# .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:** 27
**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\ **Post date:** [August 8, 2012, 9:07pm UTC](https://boards.straightdope.com/t/999-1/27517/521 "2012-08-08T21:07:22Z")
> [@Trinopus](#): > > Who says math isn’t fun? Well, certainly not me! ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 8, 2012, 9:24pm UTC](https://boards.straightdope.com/t/999-1/27517/522 "2012-08-08T21:24:55Z")
It’s not an unreasonable thing to say, on some interpretations of “math” in some contexts, though of course on others… 🙂 ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 9, 2012, 3:21am UTC](https://boards.straightdope.com/t/999-1/27517/523 "2012-08-09T03:21:55Z")
> [@Buck\_Godot](#): > > I admit that I didn’t wade through all of the pages before this so I may be repeating others arguments: > > Eric, I understand that the last 1 was written at exactly 1 hour, what time was the second to last number written? > > I am confused that you t that you accept the solution to Zeno’s paradox that 1/2+1/4+1/8+…=1, while having problems with 9/10+9/100+9/1000=1. It is exactly the same thing, except instead of each step getting you half way to the end, each step gets you 90% of the way to the end. > It seems to me that you are trying to create a new number system that is distinct from that used my mathematicians. This number system includes all of real numbers, and also includes a number, alpha, such that alpha is greater than 0, but alpha is smaller than any non-zero number. You might be able to make a consistent system based on this number, for example defining n_alpha=alpha for any non-zero n, but in the process you would likely lose some of the properties of the reals. For example, 3_alpha-2\*alpha = alpha-alpha = 0 which is not equal to (3-2)\*alpha= alpha. So you lose the commutative property of the reals. If you want to do so for you won amusement you could play with this idea and let us know what sort of number system you end up with and what properties do and don’t hold, but whatever you come up with, it isn’t the reals. RE:“Eric, I understand that the last 1 was written at exactly 1 hour, what time was the second to last number written?” -An excellent question and I don’t think I can answer that. And I don’t mean to say someone else couldn’t figure out a meaningful answer, just not I at this time. But I admit I am not going to put much effort into it if it’s even currently answerable at all. I only wished to show that the infinite’th decimal place exists and how to locate that. - Regarding my choice of series 1/2^i vs 9/10^i, I was aware of that and almost thought about changing it. The 1/2^i is the one I am most familiar with being associated with this Zeno’s Paradox. I figured it would relate better to those also familiar with it. As far as 9/10^i = 1, (I’ll just talk about that one since that has been really the heart of the debate here I think and no less the title of the thread), I have been waiting for this question. And I have to admit it makes me waiver back and forth. I have been not totally closed to the idea that it does = 1, just vehemently opposed to using limits to “prove” this. I still prefer to think it = .999… First of all as several people have been discussing, (and I haven’t talked about it too much myself, but it’s been interesting to read), what exactly does the ellipsis mean? Clearly it’s related to infinity but “just” the countable or larger? I think we have been in general accepting it as “just” the countable version, but to be honest I don’t know what the proper definition is. For now let’s stick with countable. So then does it include infinity (or what I have been referring to as the infinite’th decimal place), or does it just go up to this decimal place? I and probably most would say it includes the infinite’th place. Now notice in this series as we sum: 9/10,99/100,999/1000, … the numerator is always once less than the denominator. So when we get to the infinite’th decimal place we actually have something like this (infinity-1)/(infinity). So actual to reach point B (the end of the line segment), we must add one more little step (1/infinity). I thought I might have to adjust the proof to account for this, but was waiting for someone to actually point this out. So the complete number of elements in the series is infinity + 1. Which would only make sense in my non-standard analysis of reals. Now before everyone jumps all over my back about the fact that infinity-1=infinity and infinity+1=infinity, maybe I am choosing a poor chose of notation. Maybe I don’t really mean (infinity-1) but instead whatever number comes just before infinity. Errr.. I don’t know man. I told you have not figured this all out. I mean that is one of the ultimate kind of questions one could ask, if it makes any sense. Infinity is not a regular number, nor is an infinitesimal that is clear. But it would seem clear if we started out at 0 and ended up at infinity, there must have been some last number before we got there…? Please it’s not like I have the universe all figured out ;-). I guess that last number before 1 can best be expressed as .999…, if we had a better way to say it we wouldn’t write .999… Anyway I hope addressed your question to some satisfaction, but in any case that’s all I got for you at the moment anyway. