By the end of this chapter, you will be able to:
Mastering these skills will help you solve real-world engineering problems involving complex numbers with confidence and precision.
Complex numbers are essential in various professional fields including engineering, physics, and applied sciences. They enable the representation and manipulation of quantities that have both magnitude and direction, which cannot be fully described by real numbers alone. For example, in a county referral hospital’s biomedical equipment maintenance, complex numbers help model alternating currents in diagnostic machines. This chapter develops the fundamental concepts of complex numbers, their geometric representation, and key properties to equip students with the skills necessary for practical problem-solving across sectors such as healthcare, finance, agriculture, and ICT.
Complex geometry forms the foundation for understanding how complex numbers can be represented and manipulated visually and algebraically. It allows professionals to interpret complex quantities in two-dimensional space, which is crucial in fields like signal processing and control systems used in universities and banks.
The real part of a complex number is its projection on the horizontal axis in the complex plane. It represents the component of the complex number that lies along the real number line.
$$ z = x + yi $$
where \( x \) is the real part and \( y \) is the imaginary part.
Example 1: A SACCO’s interest rate fluctuation is modeled by the complex number \( z = 5 + 3i \). Find the real part.
Given: \( z = 5 + 3i \)
$$ \text{Real part} = x $$
$$ \text{Real part} = 5 $$
Answer: 5
Example 2: In a county government office’s signal processing unit, the complex number is \( z = -7 + 4i \). Determine the real part.
Given: \( z = -7 + 4i \)
$$ \text{Real part} = x $$
$$ \text{Real part} = -7 $$
Answer: -7
Example 3: A farm uses complex numbers to analyze irrigation patterns. Given \( z = 0 + 6i \), find the real part.
Given: \( z = 0 + 6i \)
$$ \text{Real part} = x $$
$$ \text{Real part} = 0 $$
Answer: 0
The imaginary part of a complex number is the coefficient of the imaginary unit \( i \), representing the projection on the vertical axis of the complex plane.
$$ z = x + yi $$
where \( y \) is the imaginary part.
Example 1: A retail business models electrical load as \( z = 8 + 2i \). Find the imaginary part.
Given: \( z = 8 + 2i \)
$$ \text{Imaginary part} = y $$
$$ \text{Imaginary part} = 2 $$
Answer: 2
Example 2: A hotel’s air conditioning system is analyzed with \( z = -3 - 5i \). Determine the imaginary part.
Given: \( z = -3 - 5i \)
$$ \text{Imaginary part} = y $$
$$ \text{Imaginary part} = -5 $$
Answer: -5
Example 3: An ICT company models phase shifts using \( z = 6 + 0i \). Find the imaginary part.
Given: \( z = 6 + 0i \)
$$ \text{Imaginary part} = y $$
$$ \text{Imaginary part} = 0 $$
Answer: 0
The Argand diagram is a graphical representation of complex numbers on a plane where the horizontal axis is the real part and the vertical axis is the imaginary part.
Example 1: Plot the complex number \( z = 4 + 3i \) on an Argand diagram. Find its coordinates.
Given: \( z = 4 + 3i \)
Coordinates are \( (x, y) = (4, 3) \)
Answer: (4, 3)
Example 2: For \( z = -2 + 5i \), determine the point on the Argand diagram.
Given: \( z = -2 + 5i \)
Coordinates are \( (x, y) = (-2, 5) \)
Answer: (-2, 5)
Example 3: A cooperative’s water quality monitoring system uses \( z = -3 - 4i \). Find its Argand diagram coordinates.
Given: \( z = -3 - 4i \)
Coordinates are \( (x, y) = (-3, -4) \)
Answer: (-3, -4)
The modulus or magnitude of a complex number represents its distance from the origin in the Argand diagram. It is calculated using the Pythagorean theorem.
$$ |z| = \sqrt{x^{2} + y^{2}} $$
Example 1: Find the modulus of \( z = 3 + 4i \) used in a university’s signal strength analysis.
Given: \( z = 3 + 4i \)
$$ |z| = \sqrt{3^{2} + 4^{2}} $$
$$ |z| = \sqrt{9 + 16} $$
$$ |z| = \sqrt{25} $$
Answer: 5
Example 2: A county hospital’s electrical load is \( z = -5 + 12i \). Calculate the modulus.
