I have a number of questions about complex numbers and I need your help:

  1. $z_1, z_2, z_3, z_4, z_5$ are complex numbers that fulfil |z1|=|z2|=|z3|=|z4|=|z5|=1

    prove that $|z_1+z_1+z_3+z_4+z_5| = |{1\over z1} + {1\over z2} + {1\over z3} + {1\over z4} + {1\over z5}| $

  2. find all the solutions for $(2i)^9z^3=(1+i)^{17}$

  • $\begingroup$ can you fix your $\LaTeX$ please? $\endgroup$ Oct 25 '14 at 19:06
  • $\begingroup$ im trying to put (1+i)^17 on the right side but its not working for me $\endgroup$
    – Xhero39
    Oct 25 '14 at 19:10
  • $\begingroup$ Enclose the 17 between curly parentheses {}, @FirasAliAbdelGhani $\endgroup$
    – Timbuc
    Oct 25 '14 at 19:20

Check that for $\;z\in\Bbb C\;$ :

$$|z|=1\iff \overline z=\frac1z\;$$

so that


and now just remember that $\;\overline{a+b}=\overline a+\overline b\;$ , $\;|z|=|\overline z|\;$and etc.

For two, take into account that


and etc.


To solve 1., use the fact that for every $z$ such that $|z|=1$, $$\frac1z=\overline{z},$$ hence $$\left|\sum_k\frac1{z_k}\right|=\left|\sum_k\overline{z_k}\right|=\left|\overline{\sum_kz_k}\right|=\left|\sum_kz_k\right|.$$ Part 2. is unrelated and should be asked separately.


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