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Math Help - Euclid's algorithm

  1. #1
    mx-
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    Question Euclid's algorithm

    By adapting Euclid's algorithm, and integers a; b lying between 1 and 1000
    such that their quotient a/b agrees with the constant pi up
    to 6 decimal places (3.14159265)

    Any help with be much appreciated!
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  2. #2
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    Hello, mx-!

    I don't see how to "adapt Euclid algorithm",
    . . but there is a continued fraction approach.


    By adapting Euclid's algorithm, find integers a, b between 1 and 1000 such that
    their quotient \frac{a}{b} agrees with the constant \pi up to 6 decimal places (3.14159265)

    . . \pi \;=\;3.14159262654 \;= \;3 + 0.141592654

    . . . . = \;3 + \frac{1}{7.062513306} \;=\;3 + \frac{1}{7 + 0.62513306}

    . . . . = \;3 + \frac{1}{7 + \dfrac{1}{15.99659441}} \;\approx\;3 + \frac{1}{7 + \frac{1}{16}}


    Therefore: . \pi \;\approx\:3 + \frac{1}{7 + \frac{1}{16}} \;=\; 3 + \frac{1}{\frac{113}{16}} \;=\;3 + \frac{16}{113} \;=\;\frac{355}{113}

    . . \begin{array}{ccc}\dfrac{355}{113} &=& {\color{blue}3.141592}92... \\ \\[-3mm]<br />
\pi &=& {\color{blue}3.141592}65...\end{array}


    This is the only solution to this problem.

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