# An Infinite Question **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [December 25, 2009, 10:31pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588 "2009-12-25T22:31:11Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 28, 2009, 4:55am UTC](https://boards.straightdope.com/t/an-infinite-question/522588/41 "2009-12-28T04:55:06Z")
> [@friedo](#): > > That’s basically how limits were defined in my high school calculus course. > [@Irishman](#): > > I learned about it in my high school Calculus class. Maybe you just hang out at the wrong high schools. 😉 Did either of you actually learn specifically about the completeness of the reals in your high school calculus class? That’s what I was referring to, not the notion of a limit. I don’t remember actually covering that specific idea, which is essential here. ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [December 28, 2009, 5:43am UTC](https://boards.straightdope.com/t/an-infinite-question/522588/42 "2009-12-28T05:43:25Z")
I did not learn that until Real Analysis, when I went back to get my math degree for the teaching certificate. :eek: The real numbers made a hell of a lot more sense once I understood some of this stuff better. I used to think you could have two numbers which were “next” to each other. I’m sure my dad (math/physics major) must have had days he just shook his head!!! 😛 ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [December 28, 2009, 5:57am UTC](https://boards.straightdope.com/t/an-infinite-question/522588/43 "2009-12-28T05:57:31Z")
> [@DSYoungEsq](#): > > I did not learn that until Real Analysis, when I went back to get my math degree for the teaching certificate. :eek: > > The real numbers made a hell of a lot more sense once I understood some of this stuff better. I used to think you could have two numbers which were “next” to each other. I’m sure my dad (math/physics major) must have had days he just shook his head!!! 😛 Sure you can, 2 7 Are those two numbers not next to each other? HUH? ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [December 28, 2009, 4:49pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/44 "2009-12-28T16:49:15Z")
> [@Jragon](#): > > Sure you can, > > 2 7 > > Are those two numbers not next to each other? > > HUH? Nope. I see space. 😉 ---
**Author:** ![MonkeyDavid](https://avatars.discourse-cdn.com/v4/letter/m/4bbf92/32.png) [@MonkeyDavid](https://boards.straightdope.com/u/MonkeyDavid)\ **Post date:** [December 28, 2009, 7:09pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/45 "2009-12-28T19:09:18Z")
The flaw in the logic is that 0.3~ + 0.3~ + 0.3~ = 0.9~ It does not. It equals 1. 0.9~ is a different number, obviously not achieved by multiplying 1/3 times 3. Try it with another infinite sequence of numbers, say pi. You can’t just add each individual number after the decimal. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 28, 2009, 7:22pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/46 "2009-12-28T19:22:55Z")
0.9~ does not mean something different from 1.0~; not as the terms are actually used by mathematicians. See above. What exactly is it that you want us to try with pi? It doesn’t matter. 0.9~ does not mean something different from 1.0~. See above. ---
**Author:** ![zut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zut/32/2875_2.png) [@zut](https://boards.straightdope.com/u/zut)\ **Post date:** [December 28, 2009, 7:58pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/47 "2009-12-28T19:58:19Z")
> [@MonkeyDavid](#): > > The flaw in the logic is that 0.3~ + 0.3~ + 0.3~ = 0.9~ > > It does not. It equals 1. 0.9~ is a different number, obviously not achieved by multiplying 1/3 times 3. What, then, is one-third of 0.9~? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 28, 2009, 7:58pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/48 "2009-12-28T19:58:23Z")
How mathematicians use infinite decimal notation: Given an infinite decimal string (e.g., “0.9999…”, or “3.14159…”, or “12.50000…”, or what have you), we can “round it down” by truncating everything to the right of some point (e.g., rounding down “123.4567…” to 120 or 123 or 123.45 or 123.456), and we can just as well “round it up” by following such truncation by incrementing at the digit immediately before the truncation point (e.g., rounding up “123.4567…” to 130 or 124 or 123.46 or 123.457). So what? Well, when a mathematician uses an infinite decimal string to represent a number, what they _mean_ by that string is “The number which is \>= all the ‘rounding downs’ of this string and \<= all the ‘rounding ups’ of this string”. So, to a mathematician, and this is simply what their notation _means_, a linguistic convention, holding guaranteedly true by inarguable definition (just as the three letters “dog” inarguably denote a familiar four-legged carnivore, simply because that’s what people mean by that word)… to a mathematician, the string “123.4567…” _means_ “The number which is greater than or equal to each of 100, 120, 123, 123.4, 123.45, 123.456, 123.4567, etc., and which is less than or equal to each of 200, 130, 124, 123.5, 124.6, 123.457, 123.4568, etc.” [Who’s to say there actually is a unique such number? Well, the mathematician also has a very particular meaning for “number” in mind here according as to which there is, but it’s not particularly important to go into the details of that right now] And, as concerns the object of this thread, what does the string “0.9999…” mean to a mathematician? It means “The number which is greater than or equal to each of 0, 0.9, 0.99, 0.999, etc., and less than or equal to each of 1, 1.0, 1.00, 1.000, etc.”