Complex conjugate

For $z=a+bi\in\mathbb{C}$ with $a,b\in\mathbb{R}$, the complex conjugate of $z$ is

Complex conjugation is an algebraic involution: it reverses the sign of the imaginary part and satisfies $\overline{\overline{z}}=z$. It is used to express the modulus and to compute inverses: if $z\ne 0$ then $z^{-1}=\overline{z}/(z\overline{z})$.

Examples:

• If $z=3-2i$, then $\overline{z}=3+2i$.
• $\overline{(1+i)^2}=\overline{2i}=-2i$.
• If $z=iy$ with $y\in\mathbb{R}$, then $\overline{z}=-iy$.