# .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:** 35
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 13, 2012, 1:44am UTC](https://boards.straightdope.com/t/999-1/27517/681 "2012-08-13T01:44:11Z")
> [@Frylock](#): > > You’ve been trying to argue with me that there must be a last member of the infinite series 1/2, 3/4, 7/8, … But your argument has relied completely on questions asked about what happens when you traverse the _real number line_. If you’re talking about some model other than the _real number line_ then you need to explain what model that is. As long as you’re talking (as you have been, in your arguments) about how to traverse the _real number line_, you are by definition incorrect in your claim that there must be a last member of the infinite series denoted above. I am talking about traversing an infinite set. ---
**Author:** ![The\_Hamster\_King](https://avatars.discourse-cdn.com/v4/letter/t/8edcca/32.png) [@The\_Hamster\_King](https://boards.straightdope.com/u/The_Hamster_King)\ **Post date:** [August 13, 2012, 1:46am UTC](https://boards.straightdope.com/t/999-1/27517/682 "2012-08-13T01:46:06Z")
> [@erik150x](#): > > Is 0.0…05 the same number as 0.0…5? Yes. 0.0…1 / 2 = 0.0…05 0.0…05 = 0.0…5 0.0…5 = 0.0…1 \* 5 0.0…1 / 2 = 0.0…1 \* 5 x/2 = x\*5 where x is non-zero. WTF? ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 13, 2012, 1:48am UTC](https://boards.straightdope.com/t/999-1/27517/683 "2012-08-13T01:48:17Z")
> [@Frylock](#): > > Augh. I’m sorry, but augh. (Hey, you said above we were “killing you” so a little expression of emotion is warranted.) You’re right that no one would “argue it,” but what someone \*would \* do in their right mind is _build a model in which 1+1=/=2_ and see where they can go with it. It’s not a question of whether the statement is true. It’s a question of what the system it is embedded in can model. > > Similarly, you seem to think you’re talking to us about whether it’s “really really true” that 0.999… = 1. You admit that it’s true by definition in one system, but then you go on to try to argue that it’s not “really” true or at least, not “really really” true in some sense. But this is a nonstarter. None of us is concerned about whether it’s true in some sense independent of any mathematical system you might care to build. There’s literally no answer to that question–it’s actually a senseless question that cannot have an answer. The only thing you could say that could be interesting would be to show us a mathematical system in which the two notations do not refer to the same value, even though in some sense they refer similarly to the way they refer in “normal math.” For this to be worthwhile, said mathematical system has some interesting properties that make it worth studying. As you’ve seen, someone has already built such a system–the hyperreals. Go learn about it! See if it gets you the kind of result you’re looking for. If not, go to school! Get a PhD! Build your own system! Everything else is pretty useless for your (or anyone else’s) purpose. > > There are no “deeper” truths about what equals what. There are just different models in which different statements are true. If there are “deeper” truths they concern metamathematical matters about relationships between mathematical systems, the expressive power of different systems, and so on. But identity is quite literally a shallow truth, simply stipulated by fiat in each mathematical system. There are no deeper truths about which expressions should have an equal sign placed between them. Different systems are different, and that is all. I would tend to disagree with your statements in general. The rules of math, number systems are not made arbitrarily. The are made to reflect an deeper understanding of our “real” world. One could play games with arbitrary number systems to pose what if scenarios. Calculus was born of necessity to describe real things, to figure out real answers. So I guess I just plain disagree with you there. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 13, 2012, 1:53am UTC](https://boards.straightdope.com/t/999-1/27517/684 "2012-08-13T01:53:42Z")
