Question

Find the Modulas and argument of $\frac{1+i}{1-i}$

Answer 1

Writing both numerator and denominator in Euler notation:
$1+i=\sqrt{2}(\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}})=\sqrt{2}{e}^{i\frac{\pi }{4}}$
Similarly, $1-i=\sqrt{2}{e}^{-i\frac{\pi }{4}}$
$\therefore \frac{1+i}{1-i}=e^{i\pi /2}$
Modulus = 1, Argument = $\pi /2$