What does this mean? $$\Large.\overline9 $$ I've never seen this notation before.
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12Perhaps $0.9999...$? – ajotatxe Jun 02 '15 at 23:28
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50It also means $1$, by that's a whole 'nother discussion. – Christopher King Jun 03 '15 at 00:32
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I thought it was more common to use $0.\stackrel\frown{9}$. Could anybody tell something about that? – Masclins Jun 03 '15 at 06:22
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4@AlbertMasclans Apparently this notation style depends on geography (http://en.wikipedia.org/wiki/Repeating_decimal): overline=US, overdot=China, parentheses=Europe. – user3449173 Jun 03 '15 at 08:02
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4@Nordik Yes, it depends on geography, but parentheses are used only sometimes in Europe. In Germany we typically use the overline, too. Parentheses are used to denote an uncertanty in the last digit(s). – Christoph Jun 03 '15 at 10:58
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1@Nordik Funnily enough, I was taught the dotted notation in the UK in the 1970s when I was at school. – ClickRick Jun 03 '15 at 11:34
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1@ClickRick dots in the UK in the 90s too. I've never seen brackets for this in UK materials. – Chris H Jun 03 '15 at 12:30
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5By the way, the overline is technically known as a vinculum. – Brian M. Scott Jun 03 '15 at 14:38
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I was taught underline. – Frank Vel Jun 03 '15 at 16:00
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@fvel: an underline above something? No, you haven't been taught that... – Thomas Weller Jun 03 '15 at 19:40
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1@ChrisH dots in the UK in the 90/00s too. (I think at this point we can maybe just say "dots in the UK too"!) – OJFord Jun 03 '15 at 22:10
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I was taught an overline in the UK. How do you use the dots to denote multiple repeating digits (eg 1/7), one over each? just first/last? – Holloway Jun 04 '15 at 10:00
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@Trengot Dots were a gradeschool thing for me, and I dropped them because of their ambiguity and excess space as soon as I learned the overline. For example, 2/11 = 0.1818181... and the reader has to infer which decimals repeat, which can only be easily done with relative certainty if the repeating decimals have been included at least twice. 1/7 would take at least 12 decimal places. – Izkata Jun 04 '15 at 12:57
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@izkata, I meant the dots over the numbers. – Holloway Jun 04 '15 at 13:00
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@Trengot Ah, dots over the number I'd not heard of before – Izkata Jun 04 '15 at 13:21
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@ThomasWeller no, an underline under the repeating digits. – Frank Vel Jun 04 '15 at 19:35
5 Answers
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It is called a vinculum and it denotes a repeating decimal.
Zubin Mukerjee
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13+1 for a good word I need to remember to use when the time is right. – JTP - Apologise to Monica Jun 02 '15 at 23:58
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It means a repeating decimal. One can write $\frac 16=0.1\overline 6$, or $\frac 1{14}=0.0\overline{714852}$ for example. The repeating part is whatever is under the overline.
Ross Millikan
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As other answers have said, it stands for a repeating decimal, where the digits under the line are repeated. $$0.\overline{9} = 0.9999999\ldots$$
But if you want to be a little pedantic, you might prefer to say that both $0.\overline{9}$ and $0.9999999\ldots$ are two different forms of notation for the same number. That number is the limit of the sequence formed by repeatedly appending copies the digits covered by the line. In other words, given the notation $0.\overline{9}$, you can write out the following sequence: $$\begin{align} a_1 &= 0.9 \\ a_2 &= 0.99 \\ a_3 &= 0.999 \\ a_4 &= 0.9999 \\ &\vdots \end{align}$$ As you tack on more and more repetitions, these numbers get closer and closer to some limiting value, which I'll call $A$. If you know calculus notation, $$\lim_{n\to\infty} a_n = A$$ The number represented by $0.\overline{9}$ is $A$. It happens to work out to be $1$ (or if we're being pedantic, $1$ is yet another notation for the same number).
As another example of this way of interpreting repeating decimals, consider $0.257\overline{143}$. You can write the sequence $$\begin{align} b_1 &= 0.257143 \\ b_2 &= 0.257143143 \\ b_3 &= 0.257143143143 \\ b_4 &= 0.257143143143143 \\ &\vdots \end{align}$$ and similarly, the number represented by $0.257\overline{143}$ is the value that this sequence gets closer and closer to as you add more repetitions; or $$\lim_{n\to\infty} b_n$$ This one works out to $\frac{128443}{499500}$.
David Z
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Did you just randomly pick that last fraction or did you already have it in mind? – user1717828 Jun 03 '15 at 12:19
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@user1717828 https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/conv_rep_decimals/e/converting_repeating_decimals_to_fractions_2 – Aditya Hase Jun 03 '15 at 13:06
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2Worth noting, since the 'overdot' notation being standard in UK and apparently China has come up in comments on OP, that in that variant we dot the first and last in a sequence (as opposed to each digit). – OJFord Jun 03 '15 at 22:13
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@user1717828 totally random, I didn't feel like going to the trouble of looking up a "significant" fraction. – David Z Jun 04 '15 at 05:08
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$0.\overline{9}=0.999999\ldots=1$
More generally, $0.\overline{n}=0.nnnnnnnnn\ldots$
For example, $\frac 13=0.333333333\ldots=0.\overline3$
Teoc
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This symbol is called a vinculum or a overline.
The number(s) under this symbol are repeated indefinitely.
For your question:
$0.\overline{9} = 0.99999999999999...$
It could also be used like:
$2.52\overline{346} = 2.52346346346346...$
Orange_Web
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What does this add to the existing, year-and-a-half-old answers? – Noah Schweber Dec 14 '16 at 16:14
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1I am new, so I am just using this question to get my bearings on formating. – Orange_Web Dec 14 '16 at 16:18
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1Ah. I think this would be a better place to do that. Also, note that you don't actually have to submit your answer (or question) to see how it will format - this site does as-you-type previews (scroll down from the box where you're typing your answer/question; it may take a few seconds for the preview to form). One reason not to answer old questions to test formatting, is that any change to a question bumps it to the front page. – Noah Schweber Dec 14 '16 at 16:25