Complex conjugates

Complex numbers have similar properties to surds that we have encountered in the past. The idea of the conjugate comes in very handy in many situations.

Given a complex number $z = x+y\mathrm{i},$ its complex conjugate, denoted by $z^*$ , is given by $z^* = x-y\mathrm{i}.$

Examples

If $z=1+2\mathrm{i},$ then $z^* = 1-2\mathrm{i}.$ More examples:

\begin{align*} (2-3\mathrm{i})^* &= 2+3\mathrm{i} \\ (5i)^* &= -5\mathrm{i} \\ (2)^* &= 2 \end{align*}

Formulas

\begin{align*} z+z^* &= 2x = 2 \textrm{Re}(z) \\ z-z^* &= 2y\mathrm{i} = 2 \textrm{Im}(z) \\ zz^* &= x^2 + y^2 = | z |^2 \\ (z^*)^* &= z \end{align*}