I quite liked the proof that 1/3 was actually equal to 0.333… by long division. Quite clever. But this won’t stop anyone saying its wrong because ‘you can’t finish’, even though we know that there’s no way any digit in the process can be anything but a 3. The results of infinite processes can be assumed based on obvious patterns.
Again, my mind boggles at this thread.
I think this is the core of your confusion right here, Phage. The limit concept is not ‘a way of introducing acceptable error’. It is a precisely defined mathematical concept introduced to define, not approximate, the sums of infinite series. One cannot individally add up an infinite number of terms; however, if the sequence of partial sums converges, the limit of that sequence is defined as the sum of the infinite series. To say that the ‘real’ total is actually ‘a little bit’ smaller is completely unfounded, as much so as to say that the ‘real’ limit of a convergent series is actually a little bit smaller than the limit value. There just isn’t any well-defined ‘little bit smaller number’ that it could be. The limit as n → infinity of (1 minus 1/10^n) is exactly one, by the definition of limits; and that means, by definition that the sum of 9/10^n as n goes from 1 to infinity is exactly one. It is irrelevant that the sum of any finite number of terms of this series is less than one–the sum of the infinite series is equal to one, using the only existing principled definition of ‘summing an infinite series’.
And kabbes, just quit with the formal number theory. You’re making me pine for grad school. 
I think limits are going to be a hard concept to work with as long as we have people suggesting that you cannot state, compute, or represent the results of any process that has an infinite number of terms, as if you had to compute each term individually.
The simple algebraic proof that 0.999… = 1 may at its heart have a limit of a sum of infinite converging terms, but the proof doesn’t need to go quite that in depth. We’re simply doing an infinite number of ‘operations’ in finite time by showing that there’s a repeating pattern.
Simply being able to prove that 0.999… * 10 is 9.999… and that the calculation is doable at all seems to be the big stink, at least in Phage’s posts.
Simply being able to show that 9.999… - 0.999… is equal to precisely 9 because everything after the decimal point must be zero digits is a similar process of infinite operations being done in finite time because there’s a repeating pattern.
I dunno. I guess my point is that we’re dragging out more and more complex proofs and definitions in math, and that is not going to convince those who eschew mathematical proof and a hundred years of history for intuition no matter how hard you try. They’re looking for an intuitive understanding, and maybe you’d have to go the other way, towards the simpler, not the more complex.
Tricky. Doable? Who knows.
But the algebraic proof requires far less ugly than formally using limits, provided you can get people to accept that assumptions on moving a decimal point or turning all those digits to 0’s through subtractions is valid.
Much less resorting to number theory. 
You know, I’ve had enough maths (PDE’s, DE’s, integral transforms, etc) to easily put me in the top few percent of math education (NOT skills…I’ve forgotten so very, very much… 
) here, and I’ve never thought of 1 in quite this way. And the more I think of it, the more correct and deeply profound that seems to me this afternoon, for some reason.
But that statement “(integer) 1 terminates” would be true, wouldn’t it? 
Proof that one can pass 30+ credit-hours of maths and still learn basic things about it.
ultrafilter said:
I think this is the saddest part. Do threads on the Riemann Hypothesis or the Poincaré Conjecture run to four pages? No, just this and the Balls and Gremlin thing.
For those who are interested in infinitesimal calculus, a free introductory textbook is available for download from this page.
It seems so. But, of course, you don’t have to calculate every single digit. What Phage is asserting, essentially, that somewhere there is a digit in the nth position that is not 9. If that were so, then he would be right and it would be true that 0.9999… * 10 != 9.9999…
What he has to do is show where this magic digit is and explain why it isn’t 9. And he can’t, so he just says, “I don’t like this, therefore it’s false.”
Nice to control reality that way.
As a non-math person who stumbled into this thread and never gave the OP any thought, I have to say that I’ve found it endlessly (ha!) fascinating. Kudos to the .9999… = 1 crowd for devising proofs that I could follow. It’s prompted me to look up the previously mentioned sci.math FAQ.
Neat stuff. (Mostly over my head, but neat.)
