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Apply Complex Numbers plays a crucial role in diverse professional fields in Kenya, including healthcare, 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 in hospitals, financial modeling in banks, and control systems in manufacturing.
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 universities, resource management in county government offices, and network design in ICT firms.
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:
$$ r = \sqrt{x^{2} + y^{2}} $$
$$ \theta = \tan^{-1}(\frac{y}{x}) $$
Example 1: A data analyst at a county referral hospital 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 SACCO financial officer 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 county government 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 \( \theta = 225^{\circ} \)
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 agricultural cooperatives 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.
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 university researcher 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 \( \theta = 270^{\circ} \)
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 banks 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:
$$ x = r \cos \theta $$
$$ y = r \sin \theta $$
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 cooperative 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: \( z = 10(\cos 4.069 + i \sin 4.069) \)
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 hospitals 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:
$$ z_1 - z_2 = (x_1 - x_2) + (y_1 - y_2) i $$
Example 1: A bank 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 TVET college lecturer 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: \( 2 - 2i \)
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Create a free accountThis 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.
Plot the complex number \(z = 3 + 4i\) on an Argand diagram. Calculate its modulus and argument in degrees. (3 marks)
Given the complex numbers \(z_1 = 2 + 3i\) and \(z_2 = -1 + 4i\), find \(z_1 + z_2\) and \(z_1 \times z_2\). (3 marks)
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