# .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:** 31
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 10, 2012, 10:48pm UTC](https://boards.straightdope.com/t/999-1/27517/601 "2012-08-10T22:48:58Z")
> [@Francis\_Vaughan](#): > > I’m not sure if this is a question or a statement, or a rhetorical question. > > You still didn’t answer my questions above - what does “completed” mean, and why must we perform this completion? > > Swap the viewpoint about. As I walk I cover an uncountable number of points. Any question about labelling or otherwise selecting from this set of points as I travel isn’t part of my travel. Perhaps some Platonic universe keeps track, but it has nothing to do with me or my movement. Given that it is possible to construct specifications for an uncountable number of these subset selections, I guess this Platonic universe simply has an uncountable set of subsets of the points I pass though. Makes no difference to me. It is just way of specifying the selection of points. I am not required to know about these selections in order to move. So I can’t see how they stop me from moving. Whether you are keeping track has no bearing on the issue. It’s like saying it doesn’t matter how an airplane flies, it just does. The distance you are covering contains and infinite number of intervals. It does not matter if you are aware. The issue pertains to how its possible for you to complete the series when they are infinite or endless. I find it difficult to see how one could argue that one can and regular does progress through an infinite number of points during the movement form point A to point B, but for the the 9s in .999… it’s different. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 10, 2012, 11:00pm UTC](https://boards.straightdope.com/t/999-1/27517/602 "2012-08-10T23:00:34Z")
> [@Saltire](#): > > From that same article: > > “…in which the use of limits in defining the basic notions of the calculus is replaced by nilpotent infinitesimals, that is, of quantities so small (but not actually zero) that some power—most usefully, the square—vanishes.” > > I can’t understand how you can accept a nonzero number so small that its square is zero, but still have trouble with 0.9…=1. I’ll try to explain. Limits were not a mathematical discovery per-say. Like the proof that the Square Root of 2 is irrational or that Pi is irrational. Limits were constructed, devised as a means to avoid issues such as 1/infinity. For example the Limit of 1/x as x-\> infinity = 0? Why don’t we just say 1/infinity = 0 then? Because that would lead to all kinds of contradictions. So that tells me there is something missing here. Something is not right about that assumption. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 10, 2012, 11:03pm UTC](https://boards.straightdope.com/t/999-1/27517/603 "2012-08-10T23:03:12Z")
> [@erik150x](#): > > I never argue that one does indeed arrive at B. The question… the paradox is HOW? You’re asking “how” by saying “you shouldn’t be able to, by the reasoning that you never reach the end as you go through the sequence of halfway marks. So how is it possible?” I’m answering that question by pointing out the reasoning is flawed. I don’t know how else to answer. “How?” By moving forward at a constant nonzero speed would be one way to do it. If that’s not a sufficient answer, then explain why. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 10, 2012, 11:05pm UTC](https://boards.straightdope.com/t/999-1/27517/604 "2012-08-10T23:05:25Z")
In your post 590 you very explicitly argue that you cannot reach B. You do not say you actually think B can’t be reached, but you offer an argument that B can’t be reached, and ask, in light of that argument, how it’s possible to arrive at B. I’m pointing out that the argument that B can’t be reached is flawed in the first place–and so the motivation to solve the puzzle of how one reaches B is lost. There’s no puzzle any longer, once the argument is shown to be flawed. ---
**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 10, 2012, 11:22pm UTC](https://boards.straightdope.com/t/999-1/27517/605 "2012-08-10T23:22:15Z")
For what it’s worth, I recently read a fascinating article [How Do You Count Parallel Universes?](http://io9.com/5932366/how-do-you-count-parallel-universes?tag=space) which introduced me to the concept of p-adic numbers- an alternative to the entire real number system, and one arguably more useful than the standard reals. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 10, 2012, 11:22pm UTC](https://boards.straightdope.com/t/999-1/27517/606 "2012-08-10T23:22:40Z")
