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Thread: Triangle inequality for n complex numbers

  1. #1
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    Triangle inequality for n complex numbers

    I am trying to prove that
    $\displaystyle |z_1+z_2+...+z_n| = |z_1| + |z_2| + ... |z_n|$

    iff $\displaystyle z_i/z_j$ is a positive real number $\displaystyle \forall$ integers i and j, s.t. $\displaystyle i,j \in $ $\displaystyle \left\{ 1,...,n \right\}$

    I really don't see how these two ideas imply each other. After looking at this for several hours the only thing I managed to come up with (which I'm sure could be extended to n variables) is

    $\displaystyle |z_1|^2 + m|z_2|^2 = |z_1+z_2|^2$ where $\displaystyle m \in \mathbb{R}$

    however, that real number m is not always $\displaystyle z_i/z_j$

    Can anyone offer me advice please?
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  2. #2
    MHF Contributor chisigma's Avatar
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    If $\displaystyle \forall i,j \in {1,2,...,n \}$ is $\displaystyle \displaystyle \frac{z_{i}}{z_{j}} = \alpha_{i,j}$ and any $\displaystyle \alpha_{i,j}$ is positive real, then $\displaystyle \forall i$ is $\displaystyle z_{i}= e^{\sigma\ \theta}\ \beta_{i}$, being $\displaystyle \sigma= \sqrt{-1}$ , $\displaystyle \theta$ real and any $\displaystyle \beta_{i}$ positive real. In this case is...

    $\displaystyle \displaystyle |z_{1} + z_{2} + ...+ z_{n}|= |e^{\sigma \theta}|\ |\beta_{1} + \beta_{2} +...+ \beta_{n}|= |z_{1}|+|z_{2}|+...+|z_{n}|$ (1)

    The inverse however is not true and a symple 'counterexample' is when one of the $\displaystyle z_{j}$ is zero and the terms $\displaystyle \frac{z_{i}}{z_{j}}$ for $\displaystyle i \ne j$ don't exist...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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  3. #3
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    Sorry, I meant to specify that $\displaystyle z_j \neq 0$. Thank you very much for your help though. I will try to figure out the other direction on my own.

    -Cheers
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  4. #4
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    Use the real part and imaginary part of the complex numbers to re-write the inequality then apply the Cauchy–Schwarz inequality.
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