Question

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

Answer 1

$$\begin{gathered} \frac{\left(\overline{1-i}\right)}{\left(\overline{1+i}\right)}\mathrm{i}.\mathrm{e}\mathrm{conjugate}\mathrm{of}\frac{1-i}{1+i} \\ =\frac{\left(\overline{1-i}\right)}{\left(\overline{1+i}\right)} \\ =\frac{1+i}{1-i} \\ =\frac{1+i}{1-i}\times \frac{1+i}{1+i} \\ =\frac{{\left(1+i\right)}^2}{1-i^2} \\ =\frac{{\left(1+i\right)}^2}{1+1}=\frac{1}{2}\left(1+i^2+2i\right) \\ \mathrm{Conjugate}\mathrm{of}\frac{1-i}{1+i}=\frac{1}{2}\left(2i\right)=i \end{gathered}$$