> [@Frylock](#): > > You’re asking “how” by saying “you shouldn’t be able to, by the reasoning that you never reach the end as you go through the sequence of halfway marks. So how is it possible?” I’m answering that question by pointing out the reasoning is flawed. > > I don’t know how else to answer. “How?” By moving forward at a constant nonzero speed would be one way to do it. If that’s not a sufficient answer, then explain why. I would just ask you to read about the apparent paradox of motion. I am sure you can find many good articles. I don’t know how to explain it any better than I have, sorry. 🙂 ---
**Author:** ![zombywoof](https://avatars.discourse-cdn.com/v4/letter/z/9e8a1a/32.png) [@zombywoof](https://boards.straightdope.com/u/zombywoof)\ **Post date:** [August 10, 2012, 11:26pm UTC](https://boards.straightdope.com/t/999-1/27517/607 "2012-08-10T23:26:36Z")
> [@erik150x](#): > > I would just ask you to read about the apparent paradox of motion. The operative word here, of course, being “apparent”! ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 10, 2012, 11:28pm UTC](https://boards.straightdope.com/t/999-1/27517/608 "2012-08-10T23:28:42Z")
> [@erik150x](#): > > I would just ask you to read about the apparent paradox of motion. I am sure you can find many good articles. I don’t know how to explain it any better than I have, sorry. 🙂 I know all about it. You don’t get the paradox without reasoning that leads to the conclusion that you can’t reach B. And indeed, you did offer reasoning that you can’t reach B. I pointed out the flaw in that reasoning. Hence, there is now no longer reasoning on the table which says you can’t reach B. Without that reasoning, there is no longer a paradox to solve. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 10, 2012, 11:28pm UTC](https://boards.straightdope.com/t/999-1/27517/609 "2012-08-10T23:28:47Z")
> [@Frylock](#): > > In your post 590 you very explicitly argue that you cannot reach B. You do not say you actually think B can’t be reached, but you offer an argument that B can’t be reached, and ask, in light of that argument, how it’s possible to arrive at B. I’m pointing out that the argument that B can’t be reached is flawed in the first place–and so the motivation to solve the puzzle of how one reaches B is lost. There’s no puzzle any longer, once the argument is shown to be flawed. In post 590 I am simply restating the apparent paradox and asking for your resolution. Your solution is you just keep walking. That does not really address it. Except in a most Zen like fashion. But not in any logical way. There are at least a couple solutions which show that there is no paradox. I am asking for yours. Just that you keep walking… that is not really one that makes any sense. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 10, 2012, 11:33pm UTC](https://boards.straightdope.com/t/999-1/27517/610 "2012-08-10T23:33:27Z")
> [@erik150x](#): > > In post 590 I am simply restating the apparent paradox and asking for your resolution. Your solution is you just keep walking. That does not really address it. Except in a most Zen like fashion. But not in any logical way. There are at least a couple solutions which show that there is no paradox. I am asking for yours. Just that you keep walking… that is not really one that makes any sense. My solution is, (as I said,) that the argument that you can’t reach B is invalid, because it illegitimately draws the conclusion that you “never” reach B when the premises actually only support a conclusion that you “don’t reach B at any of the points in time where you hit a 1/2^n mark on the line from A to B.” ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 10, 2012, 11:37pm UTC](https://boards.straightdope.com/t/999-1/27517/611 "2012-08-10T23:37:55Z")
> [@Frylock](#): > > I know all about it. You don’t get the paradox without reasoning that leads to the conclusion that you can’t reach B. And indeed, you did offer reasoning that you can’t reach B. I pointed out the flaw in that reasoning. Hence, there is now no longer reasoning on the table which says you can’t reach B. Without that reasoning, there is no longer a paradox to solve. LOL… The crux of this paradox is that indeed we KNOW it is possible to move. But when analyzing this motion we can see that there is an endless series of halfway points since you can always divide the remaining distance in half, so it would “SEEM” as though it should not be possible. THAT is the paradox. Yes the conclusion that we cannot move is obviously wrong, but how is it wrong. Premise A) any distance contains an infinite (endless) number of halfway points Premise B) in order to move accross/through/over these points you would have complete an infinite endless sequence of events halfway, halfway, halfway… Conclusion: Since it is endless you can never complete this motion Yes the argument is flawed, because we can move, but where is the flaw? ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [August 10, 2012, 11:45pm UTC](https://boards.straightdope.com/t/999-1/27517/612 "2012-08-10T23:45:20Z")
