How to divide a complex number by another complex number?

Question

Is this true to say that :

I don't find this definition in my book but it is the feeling that I had while looking at exercises correction.

Answer 1

Yes

Let $$z=a+bi$$

$$z_1=x+iy\implies \bar{z_1}=x-iy$$ $$\frac{z}{z_1}=\frac{a+ib}{x+iy}=\frac{(a+ib)(x-iy)}{(x+iy)(x-iy)}=\frac{z\bar{z_1}}{z_1\bar{z_1}}=\frac{(a+ib)(x-iy)}{x^2+y^2}=\frac{z\bar{z_1}}{|z_1|^2}$$

Answer 2

$\dfrac{Z}{Z'}=\dfrac{Z}{Z'}\cdot1=\dfrac{Z}{Z'}\cdot\left(\dfrac{\bar{Z'}}{\bar{Z'}}\right)=\dfrac{Z\bar{Z'}}{Z'\bar{Z'}}$