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Thread: Golden Ratio

  1. #1
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    Golden Ratio

    The Golden ratio is the ratio $\displaystyle b:a \ (b>a)$ so that $\displaystyle b:a \ =(a+b):b$ or $\displaystyle \frac{b}{a}=\frac{a+b}{b}$.

    Show that if $\displaystyle b:a$ is the Golden ratio, then $\displaystyle \phi=b/a=\frac{1+\sqrt{5}}{2}$

    I am confused on what do for this problem.
    Last edited by dwsmith; Jun 11th 2011 at 05:56 PM.
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    Super Member Random Variable's Avatar
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    $\displaystyle \frac{b}{a} = \varphi $

    $\displaystyle \frac{a+b}{b} = \frac{a}{b} + 1 = \frac{1}{\varphi} + 1 $

    so $\displaystyle \varphi = \frac{1}{\varphi} + 1 $

    Now just solve solve for $\displaystyle \varphi $ and ignore the negative root.
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    Quote Originally Posted by Random Variable View Post
    $\displaystyle \frac{b}{a} = \varphi $

    $\displaystyle \frac{a+b}{b} = \frac{a}{b} + 1 = \frac{1}{\varphi} + 1 $

    so $\displaystyle \varphi = \frac{1}{\varphi} + 1 $

    Now just solve solve for $\displaystyle \varphi $ and ignore the negative root.
    I don't understand why you even did that. With this question, I have no idea what they were even asking.
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  4. #4
    Super Member Random Variable's Avatar
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    Two numbers are in the golden ratio if $\displaystyle \frac{b}{a} = \varphi $ and $\displaystyle \frac{a+b}{b} = \varphi $

    Were interested in the ratio $\displaystyle \varphi $. But notice that $\displaystyle \varphi = \frac{a+b}{b} = \frac{a}{b} + 1 = \frac{1}{\varphi} + 1 $
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    $\displaystyle \phi=\frac{1}{\phi}+1$

    Or:


    $\displaystyle \phi^2=\phi+1$

    Or:


    $\displaystyle \phi^2-\phi-1=0$

    THE ROOTS ARE:

    $\displaystyle \phi_{1,2}=\frac{1\pm \sqrt{1+4}}{2}$
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  6. #6
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    Quote Originally Posted by Random Variable View Post
    Two numbers are in the golden ratio if $\displaystyle \frac{b}{a} = \varphi $ and $\displaystyle \frac{a+b}{b} = \varphi $

    Were interested in the ratio $\displaystyle \varphi $. But notice that $\displaystyle \varphi = \frac{a+b}{b} = \frac{a}{b} + 1 = \frac{1}{\varphi} + 1 $
    Why isn't the negative value considered in the question?
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  7. #7
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by dwsmith View Post
    Why isn't the negative value considered in the question?
    Read about this ratio here:

    Golden ratio - Wikipedia, the free encyclopedia
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