Question

The complex number z is purely imaginary if

• A. $z\tilde{z}$ is real
• B. $z=\tilde{z}$
• C. $z+\tilde{z}=0$
• D. None of these

Answer 1

Answer: (A), (C)

$z\tilde{z}$ is real
$z+\tilde{z}=0$

Let $z$ be equal to $x+iy.$

A. $z\overline{z}=(x+iy)(x-iy)=x^2+y^2$, which is always real.

B. $x+iy=x-iy$, implies $y=0$

Hence, the complex number is real.

C. $x+iy+x-iy=0$ implies $x=0.$

Hence, the complex number is purely imaginary.