# Complex numbers on unit circle

1. Aug 17, 2011

### treetheta

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. Aug 17, 2011

### micromass

Staff Emeritus
So you know

$$z_1+z_2+z_3+z_4=1+i$$

What if you take the conjugate of both sides?

3. Aug 17, 2011

### dynamicsolo

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 ?