. What number has those properties? Well, clearly, 1 does… this string means something to a mathematician which clearly means 1. If you want to regard this string to mean something different, then, fine, but then you’re not pointing out somewhere where the mathematician has gone wrong; you’re just speaking a different (and potentially less useful or coherent) language. ---
**Author:** ![Find\_Friends](https://avatars.discourse-cdn.com/v4/letter/f/c37758/32.png) [@Find\_Friends](https://boards.straightdope.com/u/Find_Friends)\ **Post date:** [December 28, 2009, 8:10pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/49 "2009-12-28T20:10:16Z")
> [@DSYoungEsq](#): > > I wanted to come back to this, not because of the discussion over why Base 10 beat out Bases 5, 8, 12, 20 or 60. I wanted to come back to it because it makes a fundamental error in understanding what it is that has the OP confused. > > The OP is not confused about the meaning of 1/3. All your discussion of using different bases does is show that “1/3” in Base 10 has different representations, depending upon what base you use for your numbering system. But all that does is help him understand that we can prove that 1/3 + 1/3 + 1/3 = 1, and that’s not his issue at all. > > No matter what base you use, there is going to be some number expressable through the concept of infinite repetition of a digit, or some string of digits. Nor is the OP really confused about what “equal to” means, though he bit on that red herring rather quickly when it was placed in front of him. One might ask him what he thinks of the number .142857142857… as a representation of 1/7! After all, if we add those up 7 times, we’ll get that same .9999~ that he is so enraptured by. So your discussion of bases is of no help to our OP friend, as it is totally off topic. > > No the real issue is highlighted here: > > What the OP isn’t understanding is the meaning of .9999999~. He thinks that, since there is some difference at every decimal place between the “9” and the “0” in 1.0000~, that the two numbers must be different. You can see this in his statment that .9999~ “will never actually reach 1.” It doesn’t have to “reach” it because it is already there! 😛 .99999~ isn’t some “process” of writing out the digit “9” infinitely many times. .9999~ is a decimal respresentation of the same concept as the digit “1” in Base 10. It only looks that way because it’s what happens if certain types of arithmetic are used to create representations of certain other numbers, like, for example, 1/3 = .3333~. > > But the OP remains unconvinced, perhaps, that this is true, that .99999~ is the same as “1”. It was here that I introduced the concept not of “infinitisemals”, but rather the concept that, for the numbers to be different, we would have to accept that there exists between “1” and “.9999~” some defineable distance (and, thus, on the real number line, the existence of an infinite amount of intervening numbers). But, of course, we cannot find any such number, or establish any such distance. If we attempt to subtract .999999~ from 1, we get “0”. If we attempt to find some place at which we can insert some other set of digits, we find that the "9"s squeeze us out indefinitely. Thus, what appears to be true (.9999~ = 1) is, indeed, actually true. You are correct, this was the wrong approach to use. I have been strongly considering since last night returning to get back on track in this thread. I’m sure that I did really “get” what the OP wasn’t seeing, though. At least, I did after reading through most of the thread yesterday. But often enough I make the mistake of using a line of argument that is distracting at best, or doesn’t really address the “speck” in someone else’s eye. I really can’t explain why I thought my post would be helpful. Further discussion of number bases, and the related history of mathematics, would naturally belong in a new thread. 🙂 ---
**Author:** ![dhkendall](https://avatars.discourse-cdn.com/v4/letter/d/a9adbd/32.png) [@dhkendall](https://boards.straightdope.com/u/dhkendall)\ **Post date:** [December 28, 2009, 8:17pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/50 "2009-12-28T20:17:59Z")
> [@True Blue Jack](#): > > And who could not know of 12 hours in each of two cycles making up a solar day? [Him](http://www.timecube.com/). 🙂 (Couldn’t resist …) ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [December 28, 2009, 9:35pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/51 "2009-12-28T21:35:32Z")
