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Math Help - Complex number equation

  1. #1
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    Complex number equation

    The question:
    Show that |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2, for all complex numbers z_1 and z_2.

    I tried getting the LHS to match the RHS, but I've done something wrong. My attempt involved factorising using the difference of two squares, but it doesn't seem to help. Any assistance would be great.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by Glitch View Post
    The question:
    Show that |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2, for all complex numbers z_1 and z_2.

    I tried getting the LHS to match the RHS, but I've done something wrong. My attempt involved factorising using the difference of two squares, but it doesn't seem to help. Any assistance would be great.
    The first page of http://www.math.ca/crux/v31/n8/public_page502-503.pdf contains the expansion of |z_1+z_2|^2. Similarly expand |z_1-z_2|^2 and add them.
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  3. #3
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    Quote Originally Posted by Glitch View Post
    The question:
    Show that |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2, for all complex numbers z_1 and z_2.

    I tried getting the LHS to match the RHS, but I've done something wrong. My attempt involved factorising using the difference of two squares, but it doesn't seem to help. Any assistance would be great.
    Write \displaystyle z_1 and \displaystyle z_2 in terms of their real and imaginary parts and simplify...
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  4. #4
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    Ahh, I forgot about that relation. Thank you.
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  5. #5
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    Quote Originally Posted by Glitch View Post
    The question:
    Show that |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2, for all complex numbers z_1 and z_2.
    Here is a different approach.
    Recall |z|^2=z\cdot\overline{z} and \overline{z\pm w}=\overline{z}\pm\overline{w}.
    So |z+w|^2=(z+w)\cdot\overline{(z+w)}=(z+w)\cdot(\ove  rline{z}+\overline{w}).
    =z\cdot\overline{z}+z\cdot\overline{w}+w\cdot\over  line{z}+w\cdot\overline{w}
    =|z|^2+z\cdot\overline{w}+w\cdot\overline{z}+|w|^2

    Do that for |z-w|^2 and add.
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  6. #6
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    Quote Originally Posted by Plato View Post
    Here is a different approach.
    That's what alexmahone suggested. I ended up solving it that way.
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