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Math Help - Intriguing question on Complex Numbers

  1. #1
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    Intriguing question on Complex Numbers

    I found a simple but interesting problem which can be solved using highschool concepts regarding complex numbers... (I posted it here since it's purely optional, those interested may try.)

    Consider three real numbers x,y,z none equal to zero.

    \alpha,\beta,\gamma are three complex numbers such that



    if x+y+z=0 , and ,

    Prove that :
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  2. #2
    Senior Member Dinkydoe's Avatar
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    Given \alpha,\beta,\gamma on the complex-unit ring in \mathbb{C}

    First we oberve that: x+y+z = 0 \Leftrightarrow x = -(y+z)

    Hence:
    \alpha x+\beta y +\gamma z = 0\Leftrightarrow
     -\alpha(y+z)+\beta y + \gamma z = 0 \Leftrightarrow
     \beta y + \gamma z = \alpha y + \alpha z \Leftrightarrow
    y+[\beta^{-1}\gamma]z = [\beta^{-1}\alpha](y+z)

    Since |\beta^{-1}\alpha| = 1 we obtain:

    |y+[\beta^{-1}\gamma]z|= |y+z| \Rightarrow \beta^{-1}\gamma = 1. Thus \beta = \gamma. And from this follows \alpha=\beta=\gamma.
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