# Complex number equation and roots of unity

1. Oct 28, 2012

### math_nuub

I have some math problems

What is the solution to this equation :

z dash(complex conjugate) = z^3 Z is complex number

I try to multiply both sides by Z in the left i get Z dash Z => |Z| but i don't see the solution

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P is primitive 9th root of unity.
Calculate the sum 1 + 2P +3P^2 + ... + 9P^8

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Thx.

2. Oct 28, 2012

### haruspex

The easiest approach is using polar notation.
Alternatively, consider the magnitude of z. Notice anything?
And you already got |z|2 = z4, right? What interesting fact does that tell you about the complex number z4?

3. Oct 28, 2012

### math_nuub

I don't know what is the interesting fact behind Z^4?

4. Oct 28, 2012

### haruspex

It's a complex number, and yet it equals |z|2, so in fact ... ?

5. Oct 28, 2012

### SammyS

Staff Emeritus
Hello math_nuub. Welcome to PF !

According to the rules for Homework Help on this Forum, you need to show some effort before we can help.

What have you tried?

Where are you stuck?

6. Oct 29, 2012

### math_nuub

I am realy trying to understand but my test is approaching and i could.t wrap my head around this material

|Z|^2 is a Real number so you mean that |z|^4 is Real number also?

I am trying from two days to understand those rots of unity.

I don't understand whiht is the 9th primitive root of unity. I think there are several of them (1,2,4,5,7,8), but how could i sum them, There are in trigonometric form.

7. Oct 29, 2012

### SammyS

Staff Emeritus
Of course |z|4 is a real number.

It means that z4 is a real number.
What is the definition of a "primitive root of unity" ?