If ${\mathrm{z}}_1, {\mathrm{z}}_2$ and ${\mathrm{z}}_3, {\mathrm{z}}_4$ are 2 pairs of complex conjugate numbers and $\notin \mathrm{R},$then $\arg \left(\frac{{\mathrm{z}}_1}{{\mathrm{z}}_4}\right)+ \arg \left(\frac{{\mathrm{z}}_2}{\mathrm{z}3}\right)$ equals

Given ${\mathrm{z}}_1, {\mathrm{z}}_2$ are conjugate complex numbers, and ${\mathrm{z}}_3,{\mathrm{z}}_4$ are conjugate complex numbers

hence, $arg\left(z_1\right)=-arg\left(z_2\right)$ and $\begin{array}{rcl}arg\left(z_3\right)& =& -arg\left(z_4\right)\end{array}$

Therefore, 

$$\begin{gathered} \arg \left(\frac{{\mathrm{z}}_1}{{\mathrm{z}}_4}\right)+ \arg \left(\frac{{\mathrm{z}}_2}{\mathrm{z}3}\right)=\arg \left({\mathrm{z}}_1\right)-\arg \left({\mathrm{z}}_4\right)+\arg \left({\mathrm{z}}_2\right)-\arg \left({\mathrm{z}}_3\right) \\ =\left(\arg \left({\mathrm{z}}_1\right)+\arg \left({\mathrm{z}}_2\right)\right)-\left(\arg \left({\mathrm{z}}_3\right)+\arg \left({\mathrm{z}}_4\right)\right)=0 \end{gathered}$$=

or

$z_2={\overline{z}}_1\text{ and }{z}_4={\overline{z}}_3$

$z_1{z}_2=z_1{\overline{z}}_1={\left|z_1\right|}^2$

$\text{ and }{z}_3{z}_4=z_3{\overline{z}}_3={\left|z_3\right|}^2$

$\text{ Now arg }\left(\frac{z_1}{z_3}\right)+\arg \left(\frac{z_2}{z_4}\right)=\arg \left(\frac{z_1}{z_3}\times \frac{z_2}{z_4}\right)$

$=\arg \left(\frac{{\left|z_1\right|}^2}{{\left|z_3\right|}^2}\right)=0(\because \text{ Argument of a positive real number is }0)$