By the end of this chapter, you will be able to:
Mastering these skills will help you confidently work with complex numbers, a key tool in many engineering calculations and designs.
Complex numbers are essential in advanced engineering calculations, including auto body repair tasks involving oscillations, vibrations, and electrical circuit analyses in vehicle systems. Understanding complex numbers enables an Auto Body Technician to solve problems involving alternating currents, signal processing for vehicle electronics, and precise measurements in alignment and diagnostics. This chapter introduces complex numbers and their geometric representation, equipping learners with the skills to apply these concepts in practical auto body engineering contexts.
Complex numbers combine a real part and an imaginary part, enabling representation of two-dimensional quantities in a single expression. They are fundamental in expressing waveforms, electrical signals, and mechanical vibrations encountered in auto body work. Understanding their geometry helps technicians visualize and manipulate these quantities effectively.
The real part of a complex number is the component that lies along the horizontal axis in the complex plane. It represents measurable quantities such as displacement or voltage magnitude without phase shift.
A complex number \( z \) is generally expressed as:
$$ z = x + yi $$
where \( x \) is the real part and \( y \) is the imaginary part.
Example 1: Determine the real part of the complex number representing an oscillation signal \( z = 7 + 3i \).
Given: \( z = 7 + 3i \)
Real part \( = x = 7 \)
Answer: 7
Example 2: Find the real part of the complex number \( z = -4 + 5i \) used in vehicle sensor signal analysis.
Given: \( z = -4 + 5i \)
Real part \( = x = -4 \)
Answer: -4
Example 3: If the complex number representing an AC voltage is \( z = 12.5 - 9.3i \), find the real part.
Given: \( z = 12.5 - 9.3i \)
Real part \( = x = 12.5 \)
Answer: 12.5
The imaginary part of a complex number is the component along the vertical axis in the complex plane. It often represents phase-shifted quantities such as reactive power or vibration components.
For the complex number \( z = x + yi \), the imaginary part is \( y \).
Example 1: Find the imaginary part of \( z = 8 + 6i \), representing a sensor output signal.
Given: \( z = 8 + 6i \)
Imaginary part \( = y = 6 \)
Answer: 6
Example 2: Determine the imaginary part of \( z = -3 - 7i \) used in vibration analysis.
Given: \( z = -3 - 7i \)
Imaginary part \( = y = -7 \)
Answer: -7
Example 3: For the signal \( z = 4.2 + 0i \), find the imaginary part.
Given: \( z = 4.2 + 0i \)
Imaginary part \( = y = 0 \)
Answer: 0
The Argand diagram is a graphical representation of complex numbers where the horizontal axis represents the real part and the vertical axis the imaginary part. It is a vital tool in visualizing complex quantities in auto body engineering, such as oscillations or electrical signals.
A point \( (x, y) \) on the Argand diagram corresponds to the complex number \( z = x + yi \).
Example 1: Plot the point corresponding to \( z = 5 + 2i \) on the Argand diagram and find its coordinates.
Given: \( z = 5 + 2i \)
Coordinates: \( (5, 2) \)
Answer: Point at (5, 2) on the Argand diagram
Example 2: For \( z = -3 + 4i \), state the coordinates on the Argand diagram.
Given: \( z = -3 + 4i \)
Coordinates: \( (-3, 4) \)
Answer: Point at (-3, 4) on the Argand diagram
Example 3: Find the Argand diagram coordinates for \( z = -2 - 6i \).
Given: \( z = -2 - 6i \)
Coordinates: \( (-2, -6) \)
Answer: Point at (-2, -6) on the Argand diagram
The modulus or magnitude of a complex number \( z = x + yi \) represents its distance from the origin on the Argand diagram. It quantifies the size or amplitude of the quantity, important in analyzing signal strength or vibration amplitudes.
The modulus is calculated by:
$$ |z| = \sqrt{x^{2} + y^{2}} $$
Example 1: Find the modulus of \( z = 3 + 4i \), representing a vibration amplitude.
Given: \( x = 3, y = 4 \)
$$ |z| = \sqrt{3^{2} + 4^{2}} $$
$$ |z| = \sqrt{9 + 16} $$
$$ |z| = \sqrt{25} $$
Answer: 5
Example 2: Calculate the magnitude of \( z = -6 + 8i \) measuring an electrical signal.
