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I was wondering is there any way to express $0.999999$ recurring as an actual fraction without equaling $1$? Because I tried to convert it into a fraction following the rules for normal recurring decimals like this:

$$\begin{align}n&=0.999\dot9\\10n&=9.999\dot9\\n&=0.999\dot9\\9n&=9\\\therefore n&=9/9\end{align}$$

But as you can see the result is $9/9$ which ultimately is equal to $1$ . And I've even tried calculating it other ways like this:

$$\begin{align}1/3&=0.333\dot3\\2/3&=0.666\dot6\\\therefore3/3&=0.999\dot9\end{align}$$

But it always ends up telling me that $0.9999999... = 1$. Is there any mistake in my logic? And I also realized this applied to other recurring decimals ending in $9$. E.g: $0.5999999...=5.4/9 = 0.6$ . So is there a way to write $0.999999...$ as a fraction so you can differentiate it from $1$?

Simply Beautiful Art
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yt.
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    Why do you expect to have $0.99\ldots \ne 1$? – John Griffin Aug 11 '17 at 13:35
  • Because they're not equal? – yt. Aug 11 '17 at 13:36
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    @Ruby: that is a common misbelief. $0.\overline{9}$ and $1$ are two representations of the same number. The decimal representation is not unique: $$\sum_{n\geq 1}\frac{9}{10^n} = 1.$$ – Jack D'Aurizio Aug 11 '17 at 13:37
  • On the other hand, if you know that $0.\overline{1}=\frac{1}{9}$ it should not come as a surprise that $$9\cdot 0.\overline{1} = 0.\overline{9} = \frac{9}{9} = 1.$$ – Jack D'Aurizio Aug 11 '17 at 13:39
  • Really? What about other recurring decimals that end in 9 like 5.999999...? – yt. Aug 11 '17 at 13:39
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    @Ruby: $5.\overline{9}=6$, simply. – Jack D'Aurizio Aug 11 '17 at 13:39
  • See https://math.stackexchange.com/questions/11/is-it-true-that-0-999999999-dots-1 – Hans Lundmark Aug 11 '17 at 13:39
  • Yeah i also tried out the 1/9 and it turned out to be equaling one so i was really keen on finding a fraction form of 0.99999.. recurring. – yt. Aug 11 '17 at 13:40
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    There is lots of good stuff about this at https://en.wikipedia.org/wiki/0.999... . Btw the fact that this equality holds is not intuitive so don't feel bad if it came as a surprise. I also remember feeling uneasy about it when I first saw it. – EHH Aug 11 '17 at 13:41
  • I knew the proof a while ago but i thought it was just like a very persuadable maths trick to mess with people's brains. – yt. Aug 11 '17 at 13:43
  • The infinite decimal representation of 1: $1.0= 1.00=...=1.00000....=1$. – Ixion Aug 11 '17 at 13:48
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    If they are not equal, what is a real number between them? "Because they are not equal" is not an argument, it is an intuition. Your intuition is wrong. – Thomas Andrews Aug 11 '17 at 13:48
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    You should be pleased with yourself! You have discovered and proved the surprising fact that $0.\dot9 = 1$. This is how science advances. – MJD Aug 11 '17 at 13:53
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    Please do not use pictures for critical portions of your post. Pictures cannot be searched and are inaccessible to those using screen readers. I have [edit]ed your question to reflect this principle. Please check to make sure I haven't changed anything. For some basic information about writing math at this site see here, here, here and here. – Simply Beautiful Art Aug 11 '17 at 14:05
  • @SimplyBeautifulArt Thank you for your edit .Sorry for the bad format, I'm still not very familiar with writing maths on here and I had to use a screenshot from word as i couldn't type out the recurring symbol. But I will make sure to write things in the correct format in the future! – yt. Aug 11 '17 at 14:09
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    @Ruby Haha, I'll tell you, $\LaTeX$ is way friendlier than that thing on word. So limited and slow! Most of the symbols you will ever used can be found here and here, and I believe you can ask how to write certain things on meta. – Simply Beautiful Art Aug 11 '17 at 14:12
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    @SimplyBeautifulArt Thanks for your help. I guess I won't be using screenshots from Word anymore... ;) – yt. Aug 11 '17 at 14:14
  • @SimplyBeautifulArt Yeah, my question was kind of pointless once I found out 0.999999..=1 – yt. Aug 11 '17 at 14:19
  • @Ruby You seem rather quick with comments, so I'll extend an invitation for you to entire one of my chats. I have most of the whole day to chat if you have more subtle questions related to this. – Simply Beautiful Art Aug 11 '17 at 14:33

2 Answers2

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As you have proven by yourself, $$ 0.999999999999999999999\dot9 = 1. $$ There are no (big) logical mistakes in your post.

Because $0.99\dots$ is equal to $1$, it also cannot be another fraction.

supinf
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Here's an alternate, more rigorous proof using series

$$ 0.9999\ldots = \frac{9}{10} + \frac{9}{10^2} + \dots = \sum_{n=1}^\infty \frac{9}{10^n} $$

This is the geometric series $9\sum \left(\frac{1}{10}\right)^n$, which has common ratio $r=\frac{1}{10}<1$. So using the formula for geometric series,

$$ 0.999\ldots = \frac{\frac{9}{10}}{1-\frac{1}{10}} = 1$$

Dando18
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