My Math Forum Is 0.9999... equal 1?

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 December 22nd, 2010, 08:42 PM #1 Member   Joined: Dec 2010 From: Viet Nam and US Posts: 36 Thanks: 0 Is 0.9999... equal 1? We all know that 9.0 is unequal 9, or 9.99 is different from 10. But there's something wrong with this problem below: 1) Simple prove Let $x=0.\bar{9}$ $\Leftrightarrow 10x-x=9.\bar{9}-0.\bar{9}$ $\Leftrightarrow 9x=9\Leftrightarrow x=1$ $\Leftrightarrow 0.\bar{9}=1$ 2) Prove using geometric sequence formula Let $x=0.\bar{9}$ $\Leftrightarrow 9\cdot 10^{-1}+9\cdot 10^{-2}+9\cdot 10^{-3}+...+9\cdot 10^{-n})$ $\Leftrightarrow 9(10^{-1}+10^{-2}+10^{-3}+...+10^{-n})$ The geometric sequence above has $\begin{cases}u_1=10^{-1}\\r=10^{-1}\end{cases}$ Since $|r|\le 1$, we have: $\Leftrightarrow 9(10^{-1}+10^{-2}+10^{-3}+...+10^{-n})$ $\Leftrightarrow 9\cdot \frac{10^{-1}}{1-10^{-1}}$ $\Leftrightarrow 9\cdot \frac{1}{9}=1$ $\Leftrightarrow 0.\bar{9}=1$ So what is the problem here
 December 22nd, 2010, 09:10 PM #2 Senior Member   Joined: Nov 2010 Posts: 502 Thanks: 0 Re: Is 0.9999... equal 1? I don't quite see what you're getting at with the second half, but I have two comments. Firstly, 0.999.... is equal to 1. Secondly, 9.0 is equal to 9, unlike your first line.
December 22nd, 2010, 09:27 PM   #3
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Re: Is 0.9999... equal 1?

Quote:
 Originally Posted by DLowry I don't quite see what you're getting at with the second half, but I have two comments. Firstly, 0.999.... is equal to 1. Secondly, 9.0 is equal to 9, unlike your first line.
Hi,
I'm just curious about the accuracy of math. I've learned that, 9.0 is different from 9, exactly only 9=9. One more digit after dot makes a difference. And then I've never known that 0.999.. equal 1, mathematics says that 0.999 < 1. So I'm just concerned about my knowledge, there must be something wrong!

 December 22nd, 2010, 09:35 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Is 0.9999... equal 1? $9.0=9+\frac{0}{10}=9+0=9$ The expression $0.\bar{9}=1$, but any truncation is less than 1. It is only when the nines go on infinitely that we have 1.
December 22nd, 2010, 09:52 PM   #5
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Re: Is 0.9999... equal 1?

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 Originally Posted by MarkFL $9.0=9+\frac{0}{10}=9+0=9$ The expression $0.\bar{9}=1$, but any truncation is less than 1. It is only when the nines go on infinitely that we have 1.
By saying 9.0 not equal 9, I mean that 9.0 is more accuracy than 9, 9.0 points out that after dot, we have a 0, so it is not a Round(9.1) or something.
Easier to imagine, let's say 9.000000000 is not exactly equal 9, because it says after dot, there are 9 more zeroes!
It's so complicated to imagine, I've learned that from school, I wish I could explain it better!

And back to my problem, basically 0.999.. is not equal 1 , so according to you, it's only equal 1 when we have a lot of nines. So there's something inaccuracy in Math, isn't it? (According to my own opinion only!)

 December 22nd, 2010, 10:02 PM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Is 0.9999... equal 1? Not when we have a lot of nines, but only when we have an infinity of nines...not a thousand, or a million, or a centillion, or a googol-plex, or any finite number.
December 22nd, 2010, 10:15 PM   #7
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Re: Is 0.9999... equal 1?

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 Originally Posted by MarkFL Not when we have a lot of nines, but only when we have an infinity of nines...not a thousand, or a million, or a centillion, or a googol-plex, or any finite number.
Hi guys,

So we have to accept that 0.999...=1 if there are infinity number of 9 after dot? Math requires highest accuracy in all subjects, I think, and Math itself claims that 0.999....<1 because $0.\bar{9}=\frac{\bar{9}}{10^n}$. It always exists a number which is the difference between 1 and 0.999... or 1 and $\frac{\bar{9}}{10^n}$ In this case, that number is $\frac{1}{10^n}$. Thus, there is no way that 0.999.. equal 1.

This problem is interesting to me ^^!

 December 22nd, 2010, 10:19 PM #8 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Is 0.9999... equal 1? What is $\lim_{n\to\infty}\frac{1}{10^n}$?
December 22nd, 2010, 10:25 PM   #9
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Re: Is 0.9999... equal 1?

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 Originally Posted by MarkFL What is $\lim_{n\to\infty}\frac{1}{10^n}$?
Zero!
So that's all the problem about, 0.999... = 1 it's about limit. I can go really close to 1 but not exactly 1, however, we consider it is 1, am I right?

 December 22nd, 2010, 10:32 PM #10 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Is 0.9999... equal 1? If we allow n to grow unbounded, then the difference is zero, and we have exactly 1.

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