1. Originally Posted by Mario F.
... we don't really have an exact representation for 1/9 or 1/3. 0.333... is merely a representation of that fraction, but itself a number that cannot be precisely expressed.
Sure we do. 1/3 is represented in decimal as zero followed by a decimal point followed by an infinite number of 3's. That's a perfectly valid representation, just like 1/2 is zero followed by a decimal point followed by 5 followed by an infinite number of 0's. Similarly, the number 0.999... is zero followed by a decimal point followed by an infinite number of 9's. Any real number that you can add to this that would be greater than 1, so 0.999... = 1.

EDIT: And this representation is really just an artifact of using base 10. For example, in base 9, 1/3 is exactly 0.3. The fact that it doesn't have a finite representation in decimal doesn't mean it's an approximation.

2. Originally Posted by Clairvoyant1332
Sure we do. 1/3 is represented in decimal as zero followed by a decimal point followed by an infinite number of 3's.
Yes. But because you can't write an infinite amount of 3, you choose to write a symbol instead (0.333...). That's a representation, not a number. The symbol of a number (it's no different from π, or e. It's the symbolic representation of a constant). And you don't perform arithmetics on the individual characters that make up a symbol. The fact that this particular symbol uses the decimal numeral system, shouldn't give you free reign to treat it as a number. It's a mere illusion it is a number.

To show you the problem of interpreting that symbol as a number, let me give you this example:

Premise: 0.333... = 1/3

Let's test it:

0.3 = 1/3 ? Answer: No
0.33 = 1/3 ? Answer: No
0.333 = 1/3 ? Answer: No
0.3333 = 1/3 ? Answer: No
0.333333333333333333333 = 1/3 ? Answer: No

And it will always be No. Towards infinity. It will never be Yes. Ever.

So, 0.333... = 1/3 ? Answer: No, if you treat it as a number.

Similarly, the number 0.999... is zero followed by a decimal point followed by an infinite number of 9's. Any real number that you can add to this that would be greater than 1, so 0.999... = 1
I'm sorry. But this debate is far beyond the point where you feel you have to tell me what does it mean 0.999... = 1. I know perfectly well what it means.

3. Originally Posted by Mario F.
Yes. But because you can't write an infinite amount of 3, you choose to write a symbol instead (0.333...). That's a representation, not a number. The symbol of a number (it's no different from π, or e. It's the symbolic representation of a constant). And you don't perform arithmetics on the individual characters that make up a symbol. The fact that this particular symbol uses the decimal numeral system, shouldn't give you free reign to treat it as a number. It's a mere illusion it is a number.
You cannot write any number. You can only write one of its representations, so I don't buy your theory.

If you really believed that "it's a mere illusion" that 0.999... "is a number", you would not have stated that 0.999... is "an unaccountable number". You would have simply stated that it is not a number.

4. Originally Posted by Yarin
So I know infinitesimility is suppose to = 0, but I was thinking, what does infinitesimility * infinity equal? I mean, I would think it would have to equal 1, which argues infinitesimility = 0.
I think we've slided a MBit off topic!
Originally Posted by Yarin
What do you guys think?

5. 0.3 = 1/3 ? Answer: No
0.33 = 1/3 ? Answer: No
0.333 = 1/3 ? Answer: No
0.3333 = 1/3 ? Answer: No
0.333333333333333333333 = 1/3 ? Answer: No

And it will always be No. Towards infinity. It will never be Yes. Ever.
Prove it. Prove that 0.33... doesn't equal 1/3.

6. Originally Posted by Shakti
Prove it. Prove that 0.33... doesn't equal 1/3.
Infinity doesn't exist! That's your proof.

EDIT: Ignore my signature...

7. Originally Posted by laserlight
You cannot write any number. You can only write one of its representations, so I don't buy your theory.

If you really believed that "it's a mere illusion" that 0.999... "is a number", you would not have stated that 0.999... is "an unaccountable number". You would have simply stated that it is not a number.
When I stated that 0.999... was an uncountable number I was in fact wrong. It is a countable number. It stemmed from my misconception of what irrational numbers are. Let's make that clear.

Anyways, to answer you, I think it's perfectly plausible to address the symbol of a number as a number during the flow of speech. 0.999... is the representation of a number, much like TT is the representation of another number. Yet, I can say "TT is an irrational number". Nobody will be shocked by that and we still intuitively know perfectly well that the two characters composing the symbol TT aren't numbers.

With that in mind, what do you get when you add pi to itself? 2TT. Similarly, 0.333... added to itself is 2 x 0.333..., and not 0.666...

