Question

The smallest integer for which $\left( \frac{1 + i}{1 - i} \right)^{n} = 1$ is

• A. n = 8
• B. n = 12
• C. n = 16
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Sol. $\left( \frac{1 + i}{1 - i} \right)^{n} = 1$

⇒ $\left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right)^{n} = \left( \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} \right)^{n}$

⇒$\left( e ^ { i \pi / 4 } \right) ^ { n } = \left( e ^ { - i \pi / 4 } \right) ^ { n }$ $e^{in\pi/4} = e^{- in\pi/4}$

⇒ $e^{in\pi/2}$ = 1 ⇒ n = 4k

where k is integer > 0