Given: \( z = -5 + 12i \)
$$ |z| = \sqrt{(-5)^{2} + 12^{2}} $$
$$ |z| = \sqrt{25 + 144} $$
$$ |z| = \sqrt{169} $$
Answer: 13
Example 3: A bank models currency fluctuations with \( z = 8 - 15i \). Find the magnitude.
Given: \( z = 8 - 15i \)
$$ |z| = \sqrt{8^{2} + (-15)^{2}} $$
$$ |z| = \sqrt{64 + 225} $$
$$ |z| = \sqrt{289} $$
Answer: 17
The argument of a complex number is the angle it makes with the positive real axis in the Argand diagram, usually measured in radians.
$$ \theta = \tan^{-1} (\frac{y}{x}) $$
Example 1: Calculate the argument of \( z = 1 + \sqrt{3}i \) used in a retail business’s power factor correction.
Given: \( z = 1 + \sqrt{3}i \)
$$ \theta = \tan^{-1} (\frac{\sqrt{3}}{1}) $$
$$ \theta = \tan^{-1} (\sqrt{3}) $$
$$ \theta = \frac{\pi}{3} \text{ radians} $$
Answer: \(\frac{\pi}{3}\) radians
Example 2: A SACCO uses \( z = -1 + 1i \) in loan risk analysis. Find the argument.
Given: \( z = -1 + 1i \)
$$ \theta = \tan^{-1} (\frac{1}{-1}) $$
$$ \theta = \tan^{-1} (-1) $$
Since \( x < 0 \) and \( y > 0 \), angle is in second quadrant:
$$ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \text{ radians} $$
Answer: \(\frac{3\pi}{4}\) radians
Example 3: For \( z = -2 - 2\sqrt{3}i \) in a county government office’s network analysis, compute the argument.
Given: \( z = -2 - 2\sqrt{3}i \)
$$ \theta = \tan^{-1} (\frac{-2\sqrt{3}}{-2}) $$
$$ \theta = \tan^{-1} (\sqrt{3}) $$
Both \( x < 0 \), \( y < 0 \), so angle in third quadrant:
$$ \theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \text{ radians} $$
Answer: \(\frac{4\pi}{3}\) radians
The conjugate of a complex number reverses the sign of the imaginary part. It is used in division and simplification of complex expressions.
$$ \overline{z} = x, yi $$
Example 1: Find the conjugate of \( z = 7 + 2i \) for a hotel’s energy monitoring system.
Given: \( z = 7 + 2i \)
$$ \overline{z} = 7 - 2i $$
Answer: \(7 - 2i\)
Example 2: A cooperative models water flow with \( z = -3 + 5i \). Determine the conjugate.
Given: \( z = -3 + 5i \)
$$ \overline{z} = -3 - 5i $$
Answer: \(-3 - 5i\)
Example 3: For \( z = 4 - 9i \) in a university’s electrical engineering lab, find the conjugate.
Given: \( z = 4 - 9i \)
$$ \overline{z} = 4 + 9i $$
Answer: \(4 + 9i\)
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Create a free accountThis chapter introduced the fundamental concepts of complex numbers, starting with their geometric representation on the Argand diagram, where the real part and imaginary part define a point in the complex plane. The modulus or magnitude measures the distance from the origin to this point, while the argument or angle indicates its direction relative to the positive real axis. The conjugate of a complex number was explained as a reflection about the real axis. The chapter then covered the four basic operations on complex numbers: addition, subtraction, multiplication, and division, with clear methods for each. It also demonstrated how to convert complex numbers between rectangular form and polar form, facilitating easier calculations. De Moivre's theorem was introduced as a powerful tool for expanding complex numbers raised to powers and for finding roots of complex numbers. Finally, the chapter showed how trigonometric identities can be derived and verified using complex number techniques.
A cooperative society has a complex number \(z = 3 + 4i\). Find the real part and imaginary part of \(z\). (2 marks)
Plot the complex number \(z = -2 + 5i\) on an Argand diagram. Calculate its modulus. (3 marks)
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