Question

Find the value of $arg(\frac{z_1}{z_3})+arg(\frac{z_2}{z_4})$

• A. 0
• B. $\pi$
• C. $\frac{\pi}{2}$
• D. $\frac{3\pi}{2}$

Answer 1

Answer: (A)

0

Answer 2

Let $z_2 = \overline{z_1}$ and $z_4 = \overline{z_3}$ since $(z_1, z_2)$ and $(z_3, z_4)$ are pairs of conjugate complex numbers. The expression to evaluate is $arg(\frac{z_1}{z_3})+arg(\frac{z_2}{z_4})$. Using the property $arg(A) + arg(B) = arg(A \cdot B)$, we get: $arg\left(\frac{z_1}{z_3}\right) + arg\left(\frac{z_2}{z_4}\right) = arg\left(\frac{z_1}{z_3} \cdot \frac{z_2}{z_4}\right)$ Substitute $z_2 = \overline{z_1}$ and $z_4 = \overline{z_3}$: $= arg\left(\frac{z_1}{z_3} \cdot \frac{\overline{z_1}}{\overline{z_3}}\right)$ $= arg\left(\frac{z_1 \cdot \overline{z_1}}{z_3 \cdot \overline{z_3}}\right)$ Using the property $z \cdot \overline{z} = |z|^2$: $= arg\left(\frac{|z_1|^2}{|z_3|^2}\right)$ Since $z_1$ and $z_3$ are non-zero complex numbers, $|z_1|^2$ and $|z_3|^2$ are positive real numbers. Therefore, $\frac{|z_1|^2}{|z_3|^2}$ is a positive real number. The argument of any positive real number is 0.