> [@erik150x](#): > > I am talking about traversing an infinite set. To do anything we might intuitively call ‘traversing’ a set, we need to know what order the elements of that set appear in. The real number line is an ordering of the reals. It is not the only ordering of the reals. It is not a uniquely or deeply ‘true’ ordering of the reals. It’s just one way to order them. (It’s an ordering that yields a model that’s very useful, but that doesn’t make it uniquely or deeply ‘true’ any more than a really good hammer is uniquely or deeply ‘true’. It’s just a tool.) The real number line is the ordering of the reals which you are implicitly relying on in your arguments–you talk in terms of hitting the halfway mark, then the next halfway mark, and so on. The most natural way to interpret what you’re saying is that you mean for us to order the reals as they are ordered on the real number line, then determine, on that ordering, what the “last” member of the set {1/2, 3/4, 7/8, …} is. But on that ordering, it is trivial to prove that there is no last member. In order for there to be a last member of a set including those elements, you have to reorder them. I can give that set both a first and a last member. Order them like this. {1/2, 7/8, 31/32, …, 15/16, 3/4}. Now the last element is 3/4. You could even embed this in a complete ordering of the reals, though it’s beyond my understanding to know how to specify any such ordering. The existence of terminal members of infinite sets depends on how you order the sets. If you order the reals as they are on the real number line, then there simply is no last member of the halfway-halfway-halfway set we’ve been talking about. If you order the reals differently, there might be a last element. (Indeed, order them in reverse and you trivially have a last element–1/2.) Go to school! Get a PhD! The kind of thing you’re trying to do is going to quickly turn into grad school level stuff. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 13, 2012, 1:54am UTC](https://boards.straightdope.com/t/999-1/27517/685 "2012-08-13T01:54:05Z")
> [@The\_Hamster\_King](#): > > 0.0…1 / 2 = 0.0…05 > 0.0…05 = 0.0…5 > 0.0…5 = 0.0…1 \* 5 > 0.0…1 / 2 = 0.0…1 \* 5 > > x/2 = x\*5 where x is non-zero. > > WTF? I guess i have not explain my notation enough. The second and third statements would be true. The first and last are incorrect: 0.0…1 / 2 = 0.0…05 should be 0.0…1 / 2 = 0.0…0.5 and again: 0.0…1 / 2 = 0.0…1 \* 5 should be: 0.0…1 / 2 = 0.0…0.5 where .5 here represents the infnite’th decimal place + 1. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 13, 2012, 2:00am UTC](https://boards.straightdope.com/t/999-1/27517/686 "2012-08-13T02:00:41Z")
One thing I’m trying to illustrate to you in my previous post is that when you think you’re doing something simple and obvious (e.g. “traversing” a set) it turns out there’s a _ton_ of stuff going on under the surface that you’re not even aware of, and which makes the actual implications of your “traversal” opaque to you. I know you know that in the abstract, but in order to pursue the questions you’re pursuing, you need to be in a situation where you don’t just know about the ton of stuff in the abstract but rather you have others who know all about that ton of stuff who can show it to you when it is relevant to your pursuit. Otherwise you will almost certainly never see it. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 13, 2012, 2:03am UTC](https://boards.straightdope.com/t/999-1/27517/687 "2012-08-13T02:03:03Z")
> [@Frylock](#): > > To do anything we might intuitively call ‘traversing’ a set, we need to know what order the elements of that set appear in. The real number line is an ordering of the reals. It is not the only ordering of the reals. It is not a uniquely or deeply ‘true’ ordering of the reals. It’s just one way to order them. (It’s an ordering that yields a model that’s very useful, but that doesn’t make it uniquely or deeply ‘true’ any more than a really good hammer is uniquely or deeply ‘true’. It’s just a tool.) > > The real number line is the ordering of the reals which you are implicitly relying on in your arguments–you talk in terms of hitting the halfway mark, then the next halfway mark, and so on. The most natural way to interpret what you’re saying is that you mean for us to order the reals as they are ordered on the real number line, then determine, on that ordering, what the “last” member of the set {1/2, 3/4, 7/8, …} is. But on that ordering, it is trivial to prove that there is no last member. In order for there to be a last member of a set including those elements, you have to reorder them. > > I can give that set both a first and a last member. Order them like this. {1/2, 7/8, 31/32, …, 15/16, 3/4}. Now the last element is 3/4. You could even embed this in a complete ordering of the reals, though it’s beyond my understanding to know how to specify any such ordering. > > The existence of terminal members of infinite sets depends on how you order the sets. If you order the reals as they are on the real number line, then there simply is no last member