Given: \( x = -6, y = 8 \)
$$ |z| = \sqrt{(-6)^{2} + 8^{2}} $$
$$ |z| = \sqrt{36 + 64} $$
$$ |z| = \sqrt{100} $$
Answer: 10
Example 3: Determine the modulus of \( z = 5 - 12i \) from a sensor reading.
Given: \( x = 5, y = -12 \)
$$ |z| = \sqrt{5^{2} + (-12)^{2}} $$
$$ |z| = \sqrt{25 + 144} $$
$$ |z| = \sqrt{169} $$
Answer: 13
The argument (or angle) of a complex number \( z = x + yi \) is the angle \( \theta \) it forms with the positive real axis on the Argand diagram. It indicates the phase shift or orientation of the complex quantity.
The argument is calculated by:
$$ \theta = \tan^{-1} (\frac{y}{x}) $$
Adjustments are required depending on the quadrant of \( z \).
Example 1: Calculate the argument of \( z = 1 + \sqrt{3}i \).
Given: \( x = 1, y = \sqrt{3} \approx 1.732 \)
$$ \theta = \tan^{-1} (\frac{1.732}{1}) $$
$$ \theta = \tan^{-1} (1.732) $$
$$ \theta = 60^\circ $$
Answer: 60 degrees
Example 2: Find the argument of \( z = -1 + \sqrt{3}i \).
Given: \( x = -1, y = 1.732 \)
Since \( x < 0 \) and \( y > 0 \), angle in second quadrant:
$$ \theta = 180^\circ - \tan^{-1} (\frac{1.732}{1}) $$
$$ \theta = 180^\circ - 60^\circ $$
$$ \theta = 120^\circ $$
Answer: 120 degrees
Example 3: Determine the argument of \( z = -1 - \sqrt{3}i \).
Given: \( x = -1, y = -1.732 \)
Third quadrant:
$$ \theta = 180^\circ + \tan^{-1} (\frac{1.732}{1}) $$
$$ \theta = 180^\circ + 60^\circ $$
$$ \theta = 240^\circ $$
Answer: 240 degrees
The conjugate of a complex number \( z = x + yi \) is \( \overline{z} = x, yi \). It reflects the number across the real axis and is used to simplify division and find magnitudes in engineering calculations.
Example 1: Find the conjugate of \( z = 6 + 2i \).
Given: \( z = 6 + 2i \)
Conjugate:
$$ \overline{z} = 6 - 2i $$
Answer: 6 - 2i
Example 2: Determine the conjugate of \( z = -3 + 5i \).
Given: \( z = -3 + 5i \)
Conjugate:
$$ \overline{z} = -3 - 5i $$
Answer: -3 - 5i
Example 3: Find the conjugate of \( z = 4 - 7i \).
Given: \( z = 4 - 7i \)
Conjugate:
$$ \overline{z} = 4 + 7i $$
Answer: 4 + 7i
Create a free account to open more of this chapter.
Free: practical guides, quick cards, workplace scenarios and more.
Create a free accountThis chapter introduced complex numbers through their geometric representation, defining key elements such as the real part, imaginary part, and how these are plotted on the Argand diagram. It covered the calculation and interpretation of a complex number’s modulus and argument, as well as the concept of the conjugate. The chapter then explained the fundamental operations on complex numbers, including addition, subtraction, multiplication, and division, and detailed the process of converting between polar and rectangular forms. The application of De Moivre’s theorem was explored for expanding complex numbers raised to powers and finding their roots. Finally, the chapter demonstrated how complex numbers can be used to derive trigonometric identities, linking algebraic and geometric perspectives in engineering mathematics.
Find the real part and imaginary part of the complex number \(z = 7 - 4i\). (2 marks)
Calculate the modulus and argument (in degrees) of the complex number \(z = 3 + 4i\). (3 marks)
A vehicle repair workshop in Nairobi uses complex numbers to model vibrations in an auto body frame. Given the complex number \(z = 6 + 8i\), find the real part and the imaginary part of \(z\). (4 marks)
Plot the complex number \(z = -3 + 4i\) on an Argand diagram and determine its modulus. (4 marks)
At the start of this chapter we promised you would be able to:
Tick each one you can genuinely do.
So, are you there yet?
You're competent when you can confidently do 50% or more of what this chapter promised.
Sign in to record how you're doing.