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Math Help - Expressing repeating decimal as fraction

  1. #1
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    Expressing repeating decimal as fraction

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

    The attempt:
    \[2 + \frac{0}{10} + \frac{1}{10^2} + \frac{1}{10^3} + \frac{0}{10^4} + ...\]
    Then I'm stuck here because I don't know how to express it as a series. Any help is appreciated, thanks
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  2. #2
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    Re: Expressing repeating decimal as fraction

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

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

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

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

    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
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  3. #3
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    Re: Expressing repeating decimal as fraction

    Quote Originally Posted by Intrusion View Post
    The problem:
    Express as a fraction: \\2.\overline{011}

    The attempt:
    \[2 + \frac{0}{10} + \frac{1}{10^2} + \frac{1}{10^3} + \frac{0}{10^4} + ...\]
    Then I'm stuck here because I don't know how to express it as a series. Any help is appreciated, thanks
    \displaystyle \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*}
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  4. #4
    Master Of Puppets
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    Re: Expressing repeating decimal as fraction

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

    etc
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