# Thread: Expressing repeating decimal as fraction

1. ## Expressing repeating decimal as fraction

The problem:
Express as a fraction: $\displaystyle \\2.\overline{011}$

The attempt:
$\displaystyle $2 + \frac{0}{10} + \frac{1}{10^2} + \frac{1}{10^3} + \frac{0}{10^4} + ...$$
Then I'm stuck here because I don't know how to express it as a series. Any help is appreciated, thanks

2. ## Re: Expressing repeating decimal as fraction

$\displaystyle 2 + \frac{11}{10^3} + \frac{11}{10^6} + \frac{11}{10^9} + ...$

$\displaystyle 2 + 11\left(\frac{1}{10^3} + \frac{1}{10^6} + \frac{1}{10^9} + ... \right)$

$\displaystyle 2 + 11 \left(\frac{\frac{1}{10^3}}{1 - \frac{1}{10^3}}\right)$

$\displaystyle 2 + 11 \left(\frac{1}{10^3 - 1}\right)$

$\displaystyle 2 + \frac{11}{999} = \frac{1998+11}{999} = \frac{2009}{999}$

and then there is the easy way ...

1000x = 2011.011011011...
- (x = 2.011011011 ...)
-------------------------
999x = 2009

x = 2009/999

3. ## Re: Expressing repeating decimal as fraction

Originally Posted by Intrusion
The problem:
Express as a fraction: $\displaystyle \\2.\overline{011}$

The attempt:
$\displaystyle $2 + \frac{0}{10} + \frac{1}{10^2} + \frac{1}{10^3} + \frac{0}{10^4} + ...$$
Then I'm stuck here because I don't know how to express it as a series. Any help is appreciated, thanks
\displaystyle \displaystyle \begin{align*} x &= 2.\overline{011} \\ 1000x &= 2011.\overline{011} \\ 1000x - x &= 2011.\overline{011} - 2.\overline{011} \\ 999x &= 2009 \\ x &= \frac{2009}{999} \end{align*}

4. ## Re: Expressing repeating decimal as fraction

1/9 =0.11111....
1/99= 0.010101....

etc