1 = 0.999999999999?

• Oct 22nd 2012, 10:34 AM
pflo
1 = 0.999999999999?
The question came up: does 0.999999999(repeating) = 1?

Several proofs were offered:

1/3 = 0.33333333333(repeating)
multiply both sides of this by 3 to get
3/3 = 0.9999999999999(repeating)
1 = 0.9999999999999(repeating)

x = 0.9999999999999(repeating)
10x = 9.999999999999(repeating)
subtracting the frist equation above from the second gives:
9x = 9
x = 1

But the idea was repulsive to many:
1-0.99999999999999(repeating) = 0????
"There MUST be some infinitesimal difference!!", they said.
The proponents of the proposition agreed, but said, "The proofs don't lie."

My response was $\displaystyle 1-0.99999999(repeating) = \lim_{x \to 0}x$
Of course, this limit = 0.

So, DOES 0.9999999999(repeating) = 1?
How could you represent the infinitesimal difference?
• Oct 22nd 2012, 04:13 PM
skeeter
Re: 1 = 0.999999999999?
$\displaystyle 0.999 ... =$

$\displaystyle 9\left(\frac{1}{10} + \frac{1}{10^2} + \frac{1}{10^3} + ... \right) =$

$\displaystyle 9 \sum_{n=1}^\infty \left(\frac{1}{10}\right)^n =$

since $\displaystyle |r|< 1$ ...

$\displaystyle 9 \cdot \frac{\frac{1}{10}}{1 - \frac{1}{10}} =$

$\displaystyle 9 \cdot \frac{\frac{1}{10}}{\frac{9}{10}} =$

$\displaystyle 9 \cdot \frac{1}{9} = 1$
• Oct 22nd 2012, 04:37 PM
Plato
Re: 1 = 0.999999999999?
I am absolutely opposed to capital punishment!
But I think that I may make an exception for someone posting this question.
This is just a stupid question. Any response is meaningless.
Any person who posts this question is brain dead.
• Oct 22nd 2012, 04:45 PM
Prove It
Re: 1 = 0.999999999999?
Quote:

Originally Posted by Plato
I am absolutely opposed to capital punishment!
But I think that I may make an exception for someone posting this question.
This is just a stupid question. Any response is meaningless.
Any person who posts this question is brain dead.

I don't see why this is such a stupid question. The OP asked "how can you succinctly write the infinitessimal difference between 1 and the 0.9999999999...?"

I would write this infinitessimal difference as \displaystyle \displaystyle \begin{align*} \left( \frac{1}{10} \right)^x \end{align*} where \displaystyle \displaystyle \begin{align*} x \end{align*} is some very large number. Clearly as \displaystyle \displaystyle \begin{align*} x \to \infty, \left( \frac{1}{10} \right)^x \to 0 \end{align*}.
• Oct 22nd 2012, 05:13 PM
pflo
Re: 1 = 0.999999999999?
Thanks Skeeter. Thanks Prove It. You two obviously aren't so brain dead that you can't see why someone would ask this question. Plato obviously is. Either that or he's just so much of a jerk that when he feels superior to a questioner he will only respond with insulting dribble making himself feel even more superior. Regardless, I hadn't seen the representations that you two posted. So thanks again.
• Oct 22nd 2012, 05:33 PM
Plato
Re: 1 = 0.999999999999?
Quote:

Originally Posted by pflo
Thanks Skeeter. Thanks Prove It. You two obviously aren't so brain dead that you can't see why someone would ask this question. Plato obviously is. Either that or he's just so much of a jerk that when he feels superior to a questioner he will only respond with insulting dribble making himself feel even more superior. Regardless, I hadn't seen the representations that you two posted. So thanks again.

Any one is brain dead who does not first do a Goolge search.
Look at the one example.
Any reply to the otherwise is evidence of brain death.
• Oct 22nd 2012, 05:35 PM
HallsofIvy
Re: 1 = 0.999999999999?
Quote:

Originally Posted by Prove It
I don't see why this is such a stupid question. The OP asked "how can you succinctly write the infinitessimal difference between 1 and the 0.9999999999...?"