of the halfway-halfway-halfway set we’ve been talking about. If you order the reals differently, there might be a last element. (Indeed, order them in reverse and you trivially have a last element–1/2.) > > Go to school! Get a PhD! The kind of thing you’re trying to do is going to quickly turn into grad school level stuff. I have already been to grad school and have my MS degree. Granted some of this stuff is beyond my expertise. So this means I am not even allowed to discuss it? I sense your frustration Frylock. Anyway, I agree from one perspective the common one at that I think. There is simply no last or final member of an infinite set. But as we have discussed in the paradox of motion it would seem there is an infinite set that must be traversed. And I am having trouble with understanding how traverse that set entirely without there being a last member? You don’t seem to have trouble with that, so just wondering if you could explain? ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 13, 2012, 2:05am UTC](https://boards.straightdope.com/t/999-1/27517/688 "2012-08-13T02:05:35Z")
> [@erik150x](#): > > I have already been to grad school and have my MS degree. In (non-applied) mathematics? > [@](#): > > Granted some of this stuff is beyond my expertise. So this means I am not even allowed to discuss it? I didn’t say or imply anything about what you’re “allowed to discuss.” I did make claims about how fruitful such discussions will be, though. > [@](#): > > I sense your frustration Frylock. Anyway, I agree from one perspective the common one at that I think. There is simply no last or final member of an infinite set. But as we have discussed in the paradox of motion it would seem there is an infinite set that must be traversed. And I am having trouble with understanding how traverse that set entirely without there being a last member? You don’t seem to have trouble with that, so just wondering if you could explain? Step 1: You formulate an argument that you need a last member in order to traverse the set. Step 2: I refute the argument. Step 3: You, as a result, stop being confused or puzzled. That’s how it’s supposed to work, anyway. 😃 ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 13, 2012, 2:06am UTC](https://boards.straightdope.com/t/999-1/27517/689 "2012-08-13T02:06:17Z")
> [@Frylock](#): > > One thing I’m trying to illustrate to you in my previous post is that when you think you’re doing something simple and obvious (e.g. “traversing” a set) it turns out there’s a _ton_ of stuff going on under the surface that you’re not even aware of, and which makes the actual implications of your “traversal” opaque to you. I know you know that in the abstract, but in order to pursue the questions you’re pursuing, you need to be in a situation where you don’t just know about the ton of stuff in the abstract but rather you have others who know all about that ton of stuff who can show it to you when it is relevant to your pursuit. Otherwise you will almost certainly never see it. It does not seem simple and obvious in this case at all. That is why I am asking you to explain, perhaps even a little of _this other stuff_? ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 13, 2012, 2:08am UTC](https://boards.straightdope.com/t/999-1/27517/690 "2012-08-13T02:08:41Z")
> [@Frylock](#): > > In (non-applied) mathematics? > > I didn’t say or imply anything about what you’re “allowed to discuss.” I did make claims about how fruitful such discussions will be, though. > > Step 1: You formulate an argument that you need a last member in order to traverse the set. > > Step 2: I refute the argument. > > Step 3: You, as a result, stop being confused or puzzled. > > That’s how it’s supposed to work, anyway. 😃 I’m sorry I see where you refuted the idea there could be a last member of an infinite set, but not how one can traverse something with no end or last member? ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 13, 2012, 2:09am UTC](https://boards.straightdope.com/t/999-1/27517/691 "2012-08-13T02:09:13Z")
> [@erik150x](#): > > It does not seem simple and obvious in this case at all. That is why I am asking you to explain, perhaps even a little of _this other stuff_? I know almost nothing more about than you do. I’ve basically told you everything I know. Others have told you more **Indistinguishable** , for example, is, AFAIK, a genuine theoretical mathematician and logician, and he’s been telling you plenty. He’s very good at communicating on a lay level, but with this particular subject matter the concepts themselves are so counterintuitive-to-the-way-we’ve-been-taught that even an extremely clear explanation can go badly misunderstood. That is sadly the case in this thread. This is why I continue to recommend you simply place yourself in the presence of experts and teachers, because I do not foresee your coming to a better understanding in any other way. Assuming this is a matter that’s that important to you, of course, which it may not be. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 13, 2012, 2:11am UTC](https://boards.straightdope.com/t/999-1/27517/692 "2012-08-13T02:11:03Z")
