Math Is Fun Forum

Ricky, sorry I miss the formula in latex and the defination. But you know people like you within the system have tons of capacity to make definations or rules.

So I need to disprove the defination of 0.999... promoted in post #51, all right.

But first allow me to distinguish some common mistake among some, though not all, young mathematicians. The point is that a ∞ simbol does not necessarily mean a infinity in practice.

For example, 0/0 and ∞/∞ problems. In fact they are not, 0/0 has no defination, so does ∞/∞.
0 should stand for an infinitesimal- a shrinking variable, as well as ∞ should represent a growing variable. And some textbooks express them with quotation marks.

Another instance might be when you say f(x) goes to infinity as x goes to infinity, you mean f(x) diverges.

∞ in the expression " limit(n->∞) An" does not mean real infinity, the reached state by its clear defination in the majority of textbooks, which can only provide a finite N-epsilon proof.

So if you interpret the infinity under a limit symbol as real, you've already make a closer-reaching imaginary leap.

Why? Involving imperfect competition, Game theory perhaps is the only way out.

Hey, I come up with an idea:

Just use two pairs of  convenient numbers both larger or both smaller than them and multiply.

If the two results you get have the same amount of digits, the same for the original pair.

Eg:
123 586
150*600=90000
100*500=50000
So 5 digits

The most Tricky Proof:

10*0.999...=9.999...
-) 0.999...=0.999...
9*0.999...=1

For the first step I get 10*0.999...=9.99... step by step, digits by digits so far. Thus the proof left a bothering remainder and is not so convincing.

An arguement against this is that 0.999... is a all developed state where are infinite 9's. Such state does exist in the lecturer's head. But it is sheer imaginery. Could I imagine  a "well developed" state like 0.999...91? They may argue it doesn't make sense. Well, I don't know if I am not permitted to think of the infiniteth digit another way, but I demand the right. Further, if everyone comes up with different ideas about infinity, infinity is an art terminology.

Another argument against it is the "The impossibility of infinite steps" one again- how do I know it's a state? So far I can do is to write 0.9, 0.99, 0.999,..., 0.9999999, which are all numbers and constant, stable, and unvarying. I don't know if infinity vary or not, and if it does not, what is it? -It's just the amount you cannot count. Okay, how complex!

One proof that 0.9+0.09+0.009+... is a number is a limit convergence proof, which could be disproved by Post 49.

The last proof that 0.9+0.09+0.009+... is a number is the " A strict increasing series must have a least upper bound" proof, definately having something to do with the Real defination created by Georg Cantor. Actually the Reals are defined as approaching something, which I should write a little bit the other day.

Yes, I dwelled into this probiblities once, but I thought you'd got a simple solution. It's a pitty to see the simple formula fail.

Or
click here
then click on the title "Has famous maths problem been solved, and in only a month?

"Or are you looking for expected meaning 50% chance of happening, in which case, I haven't done the math yet."
- this meaning more or less, though the chance probabilities are more complex. Too complex for this question!

Hey, since Ricky don't exist in the reals, he is 0-just kidding.

An infinitesimal isn't even a number at all, at least highlighted in most caculus books.

Forget it-i'm not curious about the answer any more. Guess the answer of this complex question is a waste for me.

Dross's and akademika's similar prooves involve a more complex structure. I will explain them with lots of detail the other day.

Proof via 1/9=0.111... or 1/3=0.333... is comparatively easier to defy. In previous posts the former as an example is questioned from the begining. That is to say no proof can make 1/9 equal to 0.111...  with infinite decimals.

It may be some cruel instead of fun in my proof, but a proof is either true or false, a zero-sum game. It was also cruel for those who thought 0.999...=1 was wrong previously.

for instance,
a(aa+ab+bb)-b(aa+ab+bb)= (a-b)(aa+ab+bb)= a^3-b^3

Use Matlab for a good approximation.

Please give your answer if you find one. Actually this is a hard and challenging topic indeed.