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 9, 2012, 3:30am UTC](https://boards.straightdope.com/t/999-1/27517/524 "2012-08-09T03:30:39Z")
> [@Francis\_Vaughan](#): > > **Indistinguishable** provided the proof earlier. Or rather a counterpoint to the fallacy. Of course I can move though an infinite number of points trivially at any time. I do it all the time. The fallacy is that there is some notion that you can’t have an infinite number of points in a finite interval. In Zeno’s case, the idea that you can’t cover an infinite number of points in a line in a finite time. But the paradox itself shows how you can have an infinite number of points in a finite interval. It simply blandly asserts, with no justification that whist you can have them in space, you can’t have them in time. There is no justification for such an assertion. If I can have an infinite number of points in a finite interval on the X axis, I can have an infinite number of points in a finite interval on the Y axis. That is how I am able to draw diagonal lines. This implicit, unjustified assumption is why worrying about Zeno and anything derived from it is simply not relevant. Time flows, and covers an infinitude of instants in time in any interval of time. So to does movement cover an infinitude of points. Neither prevents the other. > > I will also point out that the halving of distance algorithm for generating the infinite number of points is but one of an further infinitude of ways of generating other sets of an infinity of points in that same finite interval. Why just halving? I can come up with an infinitude of infinite series that converge to one. I cover all of the points in all of them as I cover that same finite interval from zero to one. They all converge to one. So, how does your algorithm for finding the last infinite digit of 0.9999… find that particular point amongst the infinitude of “last” points that belong to all those other infinite series? They are all the last infinite point of a series that converges to one. Given that I can choose infinite series with real terms, not just integer terms, there are more than just Aleph Null series here too. > > The issue with the inconsistency of the arithmetic has been belaboured here. You can choose a system where your particular notation and idea works. But it isn’t the real numbers. The real numbers are what we assume the decimal notation of the infinite series 0.999… is addressing. All conversations, in this and all other threads assume you mean the reals. You mean the reals too. That is clear. But the reals have a very clear set of properties. The provided notation for 0.999…9 and the operations you define upon that - in particular the property of the “infinitieth” digit can be shown to lead to a contradiction of the properties of the reals. Thus we have proven that whatever various systems of numbers your proof covers, the real numbers isn’t one of them. FV, RE: “The fallacy is that there is some notion that you can’t have an infinite number of points in a finite interval” From what your now saying I think you are misinterpreting Zeno’s Paradox here. It is not at all about the idea that there can’t be an infinite number of points in a finite interval. The paradox is how can one ever complete an infinite sequence IN TIME. Movement does not happen outside of time. When one first thinks about completing an infinite number of anything, the first reaction is it can’t be done. I have been assuming you are maybe more familiar with how it us usually presented as a paradox and not in terms of the solution as I have been talking about it mostly. The paradox I am referring to goes more or less like this, (and there are various versions): You want to move from point A to point B, but in the process you must always go half way first, but then there is always another half of the remaining distance, so how can you ever complete it? I have already presented the well know solution that each half takes half the time so in the end it is possibly as you pass more and more points faster and faster all the way up to infinite points/unit of time. You then talk about separating