# What does .9 with a line above the 9 mean?

What does this mean? $$\Large.\overline9$$ I've never seen this notation before.

• Perhaps $0.9999...$? – ajotatxe Jun 2 '15 at 23:28
• It also means $1$, by that's a whole 'nother discussion. – PyRulez Jun 3 '15 at 0:32
• @AlbertMasclans Apparently this notation style depends on geography (en.wikipedia.org/wiki/Repeating_decimal): overline=US, overdot=China, parentheses=Europe. – user3449173 Jun 3 '15 at 8:02
• @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 3 '15 at 10:58
• By the way, the overline is technically known as a vinculum. – Brian M. Scott Jun 3 '15 at 14:38

## 5 Answers

It is called a vinculum and it denotes a repeating decimal.

• +1 for a good word I need to remember to use when the time is right. – JoeTaxpayer Jun 2 '15 at 23:58
• Sounds like something in a gynecologist's office. – zhw. Jun 3 '15 at 0:36
• @JoeTaxpayer: Do you mean +1 or +0.9...? – mipadi Jun 3 '15 at 22:18

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.

$0.\overline{9}=0.999999\ldots=1$

More generally, $0.\overline{n}=0.nnnnnnnnn\ldots$

For example, $\frac 13=0.333333333\ldots=0.\overline3$

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 $\frac{128443}{499500}$.

• Did you just randomly pick that last fraction or did you already have it in mind? – user1717828 Jun 3 '15 at 12:19
• @user1717828 khanacademy.org/math/algebra/… – Aditya Hase Jun 3 '15 at 13:06
• Worth 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 3 '15 at 22:13
• @user1717828 totally random, I didn't feel like going to the trouble of looking up a "significant" fraction. – David Z Jun 4 '15 at 5:08

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:

$2.52\overline{346} = 2.52346346346346...$

• What does this add to the existing, year-and-a-half-old answers? – Noah Schweber Dec 14 '16 at 16:14
• I am new, so I am just using this question to get my bearings on formating. – Orange_Web Dec 14 '16 at 16:18
• Ah. 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