If z_1,z_2 and z_3,z_4 are 2 pairs of complex conjugate numbers, then z_1z_4+z_2z_3 equals…

Question

If $z_1,z_2$ and $z_3,z_4$ are 2 pairs of complex conjugate numbers, then $\arg\dfrac{z_1}{z_4}+\arg\dfrac{z_2}{z_3}$ equals:

Accepted Answer

Since $z_2=\bar z_1,\ z_4=\bar z_3$, and $\arg(\bar w)=-\arg(w)$:

$$\arg\frac{z_1}{z_4}+\arg\frac{z_2}{z_3}=[\arg z_1-\arg z_4]+[\arg z_2-\arg z_3]=0.$$

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