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Math Help - Repeating decimals

  1. #1
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    Repeating decimals

    I discovered that the decimal representation of 15/17 consists of an endlessly repeating sequence of 16 digits. I wonder what rational number has the longest sequence of repeating digits?
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  2. #2
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    Thinking more about this, it occurs to me that a rational number expressed as a fraction having arbitrarily long numerator and denominator that differ by 1 can have an astronomically long sequence of repeating digits in the decimal representation.
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  3. #3
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    Quote Originally Posted by StevenBrown View Post
    Thinking more about this, it occurs to me that a rational number expressed as a fraction having arbitrarily long numerator and denominator that differ by 1 can have an astronomically long sequence of repeating digits in the decimal representation.
    You're getting there (actually the sequence is unlimited as you progress through the fractions)
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    You might find useful this reference (also the other subheadings)

    Wikipedia - Repeating decimal - subheading Other properties of repetend lengths

    You might also like to think backwards; given any positive decimal expansion with a repetend, we can find unique positive integers p,q such that gcd(p,q)=1 and p/q has the given value.
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  5. #5
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    It occurred to me that a repeating decimal such as 15/17 =

    0.8823529411764705 8823529411764705 8823529411764705 ...

    is "going fractal," in the sense that the same pattern repeats endlessly on smaller and smaller scales.
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  6. #6
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    Hello, StevenBrown!

    You discovered this on your own? . . . Good for you!
    Keep exploring . . . there's some fascinating stuff out there!


    I discovered that the decimal representation of \frac{15}{17}
    consists of an endlessly repeating sequence of 16 digits.
    I wonder what rational number has the longest sequence of repeating digits?

    Considerable exploration has already been done on this subject.

    Here are some basics that I remember . . .


    If p is a prime, the decimal for \frac{1}{p} has a p-1 digit repeating cycle.

    However, some primes can have even shorter cycles.


    For example: . \begin{Bmatrix}\frac{1}{11}\text{ has a 2-digit cycle:} &0.\overline{09} \\ \\[-3mm]<br />
\frac{1}{13}\text{ has a 6-digit cycle:} & 0.\overline{076923} \\ \\[-3mm]<br />
\frac{1}{37} \text{ has a 3-digit cycle:} & 0.\overline{027}\end{Bmatrix}


    What determines the length n of the cycle for prime reciprocals?

    It is the least n for which 10^n-1 is divisible by p.


    In baby-talk, consider a string of 9's.
    Begin dividing by a prime p.
    When it "comes out even", stop.
    The number of 9's is the length of the cycle.


    Example: \frac{1}{7}

    The first time the division stops is: . 999999 \div 7 \,=\,142857
    . . Therefore, \frac{1}{7} has a 6-digit cycle.


    Example: \frac{1}{37}

    The first time the division stops is: . 999 \div 37 \,=\,27
    . . Therefore, \frac{1}{37} has a 3-digit cycle.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    Here's another interesting feature . . .

    . . \begin{array}{ccc}<br />
\frac{1}{7} &=& 0.\overline{142857} \\ \\[-3mm]<br />
\frac{2}{7} &=& 0.\overline{285714} \\ \\[-3mm]<br />
\frac{3}{7} &=& 0.\overline{428571} \\ \\[-3mm]<br />
\frac{4}{7} &=& 0.\overline{571428} \\ \\[-3mm]<br />
\frac{5}{7} &=& 0.\overline{714285} \\ \\[-3mm]<br />
\frac{6}{7} &=& 0.\overline{857142} \end{array}

    The last five decimals are cyclic arrangements of the first one.

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