Results 1 to 4 of 4

Math Help - Triangle inequality for n complex numbers

  1. #1
    Member
    Joined
    Feb 2011
    Posts
    83
    Thanks
    2

    Triangle inequality for n complex numbers

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

    iff z_i/z_j is a positive real number  \forall integers i and j, s.t. i,j \in  \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

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

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

    Can anyone offer me advice please?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    If \forall i,j \in {1,2,...,n \} is \displaystyle \frac{z_{i}}{z_{j}} = \alpha_{i,j} and any \alpha_{i,j} is positive real, then \forall i is z_{i}= e^{\sigma\ \theta}\ \beta_{i}, being \sigma= \sqrt{-1} , \theta real and any \beta_{i} positive real. In this case is...

    \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 z_{j} is zero and the terms \frac{z_{i}}{z_{j}} for i \ne j don't exist...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2011
    Posts
    83
    Thanks
    2
    Sorry, I meant to specify that 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
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23
    Use the real part and imaginary part of the complex numbers to re-write the inequality then apply the Cauchy–Schwarz inequality.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Inequality in Complex Numbers
    Posted in the Algebra Forum
    Replies: 0
    Last Post: August 31st 2010, 06:33 AM
  2. Complex numbers - two inequality questions
    Posted in the Algebra Forum
    Replies: 6
    Last Post: October 4th 2009, 04:52 AM
  3. complex numbers - Equilateral triangle
    Posted in the Math Topics Forum
    Replies: 3
    Last Post: March 28th 2009, 07:55 AM
  4. Complex numbers inequality
    Posted in the Algebra Forum
    Replies: 1
    Last Post: December 23rd 2008, 04:23 PM
  5. complex numbers inequality
    Posted in the Algebra Forum
    Replies: 1
    Last Post: May 25th 2008, 06:54 PM

Search Tags


/mathhelpforum @mathhelpforum