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Homework Help: Complex no & conjugate

  1. Apr 21, 2006 #1
    I would like to prove [itex]\mid z + \overline{z} \mid \leq 2 \mid z \mid[/itex]

    The first way I could think of:
    [tex]
    \begin{multline}
    RHS^2 - LHS^2\\
    =4\mid z \mid^2 - \mid z + \overline{z} \mid ^ 2\\
    =4z\overline{z} - (z + \overline{z})(\overline{z}+z)\\
    =4z\overline{z} - z\overline{z}-z^2-\overline{z}^2-z\overline{z}\\
    =2z\overline{z}-z^2-\overline{z}^2\\
    =-(z^2-2z\overline{z}+\overline{z}^2)\\
    =-(z-\overline{z})^2\\
    \leq 0 ???
    \end{multline}
    [/tex]

    I now know the correct proof is as follow:
    [tex]
    \begin{multline}
    \mid z + \overline{z} \mid\\
    \leq \mid z \mid + \mid \overline{z} \mid\\
    = \mid z \mid + \mid z \mid\\
    = 2 \mid z \mid\\
    \end{multline}\\
    [/tex]

    But what is wrong with my original proof?
     
    Last edited: Apr 21, 2006
  2. jcsd
  3. Apr 21, 2006 #2

    matt grime

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    Homework Helper

    Nothing is wrong with it. what is z-z*? It's not a real number, is it.....
     
  4. Apr 21, 2006 #3
    okay got it..
    lets say b is Im(z).
    then b^2 is always positive, and (ib)^2 is always negative...

    you just got the signs wrong to start with.
     
    Last edited: Apr 21, 2006
  5. Apr 21, 2006 #4
    according to my proof, instead of [itex]\mid z + \overline{z} \mid \leq 2 \mid z \mid[/itex], it is [itex]\mid z + \overline{z} \mid \geq 2 \mid z \mid[/itex]...

    z is a complex no. and [itex]\overline{z}[/itex] is its conjugate. Have I mixed up some basic rules in complex no. with those in real no.?
     
  6. Apr 21, 2006 #5
    Sorry forgoth I haven't noticed your reply when i post mine... but I do not understand... do you mean that I cannot square a complex no?
     
  7. Apr 21, 2006 #6
    [tex]-(z-z^*)^2=-(a+ib-a+ib)^2=4b^2>0[/tex]
    your last line had the wrong signs... or i just missed something.
     
    Last edited: Apr 21, 2006
  8. Apr 21, 2006 #7

    matt grime

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    When ever you do something like this, always step back and think: what happens in a simple example. For instance why not put z=i in and see what happens?

    But you are confusing some rules of real numbers with complex ones. Sure, if you square a real number it is positive, but the whole raison d'etre of complex number is that you have things that square to negative numbers.
     
  9. Apr 21, 2006 #8
    I see... thx~
    Too much used to real no... have never thought that there would be a problem on that line
    (never thought of posting a question in a forum could get immediate response too~ ^^)
     
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