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$\frac{1}{9}=0.111\dots$

$9\times \frac{1}{9} = 0.999\dots$

$1=0.999\dots$

What is the problem here? Thanks for any help.

Stahl
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bor
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    There is no problem. – Tomas Nov 04 '13 at 09:16
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    None whatsoever. $0.(1)=\frac19$ ; $0.(3)=\frac39=\frac13$ ; $0.(9)=\frac99=1$. – Lucian Nov 04 '13 at 09:19
  • There are $2$ decimal notations for $1=0.999...$. Just get used to it. – drhab Nov 04 '13 at 09:25
  • In fact, 0.9999999 = 1. Your solution is infact a proof for it. See, http://en.wikipedia.org/wiki/0.999... , for more information. – Sawarnik Nov 04 '13 at 09:26
  • This is actually a very elegant proof for http://en.wikipedia.org/wiki/0.999. The one I learned was certainly not as neat. – Newb Nov 04 '13 at 09:32
  • @user72694: Please don't add new tags without considering a meta thread at first. And certainly don't add them to 13 questions without consulting on meta first. – Asaf Karagila Nov 21 '13 at 13:32
  • @AsafKaragila, This seems like routine maintenance that helps avoid duplication in the future. If there is any problem at all with this tag I'll be happy to remove this. Do you see any problem? – Mikhail Katz Nov 21 '13 at 15:50
  • @user72694: The problem is the procedure. With a site this big we can't have people starting to use tags without some measures of control (i.e. posting a discussion on the meta site first); moreover never ever bump more than three-four questions at a time, unless some exceptional reason is at hand. Adding the tag to 13 questions is very much against the norms of the community, since it bumps 13 questions to the front page which only houses 50 questions. With 500 new every day, that's a lot of bumps at the same time. Finally, http://meta.math.stackexchange.com/q/11734 – Asaf Karagila Nov 21 '13 at 20:17
  • @AsafKaragila, thanks for mentioning these technical problems of which I was unaware. – Mikhail Katz Nov 22 '13 at 08:38
  • @user72694: Sure thing. שבת שלום! – Asaf Karagila Nov 22 '13 at 09:48
  • You could see this question from long ago – Ross Millikan Nov 25 '13 at 15:33
  • The problem, if any, is in the instruction in elementary school. While we are taught there that every decimal expansion corresponds to a (unique) number, we are not sufficiently reminded at the same time that some numbers have more than one decimal expansion. – Lubin Nov 25 '13 at 16:17

1 Answers1

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there is no problem, you just expressed the same number in two different ways or repesantations, it's just like you say $\frac{1}{2}=\frac{2}{4}$, the more rigorous way to write $0.9999999...$ is $9 \lim_{n\to +\infty}\sum_{i=1}^{n}(\frac{1}{10})^i$ and we have: $$9 \lim_{n\to +\infty}\sum_{i=1}^{n}(\frac{1}{10})^i=1 $$

Abdelmajid Khadari
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