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Math Help - Golden Ratio

  1. #1
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    Golden Ratio

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

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

    I am confused on what do for this problem.
    Last edited by dwsmith; June 11th 2011 at 06:56 PM.
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  2. #2
    Super Member Random Variable's Avatar
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     \frac{b}{a} = \varphi

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

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

    Now just solve solve for  \varphi and ignore the negative root.
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  3. #3
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    Quote Originally Posted by Random Variable View Post
     \frac{b}{a} = \varphi

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

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

    Now just solve solve for  \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  \frac{b}{a} = \varphi and \frac{a+b}{b} = \varphi

    Were interested in the ratio  \varphi . But notice that  \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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     \phi=\frac{1}{\phi}+1

    Or:


     \phi^2=\phi+1

    Or:


     \phi^2-\phi-1=0

    THE ROOTS ARE:

     \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  \frac{b}{a} = \varphi and \frac{a+b}{b} = \varphi

    Were interested in the ratio  \varphi . But notice that  \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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