First slide

Algebra of complex numbers

Question

The number of complex numbers satisfying $\bar{z} = i z^{2}$ is

Moderate

A

1
B

2
C

3
D

4

Solution

$| \bar{z} | = |i z^{2}| \Rightarrow | z | = | z |^{2}$
$\Rightarrow | z | = 0 or | z | = 1 .$
If $| z | = 0,$ then $z = 0$
If $| z | = 1,$ we get $\bar{z} = 1 / z$ so that the equation becomes
$1 / z = i z^{2}$
or $z^{3} = - i = i^{3} \Rightarrow z = i, i ω, i ω^{2}$
where $ω (\neq 1)$ ) is a cube root of unity

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