Complex fractions

If z=a+ib then we define the conjugate z^*=a-ib

This has the very useful property that zz^*=a²+b²=|z|²

This follows from our old friend a²-b²=(a+b)(a-b), substituting ib instead of b and remembering that (ib)²=-b².

Other facts about the conjugate: if w and z are complex numbers, then

(wz)^*=w^*z^*

(w+z)^*=w^*+z^*

This is very useful for fractions: if you want to get rid of a complex number in the denominator, just multiply top and bottom by the conjugate of the denominator.

For example, 2/(3+i) = 2(3-i)/(3+i)(3-i) = 2(3-i)/(3²+1²) = (3-i)/5