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Thread: Simple intro to complex variables question

  1. #1
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    Simple intro to complex variables question

    I have to show that
    $\displaystyle |(2\overline{z}+5)(\sqrt{2}-i)|=\sqrt{3}|2z+5|$

    So far, I have
    $\displaystyle |(2\overline{z}+5)(\sqrt{2}-i)|=|2\overline{z}+5||\sqrt{2}-i)|$

    My sol'n book then says this is equal to
    $\displaystyle |\overline{2z+5}|\sqrt{2+1}$

    I understand that for the second term they are just finding the length, but I don't understand how
    $\displaystyle |2\overline{z}+5|=|\overline{2z+5}|$.

    Can anyone explain the math behind that?
    Last edited by paupsers; Jan 12th 2010 at 06:53 AM. Reason: PS: Also, is this the right forum to post questions about complex variables?
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  2. #2
    Senior Member Sampras's Avatar
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    Quote Originally Posted by paupsers View Post
    I have to show that
    $\displaystyle |(2\overline{z}+5)(\sqrt{2}-i)|=\sqrt{3}|2z+5|$

    So far, I have
    $\displaystyle |(2\overline{z}+5)(\sqrt{2}-i)|=|2\overline{z}+5||\sqrt{2}-i)|$

    My sol'n book then says this is equal to
    $\displaystyle |\overline{2z+5}|\sqrt{2+1}$

    I understand that for the second term they are just finding the length, but I don't understand how
    $\displaystyle |2\overline{z}+5|=|\overline{2z+5}|$.

    Can anyone explain the math behind that?
    Suppose $\displaystyle z = a+bi $. Then $\displaystyle \overline{z} = a-bi $. So $\displaystyle \overline{2} = 2 $ and $\displaystyle \overline{5} = 5 $.
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  3. #3
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    Here is a bit more.
    Notice that $\displaystyle \sqrt {\left( x \right)^2 + \left( y \right)^2 } = \sqrt {\left( x \right)^2 + \left( { - y} \right)^2 } $. So that means that $\displaystyle \left| z \right| = \left| {\overline z } \right|$.
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