Engineering professionals across all sectors in Kenya frequently encounter problems involving complex numbers, especially when analyzing oscillations, waveforms, and alternating current circuits common in healthcare equipment, financial modelling, or ICT systems. Mastery of complex numbers and their graphical representation through Argand diagrams enables technicians to visualize and solve problems involving magnitude and phase relationships efficiently. This chapter develops skills to represent complex numbers on the Argand plane, facilitating intuitive understanding and manipulation in practical scenarios such as signal processing in hospitals or data analysis in banks.

5.1 Argand diagrams

Argand diagrams provide a geometric representation of complex numbers on a two-dimensional plane, where the horizontal axis shows the real part and the vertical axis shows the imaginary part. This visual tool is essential for professionals across sectors to interpret complex quantities and perform operations like addition, subtraction, and multiplication graphically. In contexts such as water quality monitoring at county government offices or electrical load analysis in hotels, Argand diagrams aid in simplifying complex calculations.

5.1.1 Representation of complex numbers on the Argand plane

Complex numbers are expressed as \(z = x + yi\) where \(x\) is the real part and \(y\) the imaginary part. On the Argand diagram, the horizontal axis (real axis) plots \(x\) and the vertical axis (imaginary axis) plots \(y\). The number \(z\) corresponds to the point \((x, y)\) in this plane or equivalently the vector from the origin to \((x, y)\).

$$z = x + yi$$

Worked Examples

Example 1: A water quality sensor at a county referral hospital records a complex signal \(z = 3 + 4i\). Plot the point representing this complex number on the Argand diagram.

Given: \(x = 3\), \(y = 4\)

$$z = 3 + 4i$$

Coordinates on Argand plane: \((3, 4)\)

Answer: Point at (3, 4) on Argand diagram

Example 2: A SACCO financial model involves \(z = -2 + 5i\). Determine the coordinates for plotting.

Given: \(x = -2\), \(y = 5\)

$$z = -2 + 5i$$

Coordinates on Argand plane: \((-2, 5)\)

Answer: Point at (-2, 5) on Argand diagram

Example 3: A TVET college ICT system uses \(z = -4 - 3i\) in signal processing. Identify the Argand coordinates.

Given: \(x = -4\), \(y = -3\)

$$z = -4 - 3i$$

Coordinates on Argand plane: \((-4, -3)\)

Answer: Point at (-4, -3) on Argand diagram

5.1.2 Modulus and argument of a complex number

The modulus \(r\) of complex number \(z = x + yi\) is the distance from the origin to the point \((x, y)\), calculated as

$$r = \sqrt{x^2 + y^2}$$

The argument \(\theta\) is the angle made with the positive real axis, measured counterclockwise, given by

$$\theta = \tan^{-1}(\frac{y}{x})$$

Worked Examples

Example 1: A retail business analyses a complex number \(z = 6 + 8i\). Calculate its modulus and argument.

Given: \(x = 6\), \(y = 8\)

$$r = \sqrt{6^2 + 8^2}$$

$$r = \sqrt{36 + 64}$$

$$r = \sqrt{100} = 10$$

$$\theta = \tan^{-1}(\frac{8}{6})$$

$$\theta = \tan^{-1}(1.333) = 53.13^\circ$$

Answer: Modulus \(r = 10\), Argument \(\theta = 53.13^\circ\)

Example 2: A bank's risk model uses \(z = -3 + 4i\). Find modulus and argument.

Given: \(x = -3\), \(y = 4\)

$$r = \sqrt{(-3)^2 + 4^2}$$

$$r = \sqrt{9 + 16} = \sqrt{25} = 5$$

$$\theta = \tan^{-1}(\frac{4}{-3})$$

Since \(x < 0\) and \(y > 0\), \(\theta = 180^\circ - \tan^{-1}(\frac{4}{3})\)

$$\theta = 180^\circ - 53.13^\circ = 126.87^\circ$$

Answer: Modulus \(r = 5\), Argument \(\theta = 126.87^\circ\)

Example 3: An agricultural cooperative monitors signals \(z = -5 - 12i\). Calculate modulus and argument in radians.

Given: \(x = -5\), \(y = -12\)

$$r = \sqrt{(-5)^2 + (-12)^2}$$

$$r = \sqrt{25 + 144} = \sqrt{169} = 13$$

$$\theta = \tan^{-1}(\frac{-12}{-5})$$

Since \(x < 0\), \(y < 0\), \(\theta = 180^\circ + \tan^{-1}(\frac{12}{5})\)

$$\theta = 180^\circ + 67.38^\circ = 247.38^\circ$$

Convert degrees to radians:

$$\theta = \frac{247.38 \times \pi}{180} = 4.32 \text{ radians}$$

Answer: Modulus \(r = 13\), Argument \(\theta = 4.32\) radians

5.1.3 Polar form of complex numbers and conversion

The polar form expresses a complex number \(z\) as

$$z = r(\cos\theta + i\sin\theta)$$

where \(r\) is modulus and \(\theta\) is argument. This form simplifies multiplication and division of complex numbers, important in ICT signal modulation or hotel power management systems.