space and time as easy as one separates the x and y axis. Well, you can go down that road if you want. I would say that’s even more controversial than anything I am talking about here. But none the less it’s open for debate. Its hard to conceive of you anything can “happen” outside of time. With the very well accepted Theories of Relativity most physicist would certainly tell you space and time are inseparable. That is why they call it Space-time. RE: “I will also point out that the halving of distance algorithm for generating the infinite number of points is but one of an further infinitude of ways of generating other sets of an infinity of points in that same finite interval. Why just halving? I can come up with an infinitude of infinite series that converge to one” Uhm, yes it’s completely obvious. So what? I picked this because the one most often used with this paradox? What’s your point? (no pun intended) In my opinion you make no case against anything in my proof, except maybe that space and time can be separated, and even then I am not sure what you gain??? There is no paradox; I have already showed the solution about 100 times… ok maybe 10. Claiming that there is no paradox, proves nothing. I am simply using the most overwhelming used solution as a paradigm for mapping an infinite number of decimal places and showing that the infinitely small exist. The point is this… I have heard many time on previous postings there is no end to .999… there is no way to find it. My proof is to contradict this. Your arguments seem to be totally oblivious to this? So to be honest you kind lose me in all this. Maybe I misunderstanding and you could clarify? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 9, 2012, 3:33am UTC](https://boards.straightdope.com/t/999-1/27517/525 "2012-08-09T03:33:27Z")
> [@erik150x](#): > > From what your now saying I think you are misinterpreting Zeno’s Paradox here. It is not at all about the idea that there can’t be an infinite number of points in a finite interval. The paradox is how can one ever complete an infinite sequence IN TIME. Movement does not happen outside of time. Suppose I were to say “It can never become midnight. Because first it must become 11:00, and then 11:30, and then 11:45, and then halfway between 11:45 and midnight, and then… So infinitely many things must happen before it can become midnight. But infinitely many things cannot happen in a finite amount of time, so it can never become midnight.” Would this seem a reasonable argument that it can never become midnight? Or would you say “Wait, who says infinitely many things cannot happen in a finite amount of time? You yourself have just pointed out how infinitely many things can happen in one hour”? ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 9, 2012, 3:35am UTC](https://boards.straightdope.com/t/999-1/27517/526 "2012-08-09T03:35:07Z")
> [@Indistinguishable](#): > > Right. For what it’s worth, my point isn’t just to invoke the corresponding time series. The point is to note that few people feel any discomfort with a finite distance containing infinitely many points, that they are correct in that lack of discomfort, and that there is no reason to feel any differently when speaking of finite intervals of time having the same property. > > The paradox asks us to rush from “There are infinitely many points” to “You can’t cover them all in a finite stretch”, and this is completely fallacious. Pointing out the difference in people’s naive attitudes towards space and towards time can help clarify that. > > I’m not one of those guys who’s boorishly dismissive of philosophy. But I am sort of dismissive of the amount of attention Zeno’s paradoxes of motion have gotten, particularly when people come to regard them as needing sophisticated mathematical treatment. It’s the analogue of having spent centuries continuing to debate “…But that’s only $29 between the bellhop and the guests. Where’s the missing dollar?”, and coming to a sense that the problem was only soundly resolved with the development of measure theory. Well, I think you’re a little dismissive. It’s easy for you to say in hind sight, and I can swear you were just saying a few pages ago a finite length was not infinitely divisible, something about “Bosh”? Anyway, it’s not nearly as simple at the $29 between the bellhop and the guests. The reason being one must accept not one, but two infinite series take place (of time and distance), while for someone who has not yet accepted that infinities and infinitesimals for that matter exist, it can be daunting to solve. But I guess you have always just taken those things for granted? Maybe you’re just that smart? If the first time you heard the paradox was after studying calculus, maybe it was obvious. Lastly, I don’t see why anybody is talking about that at all, I have already presented the standard solution, which is saying THERE IS NO PARADOX. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 9, 2012, 3:53am UTC](https://boards.straightdope.com/t/999-1/27517/527 "2012-08-09T03:53:51Z")
