# How do you find the repeating decimal 0.1 with 1 repeated as a fraction?

Nov 26, 2016

$0.1111111 \ldots = 0. \overline{1} = \frac{1}{9}$

#### Explanation:

$0.111111 \ldots . .$is a recurring decimal which can be written as $0. \overline{1}$

It is useful to know that the decimals which have only 1 digit repeating all come from the fractions which are ninths.

$\frac{1}{9} = 0.1111 \ldots \text{ "2/9 = 0.22222...." } \frac{3}{9} = 0.333333 \ldots$

$\frac{4}{9} = 0.444444 \ldots$

and so on... But why is this?

Let $\text{ "x= 0.111111111." } \leftarrow$ one digit recurs, multiply by 10.
$\text{ ":.10x = 1.11111111..." } \leftarrow$ subtract them. $10 x - x = 9 x$

$\text{ "9x = 1.0000000..." } \leftarrow$ all the way to infinity

Now: $\text{ } 9 x = 1 \Leftrightarrow x = \frac{1}{9}$

Dividing will confirm this: $1 \div 9 = 0.1111111 \ldots . . = 0. \overline{1}$

What do you make of 0.99999..... ?