Question

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

Answer 1

First we solve $\frac{1+i}{1-i}$

Let $z=\frac{1+i}{1-i}$

Rationalizing the same

$=\frac{1+i}{1-i}\times \frac{1+i}{1+i}$

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

Using $(a+b)(a-b)=a^2-b^2$

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

Using $(a+b{)}^2=a^2+2ab+b^2$

$=\frac{(1{)}^2+(i{)}^2+\mathrm{2.1.}i}{(1{)}^2-i^2}$

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