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Math Help - n repeating digit of decimal number

  1. #1
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    n repeating digit of decimal number

    I have a problem how to convert number 0.3333....3 where 3 repeats N times to fraction.
    for N=1 0.3
    N=2 0.33 and so on.

    I know how to convert number where digits repeats forever, but how can I convert this to fraction to get correct result for any N.

    Thanks in advance
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  2. #2
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    Re: n repeating digit of decimal number

    Quote Originally Posted by Bokas View Post
    I have a problem how to convert number 0.3333....3 where 3 repeats N times to fraction.
    for N=1 0.3
    N=2 0.33 and so on.

    I know how to convert number where digits repeats forever, but how can I convert this to fraction to get correct result for any N.

    Thanks in advance
    \displaystyle \begin{align*} x &= 0.\dot{3} \\ 10x &= 3.\dot{3} \\ 10x - x &= 3.\dot{3} - 0.\dot{3} \\ 9x &= 3 \\ x &= \frac{3}{9} \\ x &= \frac{1}{3} \\  0.\dot{3} &= \frac{1}{3} \end{align*}
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  3. #3
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    Re: n repeating digit of decimal number

    Quote Originally Posted by Bokas View Post
    I have a problem how to convert number 0.3333....3 where 3 repeats N times to fraction.
    for N=1 0.3
    N=2 0.33 and so on.

    I know how to convert number where digits repeats forever, but how can I convert this to fraction to get correct result for any N.

    Thanks in advance
    A terminating decimal is easier than a repeating decimal (which is what ProveIt did). If x= 0.33333..3 where the "3" repeats N times, then 10^Nx= 3333...3 so that x= \frac{3333...3}{10^N}. Because of that " 10^N" in the denominator, a terminating decimal, converted to a fraction, has only powers of 2 and 5 in its denominator.
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  4. #4
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    Re: n repeating digit of decimal number

    Thank you both
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