I would write this infinitessimal difference as \displaystyle \displaystyle \begin{align*} \left( \frac{1}{10} \right)^x \end{align*} where \displaystyle \displaystyle \begin{align*} x \end{align*} is some very large number. Clearly as \displaystyle \displaystyle \begin{align*} x \to \infty, \left( \frac{1}{10} \right)^x \to 0 \end{align*}.

Perhaps it would have been better to point out that there isn't any "infinitesmal difference"! 1 and 0.99999... (repeating) are exactly the same number. That is what skeeter and Plato both said. But, perhaps, Plato has seen too many times!
• Oct 22nd 2012, 05:40 PM
Prove It
Re: 1 = 0.999999999999?
Quote:

Originally Posted by HallsofIvy
Perhaps it would have been better to point out that there isn't any "infinitesmal difference"! 1 and 0.99999... (repeating) are exactly the same number. That is what skeeter and Plato both said. But, perhaps, Plato has seen too many times!

That is a good point, I was assuming that the OP had meant that if you first treated 0.999999... as a finite amount...
• Oct 22nd 2012, 06:03 PM
Plato
Re: 1 = 0.999999999999?
Quote:

Originally Posted by HallsofIvy
But, perhaps, Plato has seen too many times!

I am quit sure how to take that statement.
There is a solid logical foundation to say that $\displaystyle \overline{0.9)}=1$
If you know non-standard analysis, you understand that $\displaystyle \overline{0.9)}$ is in the mondad of (1).
• Oct 22nd 2012, 07:45 PM
pflo
Re: 1 = 0.999999999999?
Hey Plato, thanks for your 2nd and 3rd responses to my question that's not worthy of a response. Your comments are proof that you are indeed the smartest person in the room and now you can sit smugly in your own contented light. Not only have you clearly shown that you know everything better than anyone else, you've also actually provided insight so that I may make a meager attempt to understand a small bit of the knowledge you have long since mastered. You have enlightened my dead brain just a little. Thank you.

And HallsOfIvy - Plato didn't say anything about these being the same number. Instead, he attacked me for asking the question. It was Skeeter and ProveIt who offered additional proofs of the equality. After your post, however, Plato did supply a link to the Wikipedia definition of mondad [sic]. There (in Wikipedia) it clearly shows that the difference between a number and its monad "is infinitesimal". So what I'm walking away with is the fact that this is a topic that even really smart, knowledgeable folks disagree on!

Thanks to everyone!!

P.S. I'm unsubscribing from this thread since I don't feel like reading any more abuse from Plato.
• Oct 22nd 2012, 07:50 PM
Prove It
Re: 1 = 0.999999999999?
Quote:

Originally Posted by pflo
Hey Plato, thanks for your 2nd and 3rd responses to my question that's not worthy of a response. Your comments are proof that you are indeed the smartest person in the room and now you can sit smugly in your own contented light. Not only have you clearly shown that you know everything better than anyone else, you've also actually provided insight so that I may make a meager attempt to understand a small bit of the knowledge you have long since mastered. You have enlightened my dead brain just a little. Thank you.

And HallsOfIvy - Plato didn't say anything about these being the same number. Instead, he attacked me for asking the question. It was Skeeter and ProveIt who offered additional proofs of the equality. After your post, however, Plato did supply a link to the Wikipedia definition of mondad [sic]. There (in Wikipedia) it clearly shows that the difference between a number and its monad "is infinitesimal". So what I'm walking away with is the fact that this is a topic that even really smart, knowledgeable folks disagree on!

Thanks to everyone!!

P.S. I'm unsubscribing from this thread since I don't feel like reading any more abuse from Plato.

I don't think anyone disagrees on the fact that 1 = 0.999999999..., I just think people disagree with needing to show numerous proofs. But I don't see anything wrong with asking for some more reasoning, even if just to see the beauty of mathematics from different points of view.

I have to agree, Plato was out of line in this thread.