinal point and an infinite + infinite ordinal point. They all occupy the same location on the line. Which is why you can’t start at the end of the line and pick one out. You are assuming that you can locate a discrete point **for which there are no further points with a greater ordinal**. If you can locate this point you have contradicted the definition of infinity, or you have a finite ordinal point. Infinite simply doesn’t work this way. It doesn’t obey the rules that apply to normal numbers. Continuing to try to force it to does get you anywhere. You obvioulsy have a stronger math background than me, so I would in all sincereity like your help in understanding the error I have made here, which I do not yet. So can we start from the beginning of my proof and we’ll go throught it piece by piece and tell me where it goes wrong? ---
**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, 4:10pm UTC](https://boards.straightdope.com/t/999-1/27517/327 "2012-08-06T16:10:01Z")
we can simplify this some more, just 1 line - 1 mile long… As I walk down this line at 1 mph I will imagine we are able to note each half way point, the practicality of this should not really be relivent, agreed? ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 6, 2012, 4:10pm UTC](https://boards.straightdope.com/t/999-1/27517/328 "2012-08-06T16:10:13Z")
> [@erik150x](#): > > Hmmm, I don’t know. I would potenitally argue there is a last digit at the infinite’th place. > > It is common to think of infinity as never ending, I know. But a line segment can be divided into an infinite number of points. Why Can I not relate those to holding places for my numerals in the decimal expansion of .999… and 1.000…? > > Why normal math of subtraction may not apply… perhaps you are right? But I do not see why not here. > > And one of the morels of Zeno’s Pardox seems to be we can indeed reach the end of an infinite sequence, so… that seems to be not true that an infinite sequence has no end. Is it somehow there are 2 different infinites here? I am definitely right about the “normal math” subtraction you applied being illegitimate in this case. Under the rules of “normal math,” the numerical representation 0.999… does not have a final digit, and neither does the numerical representation 1.000… This is just a matter of definition–it is how the ellipses are defined in this context. But under the rules of “normal math,” the subtraction algorithm you were using requires that you start with the final digit of the numerical representations you are working on. It requires a final digit–but by definition (see above) there is no final digit. Hence, the algorithm can not be applied. ---
**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, 4:12pm UTC](https://boards.straightdope.com/t/999-1/27517/329 "2012-08-06T16:12:36Z")
> [@Frylock](#): > > I am definitely right about the “normal math” subtraction you applied being illegitimate in this case. > > Under the rules of “normal math,” the numerical representation 0.999… does not have a final digit, and neither does the numerical representation 1.000… This is just a matter of definition–it is how the ellipses are defined in this context. > > But under the rules of “normal math,” the subtraction algorithm you were using requires that you start with the final digit of the numerical representations you are working on. > > It requires a final digit–but by definition (see above) there is no final digit. Hence, the algorithm can not be applied. Well now I think that is how we all taught to think about it, but in reality all it is is an infinite sequence of numbers. Your ordinary plain jane infinity. One kind of assumes you can never find the end… but I don’t know that is stated as a requirement of the number. ---
**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, 4:16pm UTC](https://boards.straightdope.com/t/999-1/27517/330 "2012-08-06T16:16:28Z")
So I am wlaking down my line at 1 mph, when I reach 1/2 mile at 30 mins, we note this is the first 0 after the decimal of 1, ie the 1.0 \<- zero. Let just ignore the 1 in fornt of the decimal and assume we maked it at the beginning of th eline. I also place my first 9 of the .9 here. Again the physical practicality of marking is not really an issue, i dont care if we actually mark, it note the place holder maybe sounds better. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 6, 2012, 4:17pm UTC](https://boards.straightdope.com/t/999-1/27517/331 "2012-08-06T16:17:09Z")
Regarding the question of whether an infinite sequence can have an end: An infinite sequence can have an end. For example, …6, 4, 2, 0. The question isn’t whether an infinite sequence _can_ have an end, but rather, whether the particular sequence 0, .5, .75, .875, … itself has an end. By definition (see the ellipses) it does not. You’re asking, how can you reach the end of it if it doesn’t have an end? Well, there are actually two senses of “end” at play here. One sense is “last member.” The other sense is “limit.” The sequence doesn’t have a last member, and so, you can’t reach the last member. It doesn’t have one. But the sequence does have a limit. And there’s no reason to think you can’t reach the limit. Clearly you can’t reach the limit simply by enumerating the members of the sequence. But that’s not relevant. You can reach the limit by other means. (In this case, by walking the required mile at a normal pace, you’ll eventually reach it. You won’t reach the “final member of the sequence” that way, because there’s no such thing. Rather, you’ll reach the _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, 4:17pm UTC](https://boards.straightdope.com/t/999-1/27517/332 "2012-08-06T16:17:32Z")
in fact I don’t really care about noting or marking any of these points until I get to the last at the end of my line. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 6, 2012, 4:18pm UTC](https://boards.straightdope.com/t/999-1/27517/333 "2012-08-06T16:18:10Z")
> [@erik150x](#): > > Well now I think that is how we all taught to think about it, but in reality all it is is an infinite sequence of numbers. Your ordinary plain jane infinity. One kind of assumes you can never find the end… but I don’t know that is stated as a requirement of the number. It is stated as a requirement of the number. In “normal math,” the ellipses in this context means “and so on without end.” ---
