First slide

Algebra of complex numbers

Question

If $z \neq 0$ is s a complex number such that $\arg (z) = π / 4$ then

Moderate

A

$Re (z^{2}) = 0$
B

$Im (z^{2}) = 0$
C

$Re (z^{2}) = Im (z^{2})$
D

none of these

Solution

As $\arg (z) = π / 4$ we can write
$z = r (\cos \frac{π}{4} + i \sin \frac{π}{4})$ where $r = | z |$
$\Rightarrow z^{2} = r^{2} (\cos \frac{π}{2} + i \sin \frac{π}{2})$
[Using De Moivre’s Theorem]
$= r^{2} (0 + i) = i r^{2}$
$\Rightarrow Re (z^{2}) = 0$

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