Complex number equation - absolute value assumption

[pinsky]

I have a problem in understanding the procedures of a solved example. It goes like this.

$$\left ( \frac{z+i}{z-i} \right )^4 = -1$$

Therefor we can write:

$$\left | \frac{z+i}{z-i} \right | = 1$$

From that we can see that z is a real number because:

$$\left | z+i \right | = \left | z-i \right | \; \; \; , z=x+yi \; \; \Rightarrow y=0 \; \; z=x$$

So let's say that:

$$W = \frac{x+i}{x-i}=\frac{x^2-1}{x+1}+\frac{2x}{x+1}i$$

$$W^4 = -1$$

And from that they get the solutions by doing some additional steps which i understand. What i can't seem to understand is the transformation written in red. For I'm having a homework to solve:

$$\left ( \frac{z+1}{z-1} \right )^3 = -1$$

and I don't know if i can apply the same assumption as in the upper example cause i don't understand what happened.


Any help is appreciated.

 

[willem2]

because |ab| = |a||b| it's easy to prove that |a^4| = |a|^4

now take the norm of both sides of the first equation and use this

 

[pinsky]

Thank you


[SOLVED]