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Math Help - Help with complex numbers

  1. #1
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    Help with complex numbers

    Show that <br />
\left| {a + b} \right|^2  + \left| {a - b} \right|^2  = 2\left( {\left| a \right|^2  + \left| b \right|^2 } \right)<br />

    where a and b are complex numbers. Interpret this results geometrically.
    <br />
\begin{array}{l}<br />
 \left| {a + b} \right|^2  + \left| {a - b} \right|^2  \\ <br />
  = \left| a \right|^2  + 2\left| a \right|\left| b \right| + \left| b \right|^2  + \left| a \right|^2  - 2\left| a \right|\left| b \right| + \left| b \right|^2  \\ <br />
  = 2\left| a \right|^2  + 2\left| b \right|^2  \\ <br />
  = RHS \\ <br />
 \end{array}<br />

    I don't know how to interpret this result geometrically though, any help would be appreciated.
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  2. #2
    Super Member flyingsquirrel's Avatar
    Joined
    Apr 2008
    Posts
    802
    Hello,
    Quote Originally Posted by free_to_fly View Post
    <br />
\begin{array}{l}<br />
 \left| {a + b} \right|^2  + \left| {a - b} \right|^2  \\ <br />
  = \left| a \right|^2  + 2\left| a \right|\left| b \right| + \left| b \right|^2  + \left| a \right|^2  - 2\left| a \right|\left| b \right| + \left| b \right|^2  \\ <br />
 \end{array}<br />
    You're saying that |a+b|^2=\left| a \right|^2  + 2\left| a \right|\left| b \right|+|b|^2 which is wrong : if b=-a the LHS equals 0 and the RHS equals 4|b|^2. I don't think there is a rule that tells us how to expand |a+b|^2.

    However, I'm sure you know that if z is a complex number then |z|^2=z\overline{z} so

    \begin{aligned}<br />
\left| {a + b} \right|^2  + \left| {a - b} \right|^2&=(a + b)\overline{(a + b)}  +  (a - b)\overline{(a-b)}\\<br />
&=\ldots<br />
\end{aligned}

    I don't know how to interpret this result geometrically though, any help would be appreciated.
    Take a look at equation #10 of Parallelogram -- from Wolfram MathWorld. It looks like the one you're asked to show, doesn't it ?
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