Anyways, I think this whole discussion is getting a little played out from my perspective. I din’t come on here to say I have some new system of math which can deal with all the difficulties of infinitesimals. I think the beef I have with people just blatantly stating, “Oh yeah, .999… = 1” no questions asked. As I have stated a couple times already, the essence of this question, especially form the uninitiated is how is it that .9 \< 1 and .99 \< 1 and .999 \< 1 but somewhere down the line it becomes = 1? It’s a perfectly valid question. I can guarantee 9/10s of the time they do not want so explanation as to how we have rigged the real number system to ignore the question by definition of limits. Or a proof that uses math which is founded on limits like the standard: x=.999… 10x=9.999… 10x - x = 9.999… - .999… 9x = 9 x = 1 It’s a silly deception away from the persons real intuitive notion, because many don’t realize you can only justifiably multiply 10 x .999… and say it’s 9.999… because we define these numbers with limits built in. Normally you have to shift in a zero at the end. As someone on here said it best, “The 9s don’t just keep coming” - at least not necessarily. It’s possible that it is true that .999… = 1 but Limits don’t prove this, they avoid it by definition. Yeah infinities and infinitesimals can make you crazy, but that’s the fun of them. 😉 ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 9, 2012, 4:01am UTC](https://boards.straightdope.com/t/999-1/27517/528 "2012-08-09T04:01:13Z")
> [@Indistinguishable](#): > > Suppose I were to say “It can never become midnight. Because first it must become 11:00, and then 11:30, and then 11:45, and then halfway between 11:45 and midnight, and then… So infinitely many things must happen before it can become midnight. But infinitely many things cannot happen in a finite amount of time, so it can never become midnight.” > > Would this seem a reasonable argument that it can never become midnight? Or would you say “Wait, who says infinitely many things cannot happen in a finite amount of time? You yourself have just pointed out how infinitely many things can happen in one hour”? I understand your point. I was never posing that this paradox was anything anyone on this board might get tripped up on. I was using the solution as model. It is obvious to me and I am sure you. Most people are puzzled by those types of questions upon first encountering these types of questions, especially if they have never had higher math. But the fact that it is not really a problem or puzzle is fine. It doesn’t negate my proof what so ever. Personally at times when i was younger I had considered that time and/or space may not be infinitely divisible for the idea of infinity or infinitesimals just wouldn’t settle into my head. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [August 9, 2012, 4:03am UTC](https://boards.straightdope.com/t/999-1/27517/529 "2012-08-09T04:03:43Z")
> [@erik150x](#): > > I think the beef I have with people just blatantly stating, “Oh yeah, .999… = 1” no questions asked. As I have stated a couple times already, the essence of this question, especially form the uninitiated is how is it that .9 \< 1 and .99 \< 1 and .999 \< 1 but somewhere down the line it becomes = 1? It’s a perfectly valid question. I don’t think it is. I wish **Indistinquishable** would address this as confirmation, but I think .999… only results from faulty algorithms and it is a representation of the value 1. If I am correct, then using the representation .999… for anything other than indicating that arithmetic operations end up in a non-terminating loop would be imprecise. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [August 9, 2012, 4:04am UTC](https://boards.straightdope.com/t/999-1/27517/530 "2012-08-09T04:04:04Z")
> [@Indistinguishable](#): > > That’s a sharp idea. That works. But you’d need more than two coordinates once you started doing multiplication. (0, 1) \* (0, 1) = 10[sup]-∞[/sup] \* 10[sup]-∞[/sup] = 10[sup]-2∞[/sup]. We might write this as (0, 0, 1). (So the first coordinate is multiplied by 1, the second coordinate is multiplied by 10[sup]∞[/sup], the third coordinate is multiplied by 10[sup]-2∞[/sup], and so on, and so on. The whole thing is essentially written in base 10[sup]-∞[/sup], or, read the other way, as a “decimal” in base 10[sup]∞[/sup]). Or you could just call it 0. Indeed; that’s how the [dual numbers](http://en.wikipedia.org/wiki/Dual_number) work. We define an ε such that ε is nonzero but ε[sup]2[/sup] = 0, and then can do math along the lines of complex numbers. (1, 0)_(0, 1) = (0, 1). (0, 1)_(0, 1) = (0, 0). In general, (a, b)_(c, d) = (a_c, a_d+b_c). Unlike complex numbers, the b\*d term disappears. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [August 9, 2012, 4:09am UTC](https://boards.straightdope.com/t/999-1/27517/531 "2012-08-09T04:09:04Z")