Sorry I haven’t been able to keep up with this thread, but in the hundred or so posts since my last one, I have seen a few examples that suggest that people do not know what the notation 0.999… means. It’s sort of like somebody saying that 0.5 is not the same as .5 - clearly the problem is that they don’t understand what’s meant by the notation.
As I mentioned earlier while decrying infinity, it is not “0.” with an infinite number of 9’s after it; it is defined as a limit.
This is wrong.
This is wrong.
This is wrong.
THIS is right. I have come to the conclusion that the fundamental problem with explaining this is that people cannot divorce themselves from the incorrect concept of what 0.999… means. Also, even if they did accept that it was defined as the limit of a sequence, I don’t think they understand what a limit is. Case in point:
I think that there is a tendency to associate the limit of a sequence with the “infinitieth” term of that sequence, or the limit at infinity of a function f with “f(infinity)”. Both of these concepts are meaningless in real analysis, and they’re just confusing, leading to problems like this:
I’m not really sure how to address it. Oh well. Now to respond to the posts made toward me:
I meant the sequence, not the series, as ultrafilter pointed out. Sorry for the confusion, but that’s how I always represent sequences; how do you do it? So anyway, you said that the only way you know how to think of limits is as the sum of an infinite number of things. How does this apply to the above sequence?
I never said your proof was wrong; reread what I said and that will become quite clear. I said I had a problem with it, but I think my problem has been well explained by others. Although I understand that it can be instructive for people who don’t know analysis, I think it is important to make it very clear that you cannot walk away from this proof with a complete understanding of the problem.
Why is it wrong to think of .9… as an infinite number of nines after a decimal place?
Hey people! I just trisected the angle!
(Now what did I do with that proof? Oh, here it is. Over here in the margin.) 
Just to add my piss to the somking train wreck: it seems that more than a few people think that the numbers in an infinite decimal expansion are somehow being calculated and don’t really exist until you write them down. I only think that because the thread has posts that allude to “completion of the calculation” or “finishing the approximation.” This is not the case. When the ellipsis is added after the last nine the series becomes well-defined mathematically.
I was going to bring up the example of PI as a well-defined limit of an infinite series but others have done so more capably. I love these boards…
Didn’t someone famous once say that the only thing infinite in this universe is human stupidity? I think this applies quite appropriately to this topic. I am amazed that some people can still think that 0.999… does not equal 1. Every mathematician in the world can prove that they are equal.
I suppose that if the standards of mathematics do not appeal to some people’s intuition, they should develop their own rigorous, consistent system of mathematics using their own axioms. Perhaps there could be a meaningful system in which 0.999… and 1 are distinct. When they have done this let them post their findings here.
Phage and other dissenters: For clarity’s sake, please define precisely what you mean the following things:
It’s not wrong. That’s exactly what it is. However, the problem with thinking like that is that it leads to the sort of confusion that’s peppered all through this thread. Since the mind can’t grasp the concept if “an infinite number”, one tends to merely think of “a very large number” of things instead. That’s where the problem is.
It isn’t. Achernar was simply using a different notation to say pretty much the same thing. He prefers set-theoretical notation, while everyone else is using decimal notation. No biggie.
Useful information: any time you see repeating patterns of numbers, you can convert them as such:
.xxxxx… = x/9
.xyxyxyxyxyxy… = xy/99
.xyzxyzxyzxyz… = xyz/999
.wxyzwxyzwxyz… = wxyz/9999
Thus, not only is .9999… = 9/9 = 1,
but, for instance, .123412341234… = 1234/9999
This is a little numerical factoid that is of actual practical use in the real world. (Well, it has been for me!)
Trinopus
Okay, I’m willing to accept that I’m wrong. I tried to find a cite to prove it, because I realized I’m working on what I consider to be the most reasonable, not something I’ve actually read. But I can’t find a cite one way or the other. Does anyone have a cite (from someone who really, really knows what they’re talking about) for what exactly the ellipsis notation actually means?