> [@Saint\_Cad](#): > > I may have to use this next time I teach begging the question in logic. How can you show that they are actually the same? > > Using limits? By using limits I can “prove” 0/0 = 1. No you can’t, because dividing by zero is, by definition, not equal to anything. ---
**Author:** ![Irishman](https://avatars.discourse-cdn.com/v4/letter/i/b487fb/32.png) [@Irishman](https://boards.straightdope.com/u/Irishman)\ **Post date:** [December 28, 2009, 10:40pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/52 "2009-12-28T22:40:02Z")
**Canadjun** said: > [@](#): > > You don’t need the parenthetical part. I know. I debated putting it in. I included it as a comment that we’ve verified it as far as we have carried the computation, but it really is a distraction. **Indistinguishable** said: > [@](#): > > Every child comes to understand and not quibble with the linguistic convention that “1/5” and “2/10” mean the same thing, just expressed differently; two different notations that happen nonetheless to specify equal referents. There’s not some tiny difference between 1/5 and 2/10; they are exactly equal, despite my having written each differently. > > The situation is the same with “0.999…” and “1”; they mean the same thing, they just express it differently. True, but misses the point. It is fundamentally easier to see that 2/10 is the same as 1/5 than it is to see that 0.999~ = 1. The first relies on understanding the notation of fractions and the concept of division. But that is already understood when they reach the point that they are considering how the two notations are equal. But the issue of an infinitely repeating decimal is a wholely different concept, and one that does not come out of what is already known about the topic. It is easy to see how someone could assume there _must_ be a difference between 0.9~ and 1.0. I mean, they _look_ different. 0.9 is less than 1, so is 0.99, and 0.999. Why does repeating it infinitely change that? That is the concept that one has to grasp. > [@](#): > > To understand this, one need simply know what they mean, and to know what they mean, one need simply ask those who actually use them what they mean by them (i.e., any old mathematician), and then it is obvious. There is no point quibbling with the mathematician; the mathematician’s claims are true _by definition_. This is not so much a question of mathematics as of lexicography. Perhaps I misunderstand you, but mathematicians did not _define_ 0.999~ to equal 1.0. They deduced it by the same kinds of methods being presented here. It is fundamental to the nature of the infinite repetition, and had to be argued in mathematician circles. Furthermore, something is not true in mathematics just because a mathematician tells you it is. Now, if you are unable to follow the steps to see it for yourself, then you are better off accepting that mathematicians as a whole who agree on something are correct, but mathematicians themselves have to prove it to each other. **Saint Cad** said: > [@](#): > > 10 x 0.9999999… = 9.999999999… ? To justify that the numbers after the respective decimal points is to make the conceptual leap that taking a 9 from an infinite string of 9’s still leave the same number of 9’s. Some may see why that is so, but to people not familiar with the intricacies of the cardnality of infinite sets I can see why they think its a bit of handwaving. If one does not understand that, then one needs to stop and go back to what the … means, what “infinitely repeating” means. That really is a key step, no changing the way the number is written will get around that conceptual basis that must be grasped. Introducing other concepts such as exponential notation or use of n terms does not simplify the issue. **ultrafilter** said: > [@](#): > > Did either of you actually learn specifically about the completeness of the reals in your high school calculus class? That’s what I was referring to, not the notion of a limit. I don’t remember actually covering that specific idea, which is essential here. Well, then, you did a poor job of expressing yourself. I reread what you said, and still only see a discussion of limits, and nothing about “the completeness of the reals”. I’m not even sure what you mean by that. **MonkeyDavid** said: > [@](#): > > The flaw in the logic is that 0.3~ + 0.3~ + 0.3~ = 0.9~ > > It does not. It equals 1. 0.9~ is a different number, obviously not achieved by multiplying 1/3 times 3. Then what, pray tell, is 0.9~? 0.3 + 03 + 0.3 = 0.9, yes? 0.03 + 0.03 + 0.03 = 0.09, yes? Seems to work for me. > [@](#): > > Try it with another infinite sequence of numbers, say pi. You can’t just add each individual number after the decimal. That depends upon what the digits are. 0.14 + 0.17 = 0.31, not 0.211, so yes, you run into trouble when the sum exceeds 9 in any decimal place, but that isn’t a problem with 1/3. Normally, you can’'t add from the higher decimal place to the lower, but must work from the end back, but if you can establish that no decimal place will exceed the value of 9, then you can work from the front down. This is a case that does work. