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Rational Representation of a Repeating Decimal

  1. Jul 10, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the rational number representation of the repeating decimal.


    2. Relevant equations

    3. The attempt at a solution

    I know it has something to do with infinite geometric sequences but I'm not sure what.

    what would your ratio be for a repeating decimal, I've thought about it and cant seam to reason it out, however I know the answer is [tex]\frac{4}{11}[/tex] from my calculator.
  2. jcsd
  3. Jul 10, 2010 #2


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    [tex]x= 0.\overline{36}= 0.36363636...[/tex]

    [tex]100x= 36.36363636....[/tex]
    The point is that since the "repeating" portion never stops, those two numbers have exactly the same decimal part and subtracting cancels them:
    100x- x= 99x= 36. Write that as a fraction and reduce it.

    Another, more "rigorous" method (so you don't have to argue about "canceling" an infinite string of digits) is to use the fact that this is a geometric series:
    [tex]0.\overline{36}= .36+ .0036+ .000036+ \cdot\cdot\cdot= .36(1+ .36+ .0036+ \cdot\cdot\cdot)[/tex]
    a geometric series with common ratio, r, equal to .36. Its sum, by the usual formula, is
    [tex]\frac{.36}{1- .36}[/tex]
  4. Jul 10, 2010 #3
    simplifying [tex]\frac{0.36}{1-0.36}[/tex] does not come out to a fraction that equals [tex]0.\overline{36}[/tex]

    I have found the infinite geometric sequence summation to be:


    which would yield [tex]0.\overline{36}[/tex]
    I understand the formula for the sum of an infinite geometric sequence, however it doesn't seam to work in this case.

    If r is really 10 then [tex]10^\infty[/tex] dosen't tend to 0 so the formula isen't valid. However, you have the exponents, I'm, not sure how you'd deal with thoes.
    Last edited: Jul 10, 2010
  5. Jul 10, 2010 #4


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    The common ratio r isn't 0.36 (Halls misspoke) and it isn't 10 either. What is it? You've got the geometric series correct. So what's the ratio between two successive terms like a_2/a_1?
  6. Jul 10, 2010 #5


    Staff: Mentor

    The last part should be .36(1 + 10-2 + 10-4 + ... ).
    The common ratio r is 1/100, so the sum is .36(1 - 1/100) = 36/99.
  7. Jul 10, 2010 #6
    Ahh... the common ratio would be 10-2 or 1/100 which is the same thing.



  8. Jul 10, 2010 #7


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    HallsofIvy's first method was perfectly rigourous and very neat. The second method is as follows:
    0.363636... & = & 36/10^2+36/10^4+36/10^6+... = 36(10^{-2}+10^{-4}+10^{-6}+...)
    The series in brackets is a geometric progression with first term 10^{-2} and common ratio 10^{-2}, the sum for this is series is 10^{-2}/(1-10^{-2})=1/99. So the rational form is 36/99 as HallsofIvy already calculated.

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