Section 2.4 Division: Rectangular Form
We can use the concept of complex conjugate to give a strategy for dividing two complex numbers, \(z_1 = x_1 + i y_1\) and \(z_2 = x_2 + i y_2\text{.}\) The trick is to multiply by the number 1, in a special form that simplifies the denominator to be a real number and turns division into multiplication.
Consider:
\begin{align}
\frac{z_1}{z_2}\amp = \frac{z_1}{z_2} \frac{z_2^*}{z_2^*}\notag\\
\amp = \frac{z_1 z_2^*}{\vert z \vert ^2}\tag{2.4.1}
\end{align}
You should try this by writing out the real an imaginary components separately.
\begin{align}
\frac{z_1}{z_2}
\amp = \frac{x_1 + i y_1}{x_2 +i y_2}\notag\\
\amp = \left(\frac{x_1 + i y_1}{x_2 +i y_2}\right)
\left(\frac{x_2 - i y_2}{x_2 - i y_2}\right)\notag\\
\amp = \frac{(x_1 y_1 - x_2 y_2) + i (x_1 y_2 + x_2 y_1)}{x_2^2 +y_2^2}\notag\\
\amp = \left(\frac{x_1 y_1 - x_2 y_2}{x_2^2 + y_2^2}\right) + i
\left(\frac{x_1 y_2 + x_2 y_1}{x_2^2 + y_2^2}\right)\tag{2.4.2}
\end{align}