Now, depending on the requirements for the identity of TT to remain visible, I can simply say 4.283 instead of 2TT. So, by all means, take advantage of the fact the symbol for 1/3 happens to use numerals and do your arithmetic on them if you are willing to lose precision. But, as you know, the identity of TT must remain visible throughout much of the operations performed on it. Particularly to do with geometry. This requirement guarantees that the symbol TT never loses precision. So, you often get 2TT instead of 4.283 and use 2TT as the final result. I expect the same deference towards the symbol "0.333..." when we are discussing the nature and implications of this number as it approaches infinity.

8. Originally Posted by Shakti
Prove it. Prove that 0.33... doesn't equal 1/3.
I just did.

9. Where?

All i found were statements:
0.3 = 1/3 ? Answer: No
0.33 = 1/3 ? Answer: No
0.333 = 1/3 ? Answer: No
0.3333 = 1/3 ? Answer: No
0.333333333333333333333 = 1/3 ? Answer: No
But these statements say nothing about wether 0.33... = 1/3. You used your intuition to say that it is false, but you can't do that. A proof must make use only of previously proved theorems of math. Intuition and simple statements can not be used.

Those statements say nothing about wether
0.33333333333333333333333333333333 = 1/3 and so on.

10. I'm sorry. But if you don't understand that proof, there's nothing more I can do for you.

"0.333..." is 1/3 if you look at it as a symbol. The moment you look at it as an actual number and try to perform operations on its individuals characters, it ceases to be for the reasons I shown. This has been discussed before on this thread. I thought it had been already established that 0.333... or 0.999... aren't numbers. You can't add, subtract, divide, or multiply the individual characters of these symbolic representations. Check the earlier pages of this thread. It's there somewhere.

11. Originally Posted by Mario F.
With that in mind, what do you get when you add pi to itself? 2TT. Similarly, 0.333... added to itself is 2 x 0.333..., and not 0.666...
With that in mind, what do you get when you add pi to itself? 2TT. Similarly, 10 added to itself is 2 x 10, and not 20.

12. With that in mind, what do you get when you add pi to itself? 2TT. Similarly, 0.333... added to itself is 2 x 0.333..., and not 0.666...

Now, depending on the requirements for the identity of TT to remain visible, I can simply say 4.283 instead of 2TT. So, by all means, take advantage of the fact the symbol for 1/3 happens to use numerals and do your arithmetic on them if you are willing to lose precision.
Well, add pi to pi and you can also get 6,2831... (in this context ... denotes all other digits in pi) and i won't lose precision. And you state that i will lose precision by equalling 1/3 with 0.333... (infinite number of 3's). I challenge you again to prove that last sentence.

1/3 = Σ(+∞, i=0)(0.3*10^-i)

14. Originally Posted by Mario F.
"0.333..." is 1/3 if you look at it as a symbol. The moment you look at it as an actual number and try to perform operations on its individuals characters, it ceases to be for the reasons I shown. This has been discussed before on this thread. I thought it had been already established that 0.333... or 0.999... aren't numbers. You can't add, subtract, divide, or multiply the individual characters of these symbolic representations. Check the earlier pages of this thread. It's there somewhere.
You only get into trouble when operating on the individual numbers simply because we are limited in the amount of numbers we can actually print down. But to say you can't add, subtract or do anything on them just because they have a ... after is just silly. Sure you can, as long as you carry in the back of your head that you are dealing with an infinite amount of numbers. You can use it in a formula. It is perfecely valid for me to say
1+0.33... and get a result of 1.33... out of it.

Edit:
Better example:
0.33... - 0.3 which i think you will agree with me equals 0.033... (infinite amount of 3s).

15. Originally Posted by laserlight
With that in mind, what do you get when you add pi to itself? 2TT. Similarly, 10 added to itself is 2 x 10, and not 20.
"0.333..." is not a number. 10 is a number. Don't look at the symbols 3 or 0 as numbers. They aren't. "0.333..." could as well be represented with the symbol ¨ and all the troubles would go away. It's not my fault that someone chose to use numerals for this symbolic reference of a constant.

...

I'll tell you what. This has been a fascinating debate. It's been 4 days now. I have to say that haven't had so much fun in quite a while. Despite feeling that I have been rowing against the current for the past 5 pages of this debate -- and where each page has multiple posts of mine -- I never for one moment felt anything but joy in arguing with you folks. The constant need to research these topics also has been a great addition to my general (and basic) knowledge of mathematics.

But I'm tired now. I refuse to go back to this type of debate and where I feel I'm not being heard, or even respected. Where there's no effort whatsoever to decipher the meaning of what I'm saying and replies are being produced the second after a post of mine is produced. Or maybe it's my poor English, or the fact I lack the basic tools to form valid and easy to understand arguments. I don't know. But I'm tired indeed. This is where I leave the debate. It's been great though.

Thanks. Sincerely.