Question

Which of the following is correct for any two complex number $z_1$ and $z_2$ ?

• A. $|z_1{z}_2|=|z_1||z_2|$
• B. $arg\left(z_1{z}_2\right)=arg\left(z_1\right)arg\left(z_2\right)$
• C. $|z_1+z_2|=|z_1|+|z_2|$
• D. $|z_1+z_2|\ge |z_1|+|z_2|$

Answer 1

The correct option is A

$|z_1{z}_2|=|z_1||z_2|$

$\text{Let}{z}_1=r_1(cos{\theta }_1+isin{\theta }_1)\Rightarrow |z_1|=r_1\text{and}{z}_2=r_2(cos{\theta }_2+isin{\theta }_2)\Rightarrow z_1{z}_2=r_1{r}_2[cos{\theta }_{1}cos{\theta }_2+isin{\theta }_{1}cos{\theta }_2+cos{\theta }_{1}sin{\theta }_2+i2sin{\theta }_{1}sin{\theta }_2]=r_1{r}_2\left[cos\right({\theta }_1+{\theta }_2)+isin({\theta }_1+{\theta }_2\left)\right]\Rightarrow |z_1{z}_2|=r_1{r}_2\therefore |z_1{z}_2|=|z_1||z_2|$

[using eqs. (i) and (ii)]