Operations with Complex Numbers

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The purpose of this section is to review arithmetic operations with complex numbers. We use complex numbers to describe circuits. When we solve circuits to find voltages, currents, and power, we often encounter addition, subtraction, multiplication, division, and the complex conjugate of complex numbers.

Complex Conjugate

We will see complex conjugate when we discuss maximum power transfer.

$\begin{eqnarray} z^* = (x+ j y)^* = x- j y = |z| e^{-j \Theta} \end{eqnarray}$

Addition

In electrical engineering, we see complex number addition and subtraction when complex impedances are in series, and we are looking for the equivalent complex impedance. The easiest way to add two complex numbers is to find the Cartesian representation of both and then add the real parts separately and the imaginary part separately.

$\begin{eqnarray} z_1=x_1 + j y_1 \nonumber \\ z_2=x_2 + j y_2 \nonumber \\ z_1+z_2 = x_1 + x_2 + j ( y_1 + y_2) \end{eqnarray}$

You can visually explore addition of two complex numbers with the app below.

Two impedances are given $Z_1=50+j200 \Omega$ and $Z_2=50-j200 \Omega$ . If the two impedances are in series, what is the total impedance?

$Z_1+Z_2 = \answer {100}$

Subtraction

$\begin{eqnarray} z_1=x_1 + j y_1 \nonumber \\ z_2=x_2 + j y_2 \nonumber \\ z_1-z_2 = x_1 - x_2 + j ( y_1 - y_2) \end{eqnarray}$

You can visually explore subtraction of two complex numbers with the app below.

Multiplication and Division

We often see multiplication and division of complex numbers in Ohm’s law or transfer function of a circuit. Two complex numbers can be multiplied or divided in either Cartesian or Polar forms. However, the easiest way to divide or multiply two complex numbers is to find the polar representation of both and then divide or multiply the amplitudes and subtract or add the phases, as shown in equations below.

$\begin{eqnarray} z_1=|z_1| e^{j \Theta_1} \nonumber \\ z_2=|z_2| e^{j \Theta_2} \nonumber \\ \frac{z_1}{ z_2} = \frac{|z_1|}{|z_2|} e^{j \Theta_1 -\Theta_2} \end{eqnarray}$

$\begin{eqnarray} z_1=|z_1| e^{j \Theta_1} \nonumber \\ z_2=|z_2| e^{j \Theta_2} \nonumber \\ z_1 z_2 = |z_1||z_2| e^{j \Theta_1 +\Theta_2} \end{eqnarray}$

Explore the multiplication of two complex numbers visually with the app below. We see that the multiplication of numbers represent rotation.

Three complex numbers are given $Z_1=100+j50$ , $Z_2=3*e^{j40^0}$ and $Z_3=20+j100$ . Calculate $\frac {Z_1 Z_2}{Z_2+Z_3}$ . Present your answer as a complex number in Cartesian coordinates with two decimal places. For example $0.11+j0.23$ :

$\frac {Z_1 Z_2}{Z_2+Z_3} = \answer {0.36+j3.16}$