Complex Numbers
Argument Properties
Premium Question
Grade 11
Question:
If $\arg(z_1) = \arg(z_2)$ (both non-zero), then:
(1) $z_1z_2 > 0$
(2) $z_1 = kz_2$ for some $k > 0$
(3) $z_1 = z_2$
(4) $\operatorname{Re}(z_1z_2^*) > 0$
Step-by-Step Solution
Key Concept: Write $z_1 = r_1e^{i\theta}$ and $z_2 = r_2e^{i\theta}$ (same argument $\theta$). Then $z_1/z_2 = r_1/r_2 > 0$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)