# The conjugate of the complex number

Question:

The conjugate of the complex number $\frac{1-i}{1+i}$ is

Solution:

$\frac{(\overline{1-i})}{(\overline{1+i})}$ i. e conjugate of $\frac{1-i}{1+i}$

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

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

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

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

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

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