# Understanding the infinate **URL:** \ **Category:** Factual Questions\ **Created:** [September 17, 2008, 7:59pm UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023 "2008-09-17T19:59:21Z")\ **Posts on this page:** 10\ **Page:** 2
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [September 18, 2008, 11:23pm UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023/21 "2008-09-18T23:23:30Z")
> [@CurtC](#): > > Sort of along the lines of what ultrafilter said, but every one of those segments has a nonzero positive length. Every one of them. I disagree. If you have infinite points then their length is zero and they all add up to 5. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 18, 2008, 11:27pm UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023/22 "2008-09-18T23:27:26Z")
> [@sailor](#): > > I disagree. If you have infinite points then their length is zero and they all add up to 5. No one’s saying points have length; we’re saying you can have infinitely many segments (i.e., intervals), each of positive length, all within a finite interval. As in the OP’s example, as pointed out by **CurtC** and in my last post. ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [September 19, 2008, 12:42am UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023/23 "2008-09-19T00:42:03Z")
> [@Indistinguishable](#): > > No one’s saying points have length; we’re saying you can have infinitely many segments (i.e., intervals), each of positive length, all within a finite interval. As in the OP’s example, as pointed out by **CurtC** and in my last post. I disagree. If you have infinite segments then their individual length is zero and they all add up to 5. There is no way to have an infinite sum of segments of finite legth add up to a finite number. No way. The total length of the original segment is 5. If you divide it into n segments then the length of each segment is 5/n and the total sum is n \* 5/n. It will always add up to 5. At the same time n-\> infinite, 5/n -\> zero. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 19, 2008, 12:51am UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023/24 "2008-09-19T00:51:38Z")
> [@sailor](#): > > I disagree. If you have infinite segments then their individual length is zero and they all add up to 5. > > There is no way to have an infinite sum of segments of finite legth add up to a finite number. No way. > > The total length of the original segment is 5. If you divide it into n segments then the length of each segment is 5/n and the total sum is n \* 5/n. It will always add up to 5. At the same time n-\> infinite, 5/n -\> zero. That’s only if you divide it into segments of equal length. If they aren’t of equal length, then there can be infinitely many of them, all disjoint and of positive finite length, and all contained within a finite interval. Did you look at the example we’ve provided? We’ve explicitly provided an example of the very thing you say is impossible! To reiterate: consider the following segments, from the OP: ```auto Start End Length 4 4.9 0.9 4.9 4.99 0.09 4.99 4.999 0.009 ... ... ... All contained within: Start End Length 4 5 1 ``` There are infinitely many of them, all disjoint, all of positive finite length, and all contained within a finite interval. The thing you are calling impossible is actually very possible, as demonstrated by the very first post in this thread. You seem to be too hooked on analogy with the case where the segments all have equal length, but that is not the case under discussion. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 19, 2008, 1:08am UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023/25 "2008-09-19T01:08:55Z")
(I am embarrassed to note that **sailor** ’s stumbling block seems to be “No way, how can an infinite sum add up to something finite?”, the very question which I’ve so often described as but a distraction in the resolution of Zeno’s paradox, despite the oft-touted wisdom that it is the crux. Oh well…) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 19, 2008, 1:22am UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023/26 "2008-09-19T01:22:57Z")
(Also, I should say, I have no qualms with discussion of infinitesimals; indeed, I find clinging to traditional analysis for traditional analysis’s sake to be a major annoyance. It’s not at all clear that all analysis of linear measurement for all problems should be done using the very particular tool of a classical Dedekind-complete Archimedean, etc., line; math need not contort itself into one-size-fits-all. But, all the same, both within that framework and outside of it, the impossibility claims **sailor** is making simply aren’t true. Any particular answer to the question of the existence of infinitesimal lengths notwithstanding, the things **sailor** is calling impossible actually are possible) ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [September 19, 2008, 1:55pm UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023/27 "2008-09-19T13:55:46Z")
OK, I have to admit I posted in haste. I am trying to participate in several threads when I really do not have the time so I am really not giving them the thought and attention they need. I withdraw everything I have posted in this thread because, as I say, I really do not have the time to give this the attention it needs. Please carry on. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [September 19, 2008, 5:01pm UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023/28 "2008-09-19T17:01:15Z")
A couple of comments. First, infinite decimals are not the best way to represent real numbers. I defy you explain how to square, say .888888… . Even simple addition is complicated. Convergent sequences of rationals is one way of dealing with the issue. So many students identify numbers with their decimal representations, which is wrong. It is funny how many are perfectly happy with 1/2 = 2/4, but balk at .9999… = 1. Second, it is possible, maybe quite likely, that both time and space are quantized and each quantum of space is traversed in some number of time quanta (the number being, in appropriate units, what we call speed (not velocity since the number is not a vector). Where are you “in between”? There is no in-between. You are here or you are there. ---
**Author:** ![copperwindow](https://avatars.discourse-cdn.com/v4/letter/c/4af34b/32.png) [@copperwindow](https://boards.straightdope.com/u/copperwindow)\ **Post date:** [September 20, 2008, 5:42pm UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023/29 "2008-09-20T17:42:43Z")
Okay, let me ask the question like this: If the door is five feet away from me, can I in theory walk towards it, never stop moving, and never reach it by continuously walking half the distance with each step to the door? ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [September 20, 2008, 6:01pm UTC](https://boards.straightdope.com/t/understanding-the-infinate/464023/30 "2008-09-20T18:01:09Z")
Sure you can, just as long as you don’t have to walk at the same speed all the time. Take one second to walk the first half of the distance, take one second to walk the next one-quarter of the distance, take one second to walk the next one-eighth of the distance, etc. You’ll never reach the door and you’ll never stop moving. Of course, practically speaking, pretty soon you will be moving so slowly that no motion will be noticeable to anyone, and pretty soon you will be so close to the door that no one will be able to see that any distance between you and the door. [Previous page](https://boards.straightdope.com/t/understanding-the-infinate/464023.md?page=1)