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Complex numbers on unit circle

  1. Aug 17, 2011 #1
    1. The problem statement, all variables and given/known data
    Let z1; z2; z3; z4 be four complex numbers on the unit circle (i.e., |z1|=|z2|=|z3|=|z4|=1).
    It is known that z1+z2+z3+z4=1+i . Find the value of
    1/z1
    +
    1/z2
    +
    1/z3
    +
    1/z4




    2. Relevant equations

    1/z = barZ/|z|^2

    3. The attempt at a solution

    I've been trying for about a day now and i just have no clue absolutely, I understand that
    [b -a]/ [ab] = [1/a] - [1/b]

    but i think there's something to do with conjugates or something this question is quite fustrating.

    i also tried letting z1 + z2 + z3 + z4 = w
    and then w*(barW) but i just got 0(1+i)(1-i) and then i cant take the inverse
     
    Last edited: Aug 17, 2011
  2. jcsd
  3. Aug 17, 2011 #2

    micromass

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    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    So you know

    [tex]z_1+z_2+z_3+z_4=1+i[/tex]

    What if you take the conjugate of both sides?
     
  4. Aug 17, 2011 #3

    dynamicsolo

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    Homework Helper

    Have you had DeMoivre's Theorem? You know that all of the z's have unit modulus. What would 1/z equal for each z ?
    Draw the z's and (1/z)'s on an Argand diagram. How does the sum of the (1/z)'s relate to the sum of the z's ?
     
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