By the end of this chapter, you will be able to:
Mastering these skills will help you solve real-world engineering problems where complex numbers are essential.
Complex numbers provide a powerful mathematical tool to represent quantities involving two components, commonly encountered in many technical and scientific fields. These numbers extend the real number system by including an imaginary unit, enabling the solution of equations that have no real roots. In Kenyan professional contexts such as county referral hospitals for biomedical signal analysis, banks for risk modeling, and agricultural cooperatives for system dynamics, complex numbers simplify analysis and design. This chapter begins by defining complex numbers, explaining their components, and introducing the Argand diagram for their graphical representation.
A complex number is a number that consists of two parts: a real part and an imaginary part. It is expressed in the form \( z = x + yi \), where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit with the property \( i^2 = -1 \). The real part \( x \) represents the horizontal component, while the imaginary part \( y \) represents the vertical component in the complex plane.
$$ z = x + yi \quad \text{where} \quad x, y \in \mathbb{R}, \quad i^2 = -1 $$
Example 1: A cooperative society is analyzing cash flow using complex numbers where the real part represents income in Ksh and the imaginary part represents expenses in Ksh. If the cash flow is \( 5000 + 3000i \), identify the real and imaginary parts.
Given: \( z = 5000 + 3000i \)
Real part \( x = 5000 \) Ksh
Imaginary part \( y = 3000 \) Ksh
Answer: Real part = 5000 Ksh, Imaginary part = 3000 Ksh
Example 2: A county hospital models biomedical signals as \( z = -120 + 80i \). Find the real and imaginary parts.
Given: \( z = -120 + 80i \)
Real part \( x = -120 \) units
Imaginary part \( y = 80 \) units
Answer: Real part = -120 units, Imaginary part = 80 units
Example 3: In an ICT firm, a complex number \( z = 7 - 24i \) represents system parameters. Determine the real and imaginary components.
Given: \( z = 7 - 24i \)
Real part \( x = 7 \)
Imaginary part \( y = -24 \)
Answer: Real part = 7, Imaginary part = -24
The real part of a complex number is the coefficient of the real unit and lies on the horizontal axis of the complex plane. The imaginary part is the coefficient of the imaginary unit \( i \) and lies on the vertical axis. Separating these parts is essential in operations such as addition, subtraction, multiplication, and division of complex numbers.
For \( z = x + yi \),
$$ \Re(z) = x, \quad \Im(z) = y $$
Example 1: A retail business uses \( z = 250 + 150i \) to represent daily sales (real) and returns (imaginary). Find \( \Re(z) \) and \( \Im(z) \).
Given: \( z = 250 + 150i \)
$$ \Re(z) = 250 $$
$$ \Im(z) = 150 $$
Answer: Real part = 250, Imaginary part = 150
Example 2: A university models student attendance as \( z = -50 + 110i \). Compute the real and imaginary parts.
Given: \( z = -50 + 110i \)
$$ \Re(z) = -50 $$
$$ \Im(z) = 110 $$
Answer: Real part = -50, Imaginary part = 110
Example 3: A SACCO models loan disbursement and repayments as \( z = 1000 - 400i \). Determine the parts.
Given: \( z = 1000 - 400i \)
$$ \Re(z) = 1000 $$
$$ \Im(z) = -400 $$
Answer: Real part = 1000, Imaginary part = -400
The Argand diagram is a graphical representation of complex numbers on a plane where the horizontal axis (x-axis) represents the real part and the vertical axis (y-axis) represents the imaginary part. Each complex number corresponds to a point or vector from the origin to \((x, y)\).
If \( z = x + yi \), its position in the Argand diagram is at coordinates \((x, y)\).
The modulus \( r \) and argument \( \theta \) of \( z \) are given by:
$$ r = \sqrt{x^2 + y^2} $$
$$ \theta = \tan^{-1}(\frac{y}{x}) $$
Example 1: A farm cooperative represents water usage as \( z = 3 + 4i \). Find the modulus and argument.
Given: \( z = 3 + 4i \)
$$ r = \sqrt{3^2 + 4^2} $$
$$ r = \sqrt{9 + 16} $$
$$ r = \sqrt{25} = 5 $$
$$ \theta = \tan^{-1}(\frac{4}{3}) $$
$$ \theta \approx 53.13^\circ $$
Answer: Modulus = 5, Argument = 53.13°
Example 2: A hotel models noise levels as \( z = -5 + 12i \). Calculate modulus and argument.
Given: \( z = -5 + 12i \)
$$ r = \sqrt{(-5)^2 + 12^2} $$
$$ r = \sqrt{25 + 144} $$
$$ r = \sqrt{169} = 13 $$
$$ \theta = \tan^{-1}(\frac{12}{-5}) $$
Since \( x < 0 \) and \( y > 0 \), \( \theta \) lies in the second quadrant:
$$ \theta = 180^\circ - \tan^{-1}(\frac{12}{5}) $$
$$ \theta = 180^\circ - 67.38^\circ = 112.62^\circ $$
Answer: Modulus = 13, Argument = 112.62°
Example 3: A county government office uses \( z = -8 - 15i \) to represent parameters. Find modulus and argument.
Given: \( z = -8 - 15i \)
$$ r = \sqrt{(-8)^2 + (-15)^2} $$
$$ r = \sqrt{64 + 225} $$
$$ r = \sqrt{289} = 17 $$
For argument:
Since \( x < 0 \) and \( y < 0 \), \( \theta \) lies in the third quadrant:
$$ \theta = 180^\circ + \tan^{-1}(\frac{15}{8}) $$
$$ \theta = 180^\circ + 61.93^\circ = 241.93^\circ $$
Answer: Modulus = 17, Argument = 241.93°
Example 4: A bank models currency exchange rates as \( z = 7 - 24i \). Find modulus and argument.
Given: \( z = 7 - 24i \)
$$ r = \sqrt{7^2 + (-24)^2} $$
$$ r = \sqrt{49 + 576} $$
$$ r = \sqrt{625} = 25 $$
Argument:
Since \( x > 0 \), \( y < 0 \), \( \theta \) lies in the fourth quadrant:
$$ \theta = 360^\circ - \tan^{-1}(\frac{24}{7}) $$
$$ \theta = 360^\circ - 73.74^\circ = 286.26^\circ $$
Answer: Modulus = 25, Argument = 286.26°
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Create a free accountThis chapter introduced complex numbers by defining their real and imaginary parts and representing them on the complex plane using the Argand diagram. It covered arithmetic operations including addition, subtraction, multiplication, division, and the use of conjugates to simplify calculations. The chapter explained the polar form of complex numbers and demonstrated how to convert between Cartesian and polar forms, followed by the exponential form derived from Euler's formula. It explored powers and roots of complex numbers using De Moivre's Theorem and detailed complex exponential functions with their properties and applications in differential equations. The chapter also addressed solving quadratic equations, including those with complex coefficients, and discussed the nature of roots as real or complex. Finally, it showed how complex numbers can be used to solve systems of linear equations, highlighting their importance in AC circuit analysis.
Find the sum and difference of the complex numbers \(3 + 4i\) and \(1 - 2i\). (2 marks)
Calculate the product and quotient of the complex numbers \(2 + 3i\) and \(4 - i\). (3 marks)
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