Here’s the justification for why I think it’s wrong to think of it like that. Generally, for an N-digit decimal, the algebraic value of the decimal representation is the SUM from p = 1 to N of d[sub]p[/sub] × 10[sup]-p[/sup]. For an infinity-digit decimal, you might think that you can just set N = infinity. So you’d have the SUM from p = 1 to infinity. But here’s the rub. Implicit in the summation notation with an infinity on top is a limit, remember? It’s actually the limit as N increases without bound of the SUM from p = 1 to N. It’s no more the sum of an infinite number of things than all limits are.
i thought amethyst was making up words. however they are real words according to the dictionary.
what the heck does he mean by that?
Informally an asymptote is a value that a function or series approaches but never quite touches within a finite range or number of terms.
f(x) = 1/x for example has three asymptotes. As x approaches infinity, f(x) approaches 0. 0 is therefore an asymptote, and f(x) is said to approach 0 asymptotically as x approaches infinity. f(x) also approaches 0 as x approaches negative infinity. The curve, in other words, approaches the horizontal line y=0 (f(x)=0) aysymptotically.
The other two asymptotes are as x approaches 0. From the positive x side, as x approaches 0, f(x) approaches positive infinity. f(x) is undefined at exactly 0, but the curve gets arbitrarily close to the x=0 vertical line, so f(x) can be said to approach the vertical line x=0 asymptotically as x approaches 0.
So what happens as x approaches 0 from the negative x side? Excercise left to the student.
Er, I guess I should say there are 3 limits, but two asymptotes. The asymptotes are the y=0 horizontal line, and the x=0 vertical line.
Mathematicians seem to be quiet, but I’m sure my inaccuracies will raise their ire.
Er, so Amethyst’s statement
“I can accept that .9999… = 1 if I can accept that an asymptotic function will ultimately equal its asymptote as the function extends to infinity.”
means that if an asymptotic function (like f(x)=1/x) can be said to be at the asymptote (0, in my example) if you allow the range (x) to go up to infinity, then Amethyst will accept 0.999… =1.
This is, of course, is what allows me to say
lim 1/x = 0
x->infinity
The issue is clouded a little, but yes, the integer representation of 1 does terminate. However, integer representations are only suitable for (surprise!) integers, so let’s stick to decimal representations. A decimal representation is a shorthand for an infinite series, but in a lot of cases, the tail of the series is zero. In that case, why write it?
And have I ever told you that the integer 1 is not the same as the real number 1?
rsa,
Yes, I can explain that, as it is no different than the rest of the discussion. The result of (0.333…) as the areas of the three triangles is inaccurate, and will always be inaccurate because there is no point at which an infinitely repeating sequence can be complete.
RealityChuck,
You don’t have to be obnoxious, if you have a point just say it plainly.
(1/3) is represented in base-10 as (0.333…). No matter how far you go, the next digit will have to be a 3. The point is that you can never have enough 3s; you will never reach the infinite number of 3s you need. The fact that a number such as 0.999… must repeat into infinity is an acknowledgement that a completely correct decimal cannot be shown.
William_Ashbless,
As you said, Calculus itself would be entirely useless without the ability to understand and perform limiting operations to get an exact answer. What you do not realize is that such limiting operations are a compromise made so that such calculations are possible. The idea of a limit was created so that mathematicians could get around calculating with a number such as (0.999…).
All,
Many of you are maintaining that because the entirety of a sequence of numbers such as 0.999… can be predicted, it is acceptable to determine a resultant number. This is directly contrary to the fact that an infinite sequence never ends, and as such cannot have an end result.
My credentials are not of any worth; I am not a mathematician. However, my explanations are merely an attempt to explain the ideas of others more versed than I in such fields. Take a look at this address, already posted at the start of the thread: http://mathforum.org/library/drmath/view/55748.html
I think that anyone who would actually learn anything from this thread has already read it and come to a conclusion. Everyone else is set in their ways, and will not change their minds even when presented with a reasonable proof. My arguments have begun to repeat themselves, and I tire of such a pointless loop. Below is the plainest explanation of my position that I can state; if anyone wants to ask questions they can reach me at phage@charter.net. Otherwise, quit cluttering the boards, and give the hamsters some rest.