Apply Complex Numbers plays a crucial role in diverse professional fields in Kenya, including manufacturing, business, ICT, and agriculture. Complex numbers extend the real number system by including the imaginary unit, enabling the solution of equations that have no real roots. This chapter introduces the graphical representation of complex numbers on Argand diagrams, an essential skill for visualizing and manipulating complex numbers in practical applications such as signal processing at a telecoms base-station site, financial modeling in market stall blocks, and control systems in manufacturing.

5.1 Argand Diagrams

Argand diagrams provide a geometric representation of complex numbers as points or vectors in a two-dimensional plane. This visual approach is fundamental for professionals in many sectors who analyze phenomena involving oscillations, waves, or two-dimensional vector quantities. Understanding Argand diagrams helps in interpreting complex numbers intuitively, facilitating problem-solving in areas such as data analysis at a bottling plant, resource management in sawmills, and network design in ICT firms.

5.1.1 Representation of Complex Numbers on Argand Diagrams

A complex number \( z = x + yi \) can be represented as a point or a vector on a plane where the horizontal axis corresponds to the real part \( x \) and the vertical axis corresponds to the imaginary part \( y \). The Argand diagram is a coordinate plane with the real axis (Re) and the imaginary axis (Im).

The position of \( z \) on the Argand diagram is given by the ordered pair \( (x,y) \).

The modulus \( r \) and argument \( \theta \) of \( z \) relate to its Cartesian form by the formulas:

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

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

Worked Examples

Example 1: A data analyst at a geothermal power station models a signal as the complex number \( z = 3 + 4i \). Find its modulus and argument.

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

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

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

$$ r = \sqrt{25} $$

$$ r = 5 $$

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

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

$$ \theta \approx 53.13^{\circ} $$

Answer: Modulus \( r = 5 \), Argument \( \theta = 53.13^{\circ} \)

Example 2: A flour mill technician uses the complex number \( z = -2 + 2\sqrt{3}i \) to represent a vector. Calculate the modulus and argument in radians.

Given: \( x = -2 \), \( y = 2\sqrt{3} \)

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

$$ r = \sqrt{4 + 4 \times 3} $$

$$ r = \sqrt{4 + 12} $$

$$ r = \sqrt{16} $$

$$ r = 4 $$

$$ \theta = \tan^{-1}(\frac{2\sqrt{3}}{-2}) $$

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

Since \( x < 0 \) and \( y > 0 \), the point lies in the second quadrant, so:

$$ \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \approx 2.094 \text{ radians} $$

Answer: Modulus \( r = 4 \), Argument \( \theta = 2.094 \) radians

Example 3: A quarry engineer represents the complex number \( z = -5 - 5i \). Find its modulus and argument in degrees.

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

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

$$ r = \sqrt{25 + 25} $$

$$ r = \sqrt{50} $$

$$ r = 7.071 $$

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

$$ \theta = \tan^{-1}(1) = 45^{\circ} $$

Since \( x < 0 \) and \( y < 0 \), the point lies in the third quadrant:

$$ \theta = 180^{\circ} + 45^{\circ} = 225^{\circ} $$

Answer: Modulus \( r = 7.071 \), Argument \( \theta = 225^{\circ} \)

5.1.2 Plotting and Interpreting Points on Argand Diagrams

Plotting complex numbers on Argand diagrams involves locating points using their real and imaginary components. This skill is essential in sectors such as retail business for inventory control systems or in dairy processing plants for modeling growth patterns with complex variables.

The distance from the origin to the point is the modulus, and the angle between the positive real axis and the line connecting the origin to the point is the argument.

Worked Examples

Example 1: A hotel manager models customer feedback scores as the complex number \( z = 1 + i \). Plot this on an Argand diagram and determine its modulus and argument.

Given: \( x = 1 \), \( y = 1 \)

$$ r = \sqrt{1^{2} + 1^{2}} $$

$$ r = \sqrt{1 + 1} $$

$$ r = \sqrt{2} = 1.414 $$

$$ \theta = \tan^{-1}(\frac{1}{1}) = 45^{\circ} $$

Answer: Point plotted at (1,1), Modulus \( r = 1.414 \), Argument \( \theta = 45^{\circ} \)

Example 2: A research technician at a printing works represents a complex number \( z = -3 + 4i \). Plot and find the modulus and argument.

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

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

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

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

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

Since \( x < 0 \) and \( y > 0 \), quadrant II:

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

Answer: Point plotted at (-3,4), Modulus \( r = 5 \), Argument \( \theta = 126.87^{\circ} \)

Example 3: A NEMA analyst models pollution data as \( z = 0 - 5i \). Plot the point and find modulus and argument.

