Question

The argument of the complex number $\frac{1+i}{1-i}$ where $i=\sqrt{-1}$, is

• A. $-\pi$
• B. $-\frac{\pi }{2}$
• C. $\frac{\pi }{2}$
• D. $\pi$

Answer 1

The correct option is C $\frac{\pi }{2}$

$Z==\frac{1+i}{1-i}$
$\Rightarrow Z=\frac{1+i}{1-i}\times \frac{1+i}{1+i}$
$Z=\frac{(1+i{)}^2}{1^2-i^2}=\frac{1+2i+i^2}{2}$
$Z=i$
$Z=e^{i\pi /2}\approx r{e}^{i\theta }$
$\Rightarrow \theta =\frac{\pi }{2}$
Hence argument is
$\theta =\frac{\pi }{2}$