Results 1 to 7 of 7

Thread: Complex numbers - two inequality questions

  1. #1
    Nrt
    Nrt is offline
    Newbie Nrt's Avatar
    Joined
    Oct 2009
    From
    Istanbul
    Posts
    21

    Complex numbers - two inequality questions

    1)

    Prove that

    $\displaystyle ||z_1|-|z_2||\leq|z_1-z_2|$

    2)

    Show that if $\displaystyle |z|<1$

    then

    $\displaystyle |z-1|+|z+1|\leq2$


    First one looks like the triangle inequality but i cant go further.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,776
    Thanks
    2823
    Awards
    1
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    848
    Hello, Nrt!

    I believe there is a typo in #2.


    2) Show that if $\displaystyle |z|\:<\: 1$

    then: .$\displaystyle |z-1|+|z+1| \;\;{\color{red}\geq}\;\;2$

    Since $\displaystyle |z| < 1,\:z$ is in the unit circle centered at the origin.

    . . $\displaystyle \begin{array}{c}|z-1|\text{ is its distance from }A(1,0) \\ \\[-4mm] |z+1|\text{ is its distance from }B(\text{-}1,0) \end{array}\bigg\}\quad\text{ endpoints of a diameter, length 2}$


    Code:
                    |
                  o o o
              o     | z   o
            o       | ∆     o
           o       *|   *    o
                *   |     *
          o  *      |       * o
       - B∆ - - - - + - - - - ∆A -
          o         |         o
                    |
           o        |        o
            o       |       o
              o     |     o
                  o o o
                    |

    From the triangle inequality: .$\displaystyle \overline{zA} + \overline{zB} \:\geq \:\overline{AB}$

    . . Therefore: .$\displaystyle |z-1| + |z+1| \:\geq \:2$

    Follow Math Help Forum on Facebook and Google+

  4. #4
    Nrt
    Nrt is offline
    Newbie Nrt's Avatar
    Joined
    Oct 2009
    From
    Istanbul
    Posts
    21
    Thank you all for help.
    Last edited by Nrt; Oct 4th 2009 at 01:50 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Nrt
    Nrt is offline
    Newbie Nrt's Avatar
    Joined
    Oct 2009
    From
    Istanbul
    Posts
    21
    Can someone explain me the first question using complex numbers? I got the idea from Plato's link but couldn't get it done.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,776
    Thanks
    2823
    Awards
    1
    Quote Originally Posted by Nrt View Post
    Can someone explain me the first question using complex numbers? I got the idea from Plato's link but couldn't get it done.
    It is the exact same proof for complex numbers as for real real.
    You must realize that $\displaystyle |z|$ is just a real number.

    The triangle inequality holds for complex numbers: $\displaystyle \left| {z + w} \right| \leqslant \left| z \right| + \left| w \right|\;\& \,\left| {z - w} \right| = \left| {w - z} \right|$.

    So you can get $\displaystyle - \left| {z - w} \right| \leqslant \left( {\left| z \right| - \left| w \right|} \right) \leqslant \left| {z - w} \right|$.

    At this point you have nothing but real numbers and you use this fact.

    If $\displaystyle a\ge 0$ and $\displaystyle -a\le b \le a$ then $\displaystyle |b|\le |a|$ or $\displaystyle \left| {\left| z \right| - \left| w \right|} \right| \leqslant \left| {z - w} \right|$.

    See you let $\displaystyle a = \left| {z - w} \right|$ and $\displaystyle b = \left| z \right| - \left| w \right|$ noting that $\displaystyle \left| {\left| {z - w} \right|} \right| = \left| {z - w} \right|$.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Nrt
    Nrt is offline
    Newbie Nrt's Avatar
    Joined
    Oct 2009
    From
    Istanbul
    Posts
    21
    Thanks mate. Now i can see it clearly.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Triangle inequality for n complex numbers
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Feb 10th 2011, 08:02 PM
  2. Inequality in Complex Numbers
    Posted in the Algebra Forum
    Replies: 0
    Last Post: Aug 31st 2010, 06:33 AM
  3. Complex numbers questions
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Aug 31st 2010, 06:09 AM
  4. Complex numbers inequality
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Dec 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