Question

Find the least value of $n\left(n\in N\right),$ for which ${\left(\frac{1+i}{1-i}\right)}^n$ is real.

Answer 1

Let $z=\left(\frac{1+i}{1-i}\right)$

On rationalizing,

$=\frac{(1+i)(1+i)}{1-i^2}$

$=\frac{1+i^2+2i}{2}=i$

$z^n=i^n$

$n=4,8,...$

Least value of $n=4$