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

Question

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

Answer 1

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

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

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

Answer 2

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.

Answer 3

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

Answer 4

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 $\frac{128443}{499500}$.