If z_1, z_2 and z_3, z_4 are 2 pairs of complex conjugate numbers, then arg ( z_1 z_4 )+arg ( z_2 z_3 ) equals:

Consider $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$arg(z4​z1​​)+arg(z3​z2​​) $$=\arg \left(z_1\right)-\arg \left(z_4\right)+\arg \left(z_2\right)-\arg \left(z_3\right)$$=arg(z1​)−arg(z4​)+arg(z2​)−arg(z3​) $$\begin{aligned} &=\left(\arg \left(z_1\right)+\arg \left(z_2\right)\right)-\left(\arg \left(z_3\right)+\arg \left(z_4\right)\right) \\ &\quad \operatorname{given}\left(\begin{array}{l} z_2=\bar{z}_1 \ \\ z_4=\bar{z}_3 \end{array}\right) \\ &=\left(\arg \left(z_1\right)+\arg \left(\bar{z}_1\right)\right)-\left(\arg \left(z_3\right)+\arg \left(\bar{z}_3\right)\right) \\ &\qquad\left\{\begin{array}{l} \operatorname{also}\left(\arg \left(\bar{z}_1\right)=-\arg \left(z_1\right)\right. \\ \arg \left(\bar{z}_3\right)=-\arg \left(z_3\right) \end{array}\right\} \\ &=\left(\arg \left(z_1\right)-\arg \left(z_1\right)\right)-\left(\arg \left(z_3\right)-\arg \left(z_3\right)\right) \\ &=0-0=0 \end{aligned}$$​=(arg(z1​)+arg(z2​))−(arg(z3​)+arg(z4​))given(z2​=zˉ1​ z4​=zˉ3​​)=(arg(z1​)+arg(zˉ1​))−(arg(z3​)+arg(zˉ3​)){also(arg(zˉ1​)=−arg(z1​)arg(zˉ3​)=−arg(z3​)​}=(arg(z1​)−arg(z1​))−(arg(z3​)−arg(z3​))=0−0=0​