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)

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