> [@Indistinguishable](#): > > Suppose I were to say “It can never become midnight. Because first it must become 11:00, and then 11:30, and then 11:45, and then halfway between 11:45 and midnight, and then… So infinitely many things must happen before it can become midnight. But infinitely many things cannot happen in a finite amount of time, so it can never become midnight.” . . . My favorite meta-Zeno riddle is the discrete “supertask.” At eleven, I switch the light on. At 11:30, I switch it off. At 11:45, I switch it on. At 11:52 and a half, I switch it off… At midnight, exactly, is the light on or off? ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [August 9, 2012, 4:11am UTC](https://boards.straightdope.com/t/999-1/27517/532 "2012-08-09T04:11:43Z")
> [@Trinopus](#): > > My favorite meta-Zeno riddle is the discrete “supertask.” > > At eleven, I switch the light on. At 11:30, I switch it off. At 11:45, I switch it on. At 11:52 and a half, I switch it off… > > At midnight, exactly, is the light on or off? How long does it take you to change the setting of the switch? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 9, 2012, 4:12am UTC](https://boards.straightdope.com/t/999-1/27517/533 "2012-08-09T04:12:34Z")
Re: **Dr. Strangelove** Yes, that’s also a nice system. Not inserting that condition and sticking with the full polynomial amounts to setting f(ε) = the Taylor series of ε, while inserting the condition that ε[sup]n[/sup] = 0 amounts to only caring about first n derivatives. Of course, you can recover the full Taylor series from just iterating the 1st derivative, so you can get an awful lot of calculus done awfully nicely with just the dual numbers. Re: **Trinopus** : It depends on your definitions/interpretation/assumptions, like everything else. 🙂 (Given that question out of the blue, my only response is to say “There’s no well defined answer. Why should I assume the state at midnight is fixed by the states prior to midnight? If you give me a rule saying in what manner this is to be fixed, then I can give you an answer”) ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [August 9, 2012, 4:18am UTC](https://boards.straightdope.com/t/999-1/27517/534 "2012-08-09T04:18:06Z")
> [@Indistinguishable](#): > > Re: **Trinopus** : > > It depends on your definitions/interpretation/assumptions, like everything else. 🙂 > > (Given that question out of the blue, my only response is to say “There’s no well defined answer. Why should I assume the state at midnight is fixed by the states prior to midnight? If you give me a rule saying in what manner this is to be fixed, then I can give you an answer”) If it took no time at all to perform the switch, could you assume it is off because the there are the same number of infinite switches to on as there are switches to off between any two points in time? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 9, 2012, 4:19am UTC](https://boards.straightdope.com/t/999-1/27517/535 "2012-08-09T04:19:14Z")
> [@Indistinguishable](#): > > while inserting the condition that ε[sup]n[/sup] = 0 amounts to only caring about first n derivatives. Er, the condition that ε[sup]n + 1[/sup] = 0. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 9, 2012, 4:20am UTC](https://boards.straightdope.com/t/999-1/27517/536 "2012-08-09T04:20:50Z")
> [@TriPolar](#): > > If it took no time at all to perform the switch, could you assume it is off because the there are the same number of infinite switches to on as there are switches to off between any two points in time? Between _any_ two points in time? That isn’t true. There are intervals when it switches on twice and off once, for example. Overall? Yes, there is a bijection between switches on and switches off, but there is also a bijection between switches off and all but one switch on. So perhaps you could tell a story along these lines, but none strike me as particularly compelling. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [August 9, 2012, 4:22am UTC](https://boards.straightdope.com/t/999-1/27517/537 "2012-08-09T04:22:25Z")
