Question

Find all complex numbers $z$ which satisfy the following equation

$z^2+{\left|z\right|}^2=0.$

• A. $0$
• B. $0,i,1,-i,-1$
• C. $0,i,-i$
• D. $i,-i,-1,1$

Answer 1

Answer: (C)

$0,i,-i$

Let

$z=r{e}^{i\theta }$

Hence

$z^2+|z{|}^2=0$

$r^2(e^{i2\theta }+1)=0$ $r=0$ gives $z=0$

When

$|r|\ne 0$

$e^{i2\theta }=-1$

$e^{i2\theta }=e^{i(2n+1)\pi }$

$\theta =\frac{(2n+1)\pi }{2}$ ....$(n=0,1,2,3,\mathrm{5..})$

$\theta =\frac{\pi }{2},\frac{3\pi }{2},\frac{3\pi }{2}$.

Hence

Considering $|z|=1$ and by substituting the values of $\theta$

$z=0,i,-i$.