3.0 Conjugate of complex number

The complex numbers $z=a+ib$ and $\mathop z\limits^ - = a - ib$, where $a$ and $b$ are real numbers and $b \ne 0$, are the complex conjugate of each other.

Conjugate of a complex number is obtained by just replacing $i$ by $-i$.

Sum of $z$ and $\mathop z\limits^ - $ is purely real i.e., $$\begin{equation} \begin{aligned} z + \mathop {{\text{ }}z}\limits^ - = (a + ib) + (a - ib) = 2a \\ \therefore {R_e}(z) = \frac{{z + \mathop {{\text{ }}z}\limits^ - }}{2} \\\end{aligned} \end{equation} $$

Difference of $z$ and $\mathop z\limits^ - $ is purely imaginary i.e., $$\begin{equation} \begin{aligned} z - \mathop {{\text{ }}z}\limits^ - = (a + ib) - (a - ib) = 2ib \\ \therefore \operatorname{Im} (z) = \frac{{z - \mathop {{\text{ }}z}\limits^ - }}{{2i}} \\\end{aligned} \end{equation} $$

Product of $z$ and $\mathop z\limits^ - $ is also purely real i.e., $$\begin{equation} \begin{aligned} z\mathop {{\text{ }}z}\limits^ - = (a + ib)(a - ib) \\ {\text{ = }}{a^2} - {(ib)^2} = {a^2} - {i^2}{b^2} \\ {\text{ = }}{a^2} + {b^2} \\\end{aligned} \end{equation} $$