Question

Find the modulus of the complex number $\frac{i}{1-i}$.

Answer 1

Given the complex number is

$\frac{i}{1-i}$

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

$=\frac{-1+i}{2}$.

Now the modulus of the complex number is

$\sqrt{{\left(\frac{-1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2}$

$=\sqrt{\frac{1}{2}}$

$=\frac{1}{\sqrt{2}}$.