Well in my mind infinite means, not-finite/unbounded, but I am not a mathematician by training and my university level maths training only extends to what is relevant to physics and not this sort of number theory.
Right, but 0.33333333333… where the number of threes is not finite or unbounded has no definition in the usual sense. We can only make sense of it by noting that:
0.1 = 10[sup]-1[/sup]; and
0.01 = 10[sup]-2[/sup] and so on, so that:
0.xyz = x.10[sup]-1[/sup] + y.10[sup]-2[/sup] + z.10[sup]-3[/sup].
It is then clear that 0.333… = 3×10[sup]-1[/sup] + 3×10[sup]-2[/sup] + 3×10[sup]-3[/sup] + …
In other words, it’s not just that 0.333… can be represented as an infinite sum, it’s that the (most obvious) way that 0.333… is defined is as an infinite sum.
Numbers are much trickier things than people give them credit for. There’s a reason that number theory generally offers the simplest sounding proposals with the most difficult proofs.
pan
Well, I certainly do remebr doing series and thereare ways of represtening, defining and evalutaing the sum of an infinte series, where that sum is a real number.
Anyway, I suppose saying 0.333… has infinite 3’s is pretty meaningless, but saying it is the sum of an infinite series is meaningful.
Exactly.
Just for all the idiots that don’t understand humour…
QED in Mathematics stands for “Quod Erat Demonstratum” - “Because it was shown”.
Quod Erat Demonstrandum “As it was to have been shown” is a latinist joke…
tsk !
Sometimes you guys take all the fun out of being a latinist…
Although ** Kabbes ** was funny -
- obviously, Kabbes, you’re not a mathematician then…
Har har…
:rolleyes: How can anyone contradict himself so blatantly and not even notice?
You say “no matter how far you go, the next digit will be three” and in the very next sentence say “you can never have enough 3s.” But, as you said, every digit, no matter how far you go, is a three. How, then could you “never have enough 3s?” You have an infinite number of threes, don’t you? If you’re short one, you just keep going; you can’t run out.
So in two adjacent sentences you say, “you have an infinite number of threes” and “you never have enough.” Which is it?
If you don’t have enough threes, it means that at least one digit isn’t three. But you’ve already stated “no matter how far you go, the next digit will be three.” Another contradiction.
It’s pretty pathetic that you can’t even be consistent from sentence to sentence.
Actually Chuck, that’s the same contradiction - you just mentioned it twice…
I don’t fall into your the category addressed (I doubt you’ll find too many that do on these boards, maybe you should try somewhere else), but I’ll answer you anyway:
I have always understood Q.E.D to stand for quod erat demonstrandum and meaning which was to be demonstrated.
Look, if f(x) = 1/x, then the limit of f(x) as x approaches infinity is zero. HOWEVER, at no point does f(x) actually equal zero. And it’s misleading to argue that f(infinity) = 0 because infinity is a concept, not a number.
No, you don’t have to prove anything about it being a limit.
Remember, for 10 * 0.9999… != 9.99999… to be correct, there needs to be at least one digit in 10 * 0.9999… that isn’t equal to 9. If we discovered, that the result was, say 9.99…9899…, it would prove the multiplication false. Yet, limit or not, the effect of multiplying the number by ten does not change any digit to a number other than nine (assuming the rules of multiplication do not change).
Whether 0.999… expresses a limit or not is irrelevant to the proof; all I am assuming is that if you multiply by 10, all digits remain nine, and this assumption is firmly based on the rules of multiplication.
(And I know only a little about limits, but are you saying that if you have a divergent limit, you can’t multiply each of its elements by a number? And that the rules of multiplication are different from divergent limits? Doesn’t seem likely to me.)
But you get a lot of credit from me for being one of the few on the board to use “begging the question” correctly. 
Look, if f(x) = 1/x, then the limit of f(x) as x approaches infinity is zero. HOWEVER, at no point does f(x) actually equal zero. And it’s misleading to argue that f(infinity) = 0 because infinity is a concept, not a number.
