Question 9.9: If z1 = 2 + 9j and z2 = 5 − 2j find z1 / z2....

We seek  \frac{2+9 j}{5-2 j}  The complex conjugate of the denominator is 5 + 2j, so we multiply both numerator and denominator by this quantity. The effect of this is to leave the value of  \frac{z_1}{z_2}  unaltered since we have only multiplied by 1. Therefore,

\begin{aligned}\frac{z_1}{z_2}=\frac{2+9 j}{5-2 j} & =\frac{(2+9 j)}{(5-2 j)} \frac{(5+2 j)}{(5+2 j)} \\& =\frac{10+45 j+4 j+18 j^2}{25+4}=\frac{-8+49 j}{29} \\& =-\frac{8}{29}+\frac{49}{29} j\end{aligned}

The multiplication of two conjugates in the denominator allows a useful simplification. We see that the effect of multiplying by the conjugate of the denominator is to make the denominator of the solution purely real.