conjugate of complex number

conjugate of $c= a+ib$ is the complex number $\overline { c} = a-ib$.

Basic properties. Assuming that :

1. $\overline{ \overline c } = c$

2. $\overline { c+d} = \overline { c} + \overline { d}$

3. $\overline { cd} = \overline { c} \overline { d}$

4. $\overline{ c} = c$ if and only if $c$ is a real number.

5. $c \overline {c} = a^2 + b^2$ is a nonnegative real number and $c \overline {c} =0$ if and only if $c=0 .$