complex numbers

Question

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 $(2i)^9z^3=(1+i)^{17}$

Answer 1

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

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

so that

$$\left|\;\frac1{z_1}+\ldots+\frac1{z_5}\;\right|=\left|\;\overline{z_1}+\ldots+\overline{z_5}\;\right|$$

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

For two, take into account that

$$(1+i)^2=2i\;,\;\;(1+i)^{16}=\left((1+i)^2\right)^8=(2i)^8=256$$

and etc.

Answer 2

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.