Mathematics - Complex Number Question with Solution | TestHub

Consider $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$ $$=\arg \left(z_1\right)-\arg \left(z_4\right)+\arg \left(z_2\right)-\arg \left(z_3\right)$$ $$\begin{array}{rl}& =\left(\arg \left(z_1\right)+\arg \left(z_2\right)\right)-\left(\arg \left(z_3\right)+\arg \left(z_4\right)\right)\\ & \mathrm{given}\left(\begin{array}{l}{z}_2={\bar{z}}_1\\ z_4={\bar{z}}_3\end{array}\right)\\ & =\left(\arg \left(z_1\right)+\arg \left({\bar{z}}_1\right)\right)-\left(\arg \left(z_3\right)+\arg \left({\bar{z}}_3\right)\right)\\ & \left\{\begin{array}{l}\mathrm{also}(\arg \left({\bar{z}}_1\right)=-\arg \left(z_1\right)\\ \arg \left({\bar{z}}_3\right)=-\arg \left(z_3\right)\end{array}\right\}\\ & =\left(\arg \left(z_1\right)-\arg \left(z_1\right)\right)-\left(\arg \left(z_3\right)-\arg \left(z_3\right)\right)\\ & =0-0=0\end{array}$$