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Question
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
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Solution
\[\left( \frac{1 + i}{1 - i} \right)^n \]
\[ = \left( \frac{1 + i}{1 - i} \times \frac{1 + i}{1 + i} \right)^n \]
\[ = \left( \frac{1 + i^2 + 2i}{1 - i^2} \right)^n \]
\[ = \left( \frac{1 - 1 + 2i}{1 + 1} \right)^n \]
\[ \Rightarrow \left( \frac{2i}{2} \right)^n \]
\[ = i^n \]
\[\text { For } i^n \text { to be real, the smallest positive value of n will be 2 } . \]
\[\text { As }, i^2 = - 1, \text{ which is real} .\]
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