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Math Help - conjugate

  1. #1
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    conjugate

    Given that |w|=|z|=|v|=1 . Show that \frac{(w+z)(z+v)(v+w)}{wzv} is a real number .

    I know that when we need to show that a complex number is real , we need to show that s=s*(its conjugate)

    My question is how to find s* ??
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  2. #2
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    Quote Originally Posted by hooke View Post
    Given that |w|=|z|=|v|=1 . Show that \frac{(w+z)(z+v)(v+w)}{wzv} is a real number .

    I know that when we need to show that a complex number is real , we need to show that s=s*(its conjugate)

    My question is how to find s* ??

    Pay attention to the fact that z\overline{z}=|z|^2=1\Longrightarrow \frac{1}{z}=z^{-1}=\overline{z} , so:

    \overline{\left(\frac{(w+z)(z+v)(v+w)}{wzv}\right)  }=\frac{(\overline{w}+\overline{z})(\overline{z}+\  overline{v})(\overline{v}+\overline{w})}{\overline  {wzv}} =\frac{(w^{-1}+z^{-1})(z^{-1}+v^{-1})(v^{-1}+w^{-1})}{w^{-1}z^{-1}v^{-1}}=\left(\frac{w+z}{wz}\cdot\frac{z+v}{zv}\cdot\f  rac{v+w}{vw}\right) wzv

    =\frac{(w+z)(z+v)(v+w)}{wzv} and we're done

    Tonio
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