First slide

Algebra of complex numbers

Question

If $| ω | = 1$ , then the set of points $z = ω + \frac{1}{ω}$ is satisfies

Moderate

A

$| Re (z) | \leq 2$
B

$| z | \leq 1$
C

$| z | = 1$
D

$| Im (z) | \geq 2$

Solution

As $| ω | = 1 \Rightarrow | ω |^{2} = 1 \Rightarrow ω \bar{ω} = 1 \Rightarrow \frac{1}{ω} = \bar{ω}$
Thus, $z = ω + 1 / ω = ω + \bar{ω} = 2 Re (ω)$
Now, $| Re (z) | = | 2 Re (ω) | = | 2 Re (ω) | \leq 2 | ω | = 2$

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