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Math Help - Complex numbers - two inequality questions

  1. #1
    Nrt
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    Complex numbers - two inequality questions

    1)

    Prove that

    ||z_1|-|z_2||\leq|z_1-z_2|

    2)

    Show that if |z|<1

    then

    |z-1|+|z+1|\leq2


    First one looks like the triangle inequality but i cant go further.
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    Hello, Nrt!

    I believe there is a typo in #2.


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

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

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

    . . \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: . \overline{zA} + \overline{zB} \:\geq \:\overline{AB}

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

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  4. #4
    Nrt
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    Thank you all for help.
    Last edited by Nrt; October 4th 2009 at 01:50 AM.
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  5. #5
    Nrt
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    Can someone explain me the first question using complex numbers? I got the idea from Plato's link but couldn't get it done.
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  6. #6
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    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 |z| is just a real number.

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

    So you can get  - \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 a\ge 0 and -a\le b \le a then |b|\le |a| or \left| {\left| z \right| - \left| w \right|} \right| \leqslant \left| {z - w} \right|.

    See you let a = \left| {z - w} \right| and b = \left| z \right| - \left| w \right| noting that \left| {\left| {z - w} \right|} \right| = \left| {z - w} \right|.
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  7. #7
    Nrt
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    Thanks mate. Now i can see it clearly.
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