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Thread: 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 $\displaystyle x,y,z$ none equal to zero.

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



    if $\displaystyle x+y+z=0$ , and ,

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

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

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

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

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