Conversion formulas:

From rectangular to polar:

$$r = \sqrt{x^2 + y^2}$$

$$\theta = \tan^{-1}(\frac{y}{x})$$

From polar to rectangular:

$$x = r \cos\theta$$

$$y = r \sin\theta$$

Worked Examples

Example 1: A county hospital electrical system has \(z = 4 + 3i\). Convert to polar form.

Given: \(x = 4\), \(y = 3\)

$$r = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$

$$\theta = \tan^{-1}(\frac{3}{4}) = 36.87^\circ$$

Polar form:

$$z = 5(\cos 36.87^\circ + i \sin 36.87^\circ)$$

Answer: \(z = 5(\cos 36.87^\circ + i \sin 36.87^\circ)\)

Example 2: A retail business signal \(z = 7(\cos 53.13^\circ + i \sin 53.13^\circ)\). Convert to rectangular form.

Given: \(r = 7\), \(\theta = 53.13^\circ\)

$$x = 7 \cos 53.13^\circ = 7 \times 0.6 = 4.2$$

$$y = 7 \sin 53.13^\circ = 7 \times 0.8 = 5.6$$

Rectangular form:

$$z = 4.2 + 5.6i$$

Answer: \(z = 4.2 + 5.6i\)

Example 3: A SACCO financial model uses \(z = 10(\cos 150^\circ + i \sin 150^\circ)\). Convert to rectangular form.

Given: \(r = 10\), \(\theta = 150^\circ\)

$$x = 10 \cos 150^\circ = 10 \times (-0.866) = -8.66$$

$$y = 10 \sin 150^\circ = 10 \times 0.5 = 5.0$$

Rectangular form:

$$z = -8.66 + 5.0i$$

Answer: \(z = -8.66 + 5.0i\)

5.1.4 Addition and subtraction of complex numbers using Argand diagrams

On the Argand plane, addition of two complex numbers is the vector addition of their position vectors, and subtraction corresponds to vector subtraction. This geometric interpretation is useful in analyzing combined signals in data systems at universities or combined loads in hotel power supply.

Given:

$$z_1 = x_1 + y_1 i$$

$$z_2 = x_2 + y_2 i$$

Addition:

$$z_1 + z_2 = (x_1 + x_2) + (y_1 + y_2)i$$

Subtraction:

$$z_1 - z_2 = (x_1 - x_2) + (y_1 - y_2)i$$

Worked Examples

Example 1: A university ICT network has signals \(z_1 = 2 + 3i\) and \(z_2 = 4 + 5i\). Find \(z_1 + z_2\).

Given:

\(z_1 = 2 + 3i\), \(z_2 = 4 + 5i\)

$$z_1 + z_2 = (2 + 4) + (3 + 5)i = 6 + 8i$$

Answer: \(6 + 8i\)

Example 2: A hotel energy system measures \(z_1 = 7 + 4i\), \(z_2 = 3 + 6i\). Find \(z_1 - z_2\).

Given:

\(z_1 = 7 + 4i\), \(z_2 = 3 + 6i\)

$$z_1 - z_2 = (7 - 3) + (4 - 6)i = 4 - 2i$$

Answer: \(4 - 2i\)

Example 3: A county government office signal analysis uses \(z_1 = -5 + 2i\), \(z_2 = 3 - 7i\). Calculate \(z_1 + z_2\).

Given:

\(z_1 = -5 + 2i\), \(z_2 = 3 - 7i\)

$$z_1 + z_2 = (-5 + 3) + (2 - 7)i = -2 - 5i$$

Answer: \(-2 - 5i\)

Chapter Summary

This chapter introduced the graphical representation of complex numbers using Argand diagrams, where complex numbers are plotted as points in a two-dimensional plane with real and imaginary axes. It then covered the fundamental operations on complex numbers including addition, subtraction, multiplication, and division, emphasizing the algebraic manipulation of their real and imaginary parts. The chapter also explored the polar form of complex numbers which simplifies multiplication and division by representing them in terms of magnitude and angle. Finally, De Moivre's theorem was presented as a powerful tool to raise complex numbers to integer powers and to extract roots, linking trigonometry with complex number calculations. Through worked examples, students learned to apply these concepts in solving practical problems relevant to engineering contexts. The combination of graphical and algebraic techniques equips learners with a comprehensive understanding of complex numbers essential for advanced mathematical and engineering applications.