Question

Let two pairs of non-zero conjugate complex numbers $(z_1,z_2)$
and $(z_3,z_4)$ then prove value of

$arg\left(\frac{z_1}{z_4}\right)+arg\left(\frac{z_2}{z_3}\right)$ is 0.

$arg\left(z_1\right)-arg\left(z_4\right)+arg\left(z_2\right)-arg\left(z_3\right)=0$

Answer 1

$arg\left(\frac{z_1}{z_4}\right)+arg\left(\frac{z_2}{z_3}\right)$

$z_2=\overline{z_1},z_4=\overline{z_3}$

$=arg\left(\frac{z_1}{z_4}\right)\left(\frac{z_2}{z_3}\right)$

$=arg\left(\frac{z_1}{\overline{z_3}}\right)\left(\frac{\overline{z_1}}{z_3}\right)$

$=arg\left(\frac{z_1\overline{z_1}}{z_3\overline{z_3}}\right)$

$=arg\left(\frac{|z_1{|}^2}{|z_3{|}^2}\right)$ $\to$ purely real

$=0$

----------------------------------

$arg\left(z_1\right)-arg\left(z_4\right)+arg\left(z_2\right)-arg\left(z_3\right)$

$=\left[arg\right(z_1)+arg(z_2\left)\right]-\left[arg\right(z_3)+arg(z_4\left)\right]$

$=\pi -\pi$ $\to$ conjugate complex numbers

$=0$