It seems to me that this is not the fatal flaw in your proof.
The problem is that if A = B, then it does not necessarily follow that A-X = B-X. (When you are dealing with infinite quantities).
Unwashed - no, just another of the perils of not being a latinist. Consider yourself enlightened. Latinists may be nearly ALL unemployed, but we know the unless things…
I can’t believe this thread has spilled over to a sixth page because people are debating a fact. It’s as if the OP had asked “What color is the sky”, and the first reply had been “Blue” then ten people come in to say “It cannot be blue and the reason is…” Gah!
I think the Great Unwashed is right actually. As a non-mathematician, what makes you so sure what mathematicians mean by QED? Every mathematician I’ve ever spoken to (and contrary to your suggestion, that does include me) has used the “As it was to be shown” definition of QED. I can assure you that every time I wrote “QED”, I had the phrase “As it was to be shown” in my head.
Why the hell would we put “Because it was shown” at the end of a proof?
It’s only three letters after all. It means what the author wants it to mean.
I am still none the wise, however, why after “reading the whole thread” you are so sure that 0.999… != 1. Explain that, oh humour-enlightened one.
RealityChuck: it’s not that you can’t multiply each element per se, it’s more that having done so the answer you get doesn’t necessarily mean anything. After all, if the sum is divergent then it goes to infinity. What does 10 time infinity mean?
It sounds simple, but what you have to show is that the sum of 0.9 + 0.09 + 0.009 + … isn’t infinite. Until you show that, you can’t multiply it by ten and subtract what you first thought of (else you are doing no more than 10 times infinity minus infinity). It’s not good enough to point at it and assert that it isn’t infinite - you have to prove it.
How does one prove it? Why - by finding the limit of course! And the limit of 0.9 + 0.09 + 0.009 + … is one. Now, having established this we can do 10×1 - 1 and complete your proof. But since we’ve already shown that the limit is 1, there isn’t much point!
Do you see where the objection is now? [ul][li]x = 0.999… can only really be defined as an infinite sum; it’s shorthand for x = 0.9 + 0.09 + 0.009 + … [/li][li]Therefore you can’t do that 10x - x thing until you prove that x isn’t infinite…[/li][li]… but the proof that x isn’t infinite shows that x = 1 all by itself![/ul]This apparently trivial problem with the process is just one of the many little annoying things that appear in higher mathematics. I remember well the day that I had to prove that 1 + 1 = 2.[/li]
pan
Kabbes - “As it was to be shown” is simply re-stating the question. That is not what ANY mathematician means when he / she writes QED. QED is used as a terminator, - this is finished BECAUSE of what I have just written - “because it was shown”.
You could use “As it was to be shown” if you had made a mistake and you were refering back to the error you made in a previous piece of work… So did you make a mistake ? If not, you meant “quod erat demonstratum”
QED stands for Quod Erat Demonstrandum, “which was to be proved”. It was added to the end of the proofs in Euclid ( though obviously not by Euclid himself, as he wrote in Greek).
Similarly, demonstrations of constructions were ended with QEF, Quod Erat Faciendum, “Which was to be done”.
Nobody is arguing that, lucwarm. The point is that:
a) 0.999… means the sum as n goes from 1 to infinity of 9*(10^-n)
b) the value of an infinite series is the limit of the values of the partial sums
c) the partial sum of the series identified in a) after k terms is 1 - 10^-k
d) the limit of 1 - 10^-k as k goes to infinity is 1
e) therefore, 0.999… = 1.
It doesn’t matter that 10^-infinity isn’t well-defined. This is irrelevant to the claim, which relies only on the convergence of the infinite sum …
Is there any repeating decimal notation of an infinite sum that doesn’t converge, where the “10x - x =9” proof wouldn’t work? Or, do all repeating decimals converge?