**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, 4:20pm UTC](https://boards.straightdope.com/t/999-1/27517/334 "2012-08-06T16:20:44Z")
> [@Frylock](#): > > Regarding the question of whether an infinite sequence can have an end: > > An infinite sequence can have an end. For example, …6, 4, 2, 0. > > The question isn’t whether an infinite sequence _can_ have an end, but rather, whether the particular sequence 0, .5, .75, .875, … itself has an end. By definition (see the ellipses) it does not. > > You’re asking, how can you reach the end of it if it doesn’t have an end? > > Well, there are actually two senses of “end” at play here. One sense is “last member.” The other sense is “limit.” The sequence doesn’t have a last member, and so, you can’t reach the last member. It doesn’t have one. But the sequence does have a limit. And there’s no reason to think you can’t reach the limit. Clearly you can’t reach the limit simply by enumerating the members of the sequence. But that’s not relevant. You can reach the limit by other means. (In this case, by walking the required mile at a normal pace, you’ll eventually reach it. You won’t reach the “final member of the sequence” that way, because there’s no such thing. Rather, you’ll reach the _limit_.) Hmmm, interesting at which decimal place would that limit occur? Or in my walk anology what is the count we would be at to reach the limit, assuming we are counting each 1/2 + 1/4 + 1/8 + … + 1/infinity assuming the cont would go 1, 2, 3, … respectively? ---
**Author:** ![President\_Johnny\_Gentle](https://avatars.discourse-cdn.com/v4/letter/p/5f9b8f/32.png) [@President\_Johnny\_Gentle](https://boards.straightdope.com/u/President_Johnny_Gentle)\ **Post date:** [August 6, 2012, 4:21pm UTC](https://boards.straightdope.com/t/999-1/27517/335 "2012-08-06T16:21:52Z")
> [@erik150x](#): > > So how do you propose to begin your addition? > > Mine always starts at the last decimal place. Where does yours start? 😉 Not to pick on you, since I’ve seen other people make similar comments elsewhere, but it’s important to note that we typically add from right to left because of a specific algorithm that we learn in elementary school. We use it because it works, but there’s nothing exceptional about this algorithm (or the typical algorithms for subtracting, multiplying, dividing…) What I’m lecturing, for example, I typically add from left to right, since that’s the way we read the numbers. I just make sure to scan one place ahead to see if it may become necessary to carry. There are many other algorithms that can be used as well, although some aren’t as effective. ---
**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, 4:22pm UTC](https://boards.straightdope.com/t/999-1/27517/336 "2012-08-06T16:22:23Z")
> [@Frylock](#): > > It is stated as a requirement of the number. In “normal math,” the ellipses in this context means “and so on without end.” Is this a higher level of infinity (bigger) than the infinity in the halves of Zeno’s Paradox? ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 6, 2012, 4:24pm UTC](https://boards.straightdope.com/t/999-1/27517/337 "2012-08-06T16:24:35Z")
> [@erik150x](#): > > Hmmm, interesting at which decimal place would that limit occur? Or in my walk anology what is the count we would be at to reach the limit, assuming we are counting each 1/2 + 1/4 + 1/8 + … + 1/infinity > assuming the cont would go 1, 2, 3, … respectively? You don’t reach the limit at any step on the walk. You reach the limit after all the steps are completed. (I think! Someone should be along shortly to correct me if I’m wrong.) ---
**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, 4:25pm UTC](https://boards.straightdope.com/t/999-1/27517/338 "2012-08-06T16:25:37Z")
> [@President\_Johnny\_Gentle](#): > > Not to pick on you, since I’ve seen other people make similar comments elsewhere, but it’s important to note that we typically add from right to left because of a specific algorithm that we learn in elementary school. We use it because it works, but there’s nothing exceptional about this algorithm (or the typical algorithms for subtracting, multiplying, dividing…) What I’m lecturing, for example, I typically add from left to right, since that’s the way we read the numbers. I just make sure to scan one place ahead to see if it may become necessary to carry. There are many other algorithms that can be used as well, although some aren’t as effective. Interesting point. I never thought of that. I’m not sure how it relates? One must clearly reach the end for either direction to get an answer though, or no? ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 6, 2012, 4:26pm UTC](https://boards.straightdope.com/t/999-1/27517/339 "2012-08-06T16:26:47Z")
> [@erik150x](#): > > Is this a higher level of infinity (bigger) than the infinity in the halves of Zeno’s Paradox? No, it’s the same infinity. Both are aleph null–simple, countable infinity. That’s in terms of cardinality. I now forget how to compare infinities for ordinality. Aleph-null plus one is the same cardinality as aleph null (namely, aleph null) but has a higher ordinality (it is one greater). I forget how to determine the ordinality of an infinite number in general. So I can not say for certain whether the two sequences have the same ordinality. But they have the same cardinality, and that is almost certainly what’s relevant to this discussion. ---
**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, 4:27pm UTC](https://boards.straightdope.com/t/999-1/27517/340 "2012-08-06T16:27:21Z")
> [@Frylock](#): > > You don’t reach the limit at any step on the walk. You reach the limit after all the steps are completed. (I think! Someone should be along shortly to correct me if I’m wrong.) Well I would think that would be at the Infnite’th half, when we finally reach the end? or no? [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=16) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=18)