re Zeno’s Paradox, I’ve often wondered why it is granted that we can move half the distance from A to B. Shouldn’t the paradox recursively prohibit the first step, which is usually granted? Not only can Herakles not outrun the tortoise, he can’t even take the first step in the race. The starting pistol (sorry, sling) goes off, and mighty Herakles is frozen in amber, unable even to take half a step… or half of a half-step… or half of half of a half-step… Why do we grant that he can take the first great many steps, but only invoke the paradox on the “last” one? ETA: anyway, I take a great big stride – from my desk toward my refrigerator – and shout, "I refute it thus! ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 10, 2012, 11:48pm UTC](https://boards.straightdope.com/t/999-1/27517/613 "2012-08-10T23:48:55Z")
> [@erik150x](#): > > LOL… > > The crux of this paradox is that indeed we KNOW it is possible to move. But when analyzing this motion we can see that there is an endless series of halfway points since you can always divide the remaining distance in half, so it would “SEEM” as though it should not be possible. THAT is the paradox. Yes the conclusion that we cannot move is obviously wrong, but how is it wrong. > > Premise A) any distance contains an infinite (endless) number of halfway points > Premise B) in order to move accross/through/over these points you would have complete an infinite endless sequence of events halfway, halfway, halfway… > Conclusion: Since it is endless you can never complete this motion > > Yes the argument is flawed, because we can move, but where is the flaw? The flaw is that your conclusion does not follow from the premises. You are equivocating between different senses of “endless”, as **Frylock** has pointed out. What more is there to say? Premise B) is true in the sense that every event in the sequence of events is followed by another one in that sequence. But this does not mean that the entire sequence cannot be followed by some other event (as indeed it is). Consider this analogous argument: Premise: Any ruler contains an infinite (endless) number of halfway points: the 1 inch mark, the 1/2 inch mark, the 1/4 inch mark, etc. Conclusion: Since it is endless, the stick cannot have an end. There cannot be a 0 inch mark on the stick. Who would buy that? ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 10, 2012, 11:48pm UTC](https://boards.straightdope.com/t/999-1/27517/614 "2012-08-10T23:48:58Z")
> [@Trinopus](#): > > re Zeno’s Paradox, I’ve often wondered why it is granted that we can move half the distance from A to B. Shouldn’t the paradox recursively prohibit the first step, which is usually granted? > > Not only can Herakles not outrun the tortoise, he can’t even take the first step in the race. The starting pistol (sorry, sling) goes off, and mighty Herakles is frozen in amber, unable even to take half a step… or half of a half-step… or half of half of a half-step… > > Why do we grant that he can take the first great many steps, but only invoke the paradox on the “last” one? It is usually easier to make the argument about not reaching B, first. But yes the further conclusion is that you can make A to B shorter and shorter as to prohibit any movement at all. It’s too confusing to start out with, okay let’s say you have an infinitely small distance A-B… people start thinking about what that means… and you never get anywhere… pun intended 😃 ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 10, 2012, 11:49pm UTC](https://boards.straightdope.com/t/999-1/27517/615 "2012-08-10T23:49:12Z")
> [@erik150x](#): > > LOL… > > The crux of this paradox is that indeed we KNOW it is possible to move. But when analyzing this motion we can see that there is an endless series of halfway points since you can always divide the remaining distance in half, so it would “SEEM” as though it should not be possible. THAT is the paradox. Yes the conclusion that we cannot move is obviously wrong, but how is it wrong. > > Premise A) any distance contains an infinite (endless) number of halfway points > Premise B) in order to move accross/through/over these points you would have complete an infinite endless sequence of events halfway, halfway, halfway… > Conclusion: Since it is endless you can never complete this motion > > Yes the argument is flawed, because we can move, but where is the flaw? See my previous post–post 610–where I repeat what the flaw is. No, the flaw is not “We move by stepping forward at a positive speed.” That is the reasoning that we can move, but it is not the flaw in the reasoning that we can’t move. For the flaw in the reasoning that we can’t move, see post 610 (and two previous posts in this conversation but I won’t hammer on that point…) ETA And now, notice, **Indistinguishable** is making the very same point I make in post 610 and elsewhere. The flaw is that the argument against motion is invalid. We explain why above. That is what you asked for. We have delivered it. Several times. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 10, 2012, 11:55pm UTC](https://boards.straightdope.com/t/999-1/27517/616 "2012-08-10T23:55:40Z")
