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Thread: Finite Continued Fraction Question

  1. December 7th 2010, 08:05 PM #1
    Janu42
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    Finite Continued Fraction Question

    Show that every rational number has exactly two finite simple continued fraction expansions.

    (Does this have something to do with how you handle the end of the continued fraction?)
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  2. December 11th 2010, 06:10 PM #2
    dwsmith
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    \displaystyle \frac{225}{157}

    This can be represented as <b_0;b_1,b_2,\cdots, b_k> \ \mbox{and} \ <b_0;b_1,b_2,\cdots, b_k-1,1>

    Therefore, our fraction can be represented like

    \displaystyle <1; 2,3,4,5>

    \displaystyle 1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{5}}}}

    or

    \displaystyle <1;2,3,4,4,1>

    \displaystyle 1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{4+\fra c{1}{1}}}}}}
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