First slide

Algebra of complex numbers

Question

The equation $z^{3} = \bar{z}$ has

Moderate

A

no solution
B

two solutions
C

five solutions
D

infinite number of solutions

Solution

$\begin{array}{l} z^{3} = \bar{z} \Rightarrow |z^{3}| = | \bar{z} | \Rightarrow |z^{3}| = | \bar{z} | \\ \Rightarrow | z | = 0 or | z | = 1 \Rightarrow z = 0 or \bar{z} = 1 / z . \\ ∴ z^{3} = \bar{z} \Rightarrow z^{3} = \frac{1}{z} \Rightarrow z^{4} = 1 \\ \Rightarrow z = 1, - 1, i, - i . \end{array}$
Thus, solutions of $z^{3} = z$ are $0, 1, - 1, i, - i$

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