Given: \( x = 0 \), \( y = -5 \)

$$ r = \sqrt{0^{2} + (-5)^{2}} = 5 $$

Since \( x = 0 \) and \( y < 0 \), angle \( \theta = 270^{\circ} \) or \( 3\pi/2 \) radians.

Answer: Point plotted at (0,-5), Modulus \( r = 5 \), Argument \( \theta = 270^{\circ} \)

5.1.3 Conversion Between Cartesian and Polar Forms Using Argand Diagrams

Converting complex numbers between Cartesian form \( x + yi \) and polar form \( r(\cos \theta + i \sin \theta) \) is fundamental in fields such as ICT for signal modulation and finance for risk modeling.

The formulas are:

From Cartesian to Polar:

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

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

From Polar to Cartesian:

$$ x = r \cos \theta $$

$$ y = r \sin \theta $$

Worked Examples

Example 1: A retail business analyst has a complex number \( z = 4 + 3i \). Convert it 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: An ICT engineer has a complex number in polar form \( z = 10(\cos 60^{\circ} + i \sin 60^{\circ}) \). Convert to Cartesian form.

Given: \( r = 10 \), \( \theta = 60^{\circ} \)

$$ x = 10 \cos 60^{\circ} = 10 \times 0.5 = 5 $$

$$ y = 10 \sin 60^{\circ} = 10 \times 0.866 = 8.66 $$

Cartesian form:

$$ z = 5 + 8.66 i $$

Answer: \( z = 5 + 8.66 i \)

Example 3: A bakery manager has a complex number \( z = -6 - 8i \). Convert to polar form and find \( r \) and \( \theta \) in radians.

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

$$ r = \sqrt{(-6)^{2} + (-8)^{2}} = \sqrt{36 + 64} = \sqrt{100} = 10 $$

$$ \theta = \tan^{-1}(\frac{-8}{-6}) = \tan^{-1}(1.333) = 53.13^{\circ} = 0.927 \text{ radians} $$

Since \( x < 0 \) and \( y < 0 \), point lies in the third quadrant:

$$ \theta = \pi + 0.927 = 4.069 \text{ radians} $$

Polar form:

$$ z = 10(\cos 4.069 + i \sin 4.069) $$

Answer: \( z = 10(\cos 4.069 + i \sin 4.069) \)

5.1.4 Addition and Subtraction of Complex Numbers Using Argand Diagrams

Adding or subtracting complex numbers corresponds to vector addition or subtraction on the Argand diagram, a practical method used in counties for resource allocation modeling and in radio broadcasting transmitter sites for signal combination.

Given two complex numbers \( z_1 = x_1 + y_1 i \) and \( z_2 = x_2 + y_2 i \):

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 tea factory analyst adds \( z_1 = 2 + 3i \) and \( z_2 = 4 + 5i \).

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

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

$$ z_1 + z_2 = 6 + 8i $$

Answer: \( 6 + 8i \)

Example 2: A trainer at a furniture workshop subtracts \( z_2 = 6 + 2i \) from \( z_1 = 9 + 7i \).

Given: \( z_1 = 9 + 7i \), \( z_2 = 6 + 2i \)

$$ z_1 - z_2 = (9 - 6) + (7 - 2)i $$

$$ z_1 - z_2 = 3 + 5i $$

Answer: \( 3 + 5i \)

Example 3: A hotel manager combines feedback scores \( z_1 = -3 + 4i \) and \( z_2 = 5 - 6i \).

Given: \( z_1 = -3 + 4i \), \( z_2 = 5 - 6i \)

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

$$ z_1 + z_2 = 2 - 2i $$

Answer: \( 2 - 2i \)

Chapter Summary

This chapter introduced the representation of complex numbers on Argand diagrams, where each complex number corresponds to a point or vector in the plane defined by its real and imaginary parts. It explained how to plot complex numbers and interpret their geometric meaning, including magnitude and direction. The chapter then covered the fundamental operations on complex numbers such as addition, subtraction, multiplication, and division, demonstrating how these operations translate both algebraically and geometrically. Emphasis was placed on the use of complex conjugates and the polar form to simplify calculations. Finally, the chapter explored De Moivre’s Theorem, which provides a powerful method for finding powers and roots of complex numbers using trigonometric expressions. The theorem was applied to solve problems involving complex number powers and roots, reinforcing the connection between algebraic and geometric perspectives. Overall, the chapter developed essential skills for manipulating complex numbers in engineering mathematics contexts.