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 28, 2009, 10:57pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/53 "2009-12-28T22:57:42Z")
[quote=“Irishman, post:52, topic:522588”] Perhaps I misunderstand you, but mathematicians did not _define_ 0.999~ to equal 1.0. They deduced it by the same kinds of methods being presented here./QUOTE] Fine. They have defined it to mean “The number which is greater than or equal to each of 0.9, 0.99, 0.999, etc., and less than or equal to each of 0.9 + 0.1, 0.99 + 0.01, 0.999 + 0.001, etc.”, and then trivially deduce that 1 satisfies this description. But that last tiny step is not a controversial one; even our OP will agree that 1 satisfies this description. The issue is entirely in the understanding that the conventional definition of the referent of an infinite decimal string is given this way, as described in post [#48](http://boards.straightdope.com/sdmb/showpost.php?p=11931109&postcount=48). And that _is_ something there’s nothing more to than to take the mathematician’s word for it: “Yup. That’s what I mean by that notation”. (I feign no claim to tracking any (potentially irrelevant and distracting) account of the _history_ of this notation’s development; I only care to give a simple and accurate account of what the notation actually is) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 28, 2009, 11:10pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/54 "2009-12-28T23:10:10Z")
Incidentally, **ultrafilter** also spelt out fairly clearly what he meant by “the completeness of the reals”: namely, [Cauchy completeness](http://en.wikipedia.org/wiki/Complete_metric_space). That having been said, I don’t think such general notions as Cauchy sequences or limits are necessary here; simple sequences of sharpening lower and upper bounds as in post #48 suffice to explain the notation. ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [December 29, 2009, 2:29am UTC](https://boards.straightdope.com/t/an-infinite-question/522588/55 "2009-12-29T02:29:39Z")
> [@ultrafilter](#): > > Did either of you actually learn specifically about the completeness of the reals in your high school calculus class? That’s what I was referring to, not the notion of a limit. I don’t remember actually covering that specific idea, which is essential here. I have no idea what “completeness of the reals” means, so I doubt it. I was referring specifically to the idea of a progression approaching a finite value, like 0.9 + 0.09 + 0.009 … approaches 1. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [December 30, 2009, 11:50am UTC](https://boards.straightdope.com/t/an-infinite-question/522588/56 "2009-12-30T11:50:08Z")
> [@John\_W.Kennedy](#): > > No you can’t, because dividing by zero is, by definition, not equal to anything. [Not always.](http://en.wikipedia.org/wiki/Division_by_zero) But I think the guy’s point is that it would appear to be true, although it is actually invalid. Although lim[sub]x-\>0[/sub] x/x = 1, 0/0 ≠ 1. I believe he is incorrect, though, because 0/0 is not undefined, but indeterminate. This means you can make it equal anything you want, really, as lim[sub]x-\>0[/sub] ax/x = a. ---
**Author:** ![Gyrate](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gyrate/32/3588_2.png) [@Gyrate](https://boards.straightdope.com/u/Gyrate)\ **Post date:** [December 30, 2009, 12:20pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/57 "2009-12-30T12:20:27Z")
To sum up: there is the number, and then there is the way we write down the number. The number exists whether we can physically write down its exact and complete value or not. And if we can’t, the problem is with the notation, not the number. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 30, 2009, 4:11pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/58 "2009-12-30T16:11:13Z")
> [@BigT](#): > > But I think the guy’s point is that it would appear to be true, although it is actually invalid. Although lim[sub]x-\>0[/sub] x/x = 1, 0/0 ≠ 1. > > I believe he is incorrect, though, because 0/0 is not undefined, but indeterminate. This means you can make it equal anything you want, really, as lim[sub]x-\>0[/sub] ax/x = a. Furthermore, the limit as x goes to zero of (1 - cos(x))/x is zero. ---
**Author:** ![Irishman](https://avatars.discourse-cdn.com/v4/letter/i/b487fb/32.png) [@Irishman](https://boards.straightdope.com/u/Irishman)\ **Post date:** [December 30, 2009, 10:17pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/59 "2009-12-30T22:17:56Z")
**Gyrate** , the problem is not so much that the notation doesn’t represent the complete value, the problem is understanding the meaning of the notation. 3 \* 1/3 = 1 1/3 = 0.3~ Therefore, 3 \* 0.3~ = 1 But, 3 \* 0.3~ also = 0.9~ Ergo 0.9~ = 1 The step that seems to throw people is the one after the But. ---
**Author:** ![Gyrate](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gyrate/32/3588_2.png) [@Gyrate](https://boards.straightdope.com/u/Gyrate)\ **Post date:** [December 30, 2009, 10:32pm UTC](https://boards.straightdope.com/t/an-infinite-question/522588/60 "2009-12-30T22:32:07Z")
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