Division: Rectangular Form

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Section 2.4 Division: Rectangular Form

Method for Dividing Two Complex Numbers in 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}

The final expression is straightforward to divide into its real and imaginary parts since the denominator is pure real.

In the following activity, you will find the real and imaginary parts of a quotient for the general case. Rather than memorizing the final answer, it will be easier to just repeat this method every time you are dividing two complex numbers.

Activity 2.2. Divide Two Complex Numbers in Rectangular Form.

Find the real and imaginary components of the complex number \(\frac{z_1}{z_2}\) where \(z_1=x_1+iy_1\) and \(z_2=x_2+iy_2\text{.}\)

Solution.

\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}