> [@erik150x](#): > > I’m sorry I see where you refuted the idea there could be a last member of an infinite set, but not how one can traverse something with no end or last member? You tell me the argument. What is the argument which you cannot find a flaw in, which starts from clearly true premises, and which ends with the conclusion “Something must have a last member in order to be traversable”? ---
**Author:** ![Ludovic](https://avatars.discourse-cdn.com/v4/letter/l/7ab992/32.png) [@Ludovic](https://boards.straightdope.com/u/Ludovic)\ **Post date:** [August 13, 2012, 2:11am UTC](https://boards.straightdope.com/t/999-1/27517/693 "2012-08-13T02:11:16Z")
> [@erik150x](#): > > I would tend to disagree with your statements in general. The rules of math, number systems are not made arbitrarily. The are made to reflect an deeper understanding of our “real” world. Then there’s no need to talk about infinitesimals. With the bytes I wasted on this post, I could just have as easily described a specific location in the universe down to the Planck Distance. The rest of the posters are saying that their mathematical system fits the real world better – i.e. has the most practical applications. From what I understand, the systems in which you can do what you are proposing provide less insight into the workings of our world than the standard view. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [August 13, 2012, 2:20am UTC](https://boards.straightdope.com/t/999-1/27517/694 "2012-08-13T02:20:02Z")
> [@erik150x](#): > > . . . \*\*"Another way to model it is to declare that “…” means “one trillion decimal places.” > \*\*is about as ridiculous as I can fathom. Why on earth would I want that. \*\* Well, for one thing, it’s rigorously defined, something you have refused to do for this entire thread. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 13, 2012, 2:26am UTC](https://boards.straightdope.com/t/999-1/27517/695 "2012-08-13T02:26:12Z")
> [@Frylock](#): > > You tell me the argument. What is the argument which you cannot find a flaw in, which starts from clearly true premises, and which ends with the conclusion “Something must have a last member in order to be traversable”? Well it just seems common sense to me? It does not to you? Infinite = endless, yes? An infinite set, has no end, no final member. e.g. {1/2, 1/4, 1/8, … } To traverse I would define to iterate through each item or element in the set. Even assuming it takes no time at all, that is we can iterate through these elements infinitely fast, how does one complete this and by complete I mean iterate through all elements without coming to some last member. If you have iterated through all of them, then shouldn’t there logically be a last element that was iterated just before completion? At this moment I see no other logical conclusion? ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 13, 2012, 2:27am UTC](https://boards.straightdope.com/t/999-1/27517/696 "2012-08-13T02:27:53Z")
> [@Trinopus](#): > > Well, for one thing, it’s rigorously defined, something you have refused to do for this entire thread. LOL, Well it’s got that, I’ll give you that. Not much else though. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 13, 2012, 2:29am UTC](https://boards.straightdope.com/t/999-1/27517/697 "2012-08-13T02:29:36Z")
> [@Ludovic](#): > > Then there’s no need to talk about infinitesimals. With the bytes I wasted on this post, I could just have as easily described a specific location in the universe down to the Planck Distance. > > The rest of the posters are saying that their mathematical system fits the real world better – i.e. has the most practical applications. From what I understand, the systems in which you can do what you are proposing provide less insight into the workings of our world than the standard view. Actually by there definition the hyperreals provide all of the insights as reals and then some. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 13, 2012, 2:43am UTC](https://boards.straightdope.com/t/999-1/27517/698 "2012-08-13T02:43:45Z")
