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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.


\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}