First slide

Algebra of complex numbers

Question

Suppose $z$ is a complex number and $n \in N$ be such that $z^{n} = (1 + z)^{n} = 1$ , then the least value of $n$ is

Moderate

A

3
B

6
C

9
D

18

Solution

$| z |^{n} = | 1 + z |^{n} = 1 \Rightarrow | z | = | z + 1 | = 1$
$\Rightarrow | z | = 1$ and $z$ lies on the perpendicular bisector of the segment joining $0 + 0 i$ and $- 1 + 0 i,$ that $z$ lies on the line $Re (z) = - 1 / 2$
Let $z = - \frac{1}{2} + i y,$ then $| z | = 1$
$\frac{1}{4} + y^{2} = 1 \Rightarrow y = \pm \frac{\sqrt{3}}{2}$
$∴ z = - \frac{1}{2} \pm i \frac{\sqrt{3}}{2} = ω, ω^{2}$
where $ϖ$ is complex cube root of unit
If $z = ω, ω^{n} = (1 + ω)^{n} = (- 1)^{n} ω^{2 n}$
$\Rightarrow n$ is even and multiple of 3. Thus, least value of n is 6. Similarly, for $z = ϖ^{2}$

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