> [@Indistinguishable](#): > > The flaw is that your conclusion does not follow from the premises. You are equivocating between different senses of “endless”, as **Frylock** has pointed out. What more is there to say? Premise B) is true in the sense that every event in the sequence of events is followed by another one in that sequence. But this does not mean that the entire sequence cannot be followed by some other event (as indeed it is). > > Consider this analogous argument: > Premise: Any ruler contains an infinite (endless) number of halfway points: the 1 inch mark, the 1/2 inch mark, the 1/4 inch mark, etc. > Conclusion: Since it is endless, the stick cannot have an end. There cannot be a 0 inch mark on the stick. > > Who would buy that? Well, first lets be clear this is not MY conclusion… I merely stating the apparent paradox. RE"But this does not mean that the entire sequence cannot be followed by some other even" … indeed it could move over an endless/infinite set of points… but how do you ever get to the end/complete moving over and endless set of points? Getting to the end, implies an end. But that contradicts the idea that they are endless… ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 11, 2012, 12:00am UTC](https://boards.straightdope.com/t/999-1/27517/617 "2012-08-11T00:00:02Z")
> [@Frylock](#): > > My solution is, (as I said,) that the argument that you can’t reach B is invalid, because it illegitimately draws the conclusion that you “never” reach B when the premises actually only support a conclusion that you “don’t reach B at any of the points in time where you hit a 1/2^n mark on the line from A to B.” Who said anything about time? There are endless points… you must reach the end to get to B? But if they are endless how do you reach the end? You have explained nothing. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 11, 2012, 12:02am UTC](https://boards.straightdope.com/t/999-1/27517/618 "2012-08-11T00:02:13Z")
There is no end within the sequence. But there is an item _outside the sequence_ which comes after every item in the sequence. You can get to this item which comes after the entire sequence, because it exists. You can’t get to a last item within the sequence, because there’s no such thing. But the goal isn’t to get to a last item within the sequence. To goal is to get to each particular item within the sequence at some time, which you can do. And you can even, after having done all that, get to another item (outside the sequence) which comes every item in the sequence. In lexicographic order, there is no end to the strings beginning with the character A; I can construct the sequence A, AA, AAA, AAAA, in which each string would be placed in a dictionary before the next one. This is an infinite, endless sequence. But that doesn’t mean there’s no string which comes after all of them; AB comes after all of them. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 11, 2012, 12:04am UTC](https://boards.straightdope.com/t/999-1/27517/619 "2012-08-11T00:04:11Z")
> [@erik150x](#): > > Well, first lets be clear this is not MY conclusion… We know this. > [@](#): > > I merely stating the apparent paradox. . We know this as well. We’ve resolved it by showing how the paradox’s reasoning is invalid. > [@](#): > > RE"But this does not mean that the entire sequence cannot be followed by some other even" … indeed it could move over an endless/infinite set of points… but how do you ever get to the end/complete moving over and endless set of points? Getting to the end, implies an end. But that contradicts the idea that they are endless… We don’t need to know how, because you’ve offered no successful argument against the view that we can do it. We showed how the argument you offered turns out to be invalid. Once we’ve shown that the argument against motion is invalid, there is no longer a “how” question to answer. “How?” in this context _means_ “How is it possible in the face of its apparent impossibility?” Once we’ve shown that the apparent impossibility is based on flawed reasoning (as we have done above) we have thereby _answered_ your “how” question. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 11, 2012, 12:05am UTC](https://boards.straightdope.com/t/999-1/27517/620 "2012-08-11T00:05:52Z")
> [@erik150x](#): > > Who said anything about time? . I did. What is illegitimate about it? [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=30) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=32)