> [@erik150x](#): > > Well it just seems common sense to me? It does not to you? Okay, I am going to tell you a rule. 😉 Never, ever say, “It’s just common sense.” If you are thinking “it’s just common sense,” you are missing the “ton of stuff” I was talking about earlier. Basically, you’re no longer making mathematical inquiry. It’s _never_ common sense. > [@](#): > > Infinite = endless, yes? > > An infinite set, has no end, no final member. > > e.g. {1/2, 1/4, 1/8, … } > > To traverse I would define to iterate through each item or element in the set. > > Even assuming it takes no time at all, that is we can iterate through these elements infinitely fast, how does one complete this and by complete I mean iterate through all elements without coming to some last member. If you have iterated through all of them, then shouldn’t there logically be a last element that was iterated just before completion? At this moment I see no other logical conclusion? I don’t mean to be vexatious here, but the above is almost not an argument at all. For progress to occur, you need to look deep within, figure out what your premises are, figure out how you think those premises lead to that conclusion, and lay it out step by step. Until you’ve done that, you haven’t given me or anyone else anything to work with. All we can say is ‘your conclusion seems implausible to me.’ Hardly progress! We need an argument. A limitation you should place on yourself is this: In the formulation you eventually come up with, there should be no question marks. “How does one complete this without…?” (as you wrote above) is not a proper part of any argument. It’s just a raincheck, promising the reader that you’ll be developing an argument for the claim that “you can’t complete this without…” ---
**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:** [August 13, 2012, 3:00am UTC](https://boards.straightdope.com/t/999-1/27517/699 "2012-08-13T03:00:50Z")
Actually, erik150x, you don’t like to debate. You like to make assertions and then get annoyed when other people don’t agree with them. You’re not convincing anyone here (and you’re not even explaining your ideas well), and thus you’re wasting your time and ours. In any case, if you want to debate, this thread should be in the forum Great Debates rather than General Questions. ---
**Author:** ![Saltire](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saltire/32/553_2.png) [@Saltire](https://boards.straightdope.com/u/Saltire)\ **Post date:** [August 13, 2012, 6:18am UTC](https://boards.straightdope.com/t/999-1/27517/700 "2012-08-13T06:18:23Z")
> [@erik150x](#): > > Show me: “You have insisted that someone solve this problem without limits (which has been done, though you ignore it)” Way back in [post #101](http://boards.straightdope.com/sdmb/showpost.php?p=15348296&postcount=101), by **Lumpy** , for starters. No limits, just showing that you can do the same math two different ways, so that you get either 0.999… or 1. This shows they are the same number, does it not? If not, why not? > [@erik150x](#): > > RE:" it’s an artifact of decimal notation." - I’ll do you better than that. It’s pretty much a definition of the decimal system. There is no argument about what .999… = 1 within the rules and definitions of the real number system. I would not argue that, it would be pointless. The question is deeper than that. The real number system does not necessarily give us some profound insight into the matter of what > > 9/10 + 9/100 + 9/1000 + … > > ACTUALLY sums to. The limit tells us that it = 1 and it also has the limit of .999… > > This is no big deal as the real number system defines them to be the same. No where in the real number system is this ever proven to be the case. And frankly, I am not concerned with what the real number system says precisely because it starts out with the assumption they are the same. You seem to use ‘decimal notation’ and ‘real number system’ as if they were synonomous. They aren’t. My statement was not about the reals, it was about notation. Refer back to [my post #658](http://boards.straightdope.com/sdmb/showpost.php?p=15376183&postcount=658)to which you are replying, and answer the question in the last paragraph. If you do have such reasons, state them, with examples if necessary. If you don’t have such reasons, can I assume you agree with me about the topic of the thread, and your continued chatter about reals and limits is not actually about the topic itself, but instead just about some stuff that came to you while you were thinking about the topic? [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=34) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=36)