# Write the Least Positive Integral Value of N for Which ( 1 + I 1 − I ) N is Real. - Mathematics

Write the least positive integral value of n for which  $\left( \frac{1 + i}{1 - i} \right)^n$ is real.

#### 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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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Q 10 | Page 62