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 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?
Nope. I see space. 
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.
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.
What, then, is one-third of 0.9~?
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.
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. 
No you can’t, because dividing by zero is, by definition, not equal to anything.
Canadjun said:
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:
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.
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:
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:
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:
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.
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.
[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. 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)
Incidentally, ultrafilter also spelt out fairly clearly what he meant by “the completeness of the reals”: namely, Cauchy completeness. 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.
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.
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.
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.
Furthermore, the limit as x goes to zero of (1 - cos(x))/x is zero.
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.
Some people just don’t know their limits.