Question

$ \left(\frac{1+i}{1-i}\right)^{n}$ is real

• A. for every real number $n$
• B. for every odd integer $n$
• C. for every rational $n$
• D. for every even positive integer $n$

Answer 1

Answer: (D)

$\left(\frac{1+i}{1-i}\right)^{n}=\left[\frac{(1+i)^{2}}{1^{2}+1^{2}}\right]^{n} $
$=\left[\frac{1-1+2 i}{2}\right]^{n}$
$=[i]^{n}$
For $(i)^{n}$ is real, $n$ should be even integer.