Complex division

In the previous section, we saw that $zz^*$ gives us a real number, $x^2 + y^2.$ We will use this property to carry out complex division.

To evaluate $\frac{a+b\mathrm{i}}{x+y\mathrm{i}},$ we multiply both numerator and denominator with the conjugate of the denominator to get $\displaystyle {\frac{a+b\mathrm{i}}{x+y\mathrm{i}} \cdot \frac{x-y\mathrm{i}}{x-y\mathrm{i}}} \allowbreak = \allowbreak {\frac{(ax+by)+(bx-ay)\mathrm{i}}{x^2+y^2}.}$

Examples

\begin{aligned} & \frac{1 + 2 \mathrm{i}}{3 - 4 \mathrm{i}} \\ =& \frac{1 + 2 \mathrm{i}}{3 - 4 \mathrm{i}} \times \frac{3 + 4 \mathrm{i}}{3 + 4 \mathrm{i}} \end{aligned}