> [@Indistinguishable](#): > > Between _any_ two points in time? That isn’t true. There are intervals when it switches on twice and off once, for example. > > Overall? Yes, there is a bijection between switches on and switches off, but there is also a bijection between switches off and all but one switch on. > > So perhaps you could tell a story along these lines, but none strike me as particularly compelling. Oh well, I gave it a shot. Could you address my previous post? I’d like to know if that holds up. And ‘bijection’, cool word. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 9, 2012, 4:28am UTC](https://boards.straightdope.com/t/999-1/27517/538 "2012-08-09T04:28:37Z")
> [@Buck\_Godot](#): > > I admit that I didn’t wade through all of the pages before this so I may be repeating others arguments: > > Eric, I understand that the last 1 was written at exactly 1 hour, what time was the second to last number written? > > I am confused that you t that you accept the solution to Zeno’s paradox that 1/2+1/4+1/8+…=1, while having problems with 9/10+9/100+9/1000=1. It is exactly the same thing, except instead of each step getting you half way to the end, each step gets you 90% of the way to the end. > It seems to me that you are trying to create a new number system that is distinct from that used my mathematicians. This number system includes all of real numbers, and also includes a number, alpha, such that alpha is greater than 0, but alpha is smaller than any non-zero number. You might be able to make a consistent system based on this number, for example defining n_alpha=alpha for any non-zero n, but in the process you would likely lose some of the properties of the reals. For example, 3_alpha-2\*alpha = alpha-alpha = 0 which is not equal to (3-2)\*alpha= alpha. So you lose the commutative property of the reals. If you want to do so for you won amusement you could play with this idea and let us know what sort of number system you end up with and what properties do and don’t hold, but whatever you come up with, it isn’t the reals. RE: “Eric, I understand that the last 1 was written at exactly 1 hour, what time was the second to last number written?” At first I misread this I thought you were asking basically what was the decimal place or sum of the series at 1 sec before the last number was written. I don’t know why I misread it so badly… LOL, long day. Anyway what you are asking… the second to last number? (in the .000…1) series? or the distance series? or the time series? okay, so for .000…1 that’s obvious 0 For the other two questions that’s “like” asking what’s the last number before infinity. I think you know darn well I can’t answer that 😉 I know how many licks to get to the center of a tootsie-pop though! ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [August 9, 2012, 4:39am UTC](https://boards.straightdope.com/t/999-1/27517/539 "2012-08-09T04:39:35Z")
> [@Trinopus](#): > > At midnight, exactly, is the light on or off? Well. If we flip it once, the light is on. And if we flip it 1+1 times, the light is off. 1+1+1 times and the light is on again. Clearly, the light is on if we flip it an odd number of times, and off if even. Since [1+1+1+1+… = -1/2](http://en.wikipedia.org/wiki/1_%2B_1_%2B_1_%2B_1_%2B_%C2%B7_%C2%B7_%C2%B7), clearly at the end the switch will be perfectly balanced between the off and on states. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 9, 2012, 4:53am UTC](https://boards.straightdope.com/t/999-1/27517/540 "2012-08-09T04:53:41Z")
In case anyone wants a layperson’s understanding of the sense in which 1 + 1 + 1 + … = -1/2 (as always, it’s interpretation-dependent, of course. No one’s denying that there aren’t other senses in which this sum is of course positively infinite. It’s up to you what you want to model and what rules would be useful for that application): Let A = -1 + 1 - 1 + 1 - … . Note that A + (A shifted over one spot) = -1 + 0 + 0 + 0 + … = -1. Thus, 2A = -1. Thus A = -1/2. Let B = 1 + 1 + 1 + 1 + … . Note that A + B = 0 + 2 + 0 + 2 + … = zeros interleaved with 2B. Thus, A + B = 2B. Thus B = A = -1/2. [The result for the alternating series A is actually rather more robust than the result for B; there are many more contexts in which one would want to say A = -1/2 than B = -1/2, essentially because the business about interleaving works out less nicely in many contexts than the business about shifting. But there are some where all this reasoning hits the spot.] In the same sense: Let C = -1 + 2 - 3 + 4 - … . Note that C + (C shifted over one spot) = -1 + 1 - 1 + 1 - … = A. Thus, 2C = A = -1/2. Thus, C = -1/4. Let D = 1 + 2 + 3 + 4 + … . Note that C + D = 0 + 4 + 0 + 8 + … = zeros interleaved with 4D. Thus, C + D = 4D. Thus, 3D = C = -1/4. Thus, D = -1/12. [Again, the result for the alternating series C is more robust than that for D] [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=26) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=28)