Question

Let z be a complex number. If $z\overline{z}=0$ then:

• A. Re(z)=0
• B. Im(z)=0
• C. Re(z)+Im(z)=0
• D. z is not equal to 0

Answer 1

Answer: (A), (B), (C)

Re(z)=0
Im(z)=0
Re(z)+Im(z)=0

Let $z=r{e}^{i\theta }$
$\overline{z}=r{e}^{-i\theta }$
Hence
$z.\overline{z}$
$=\left(r{e}^{i\theta }\right)\left(r{e}^{-i\theta }\right)$
$=r^2{e}^0$
$=r^2$
$z.\overline{z}=0$
$r^2=0$
$Re(z{)}^2+Im(z{)}^2=0$
This can be true if
$Re\left(z\right)=Im\left(z\right)=0$
Or
$|z|=0$
Or
$z=0+0i$.