By the end of this chapter, you will be able to:
Mastering these skills will help you solve real-world engineering problems efficiently and confidently using matrices.
Matrices and complex numbers are fundamental in Mechatronics Technology, especially in analyzing control systems, signal processing, and robotics. Complex geometry provides a way to represent and manipulate signals and system states that have both magnitude and phase components. This chapter focuses on the application of complex numbers, particularly their geometric interpretation and algebraic properties, which are essential for understanding system behaviors in Kenyan industries such as automation in manufacturing plants or control systems in water treatment facilities.
Complex geometry is the study of complex numbers represented geometrically in the plane, combining algebraic and geometric methods. This is crucial in Mechatronics for tasks such as analyzing AC circuits, control system stability, and signal modulation where both magnitude and phase angle matter.
The real part of a complex number represents its horizontal coordinate on the complex plane. For a complex number \( z = x + yi \), the real part is \( x \).
$$\text{Real part} = \Re(z) = x$$
Example 1: A signal in a control system is represented as \( z = 4 + 3i \). Find the real part.
Given: \( z = 4 + 3i \)
$$\Re(z) = 4$$
Answer: 4
Example 2: In a sensor output, the complex reading is \( z = -7 + 5i \). Determine the real component.
Given: \( z = -7 + 5i \)
$$\Re(z) = -7$$
Answer: -7
Example 3: A signal \( z \) has a real part of 2.5 and an imaginary part of -1.8. Write the complex number and identify the real part.
Given: \( \Re(z) = 2.5 \), \( \Im(z) = -1.8 \)
Complex number: \( z = 2.5 - 1.8i \)
$$\Re(z) = 2.5$$
Answer: 2.5
The imaginary part of a complex number represents its vertical coordinate on the complex plane. For \( z = x + yi \), the imaginary part is \( y \).
$$\text{Imaginary part} = \Im(z) = y$$
Example 1: For \( z = 6 + 9i \), find the imaginary part.
Given: \( z = 6 + 9i \)
$$\Im(z) = 9$$
Answer: 9
Example 2: A voltage phasor is \( z = -3 - 4i \). Determine the imaginary part.
Given: \( z = -3 - 4i \)
$$\Im(z) = -4$$
Answer: -4
Example 3: A complex number has real part 0 and imaginary part 7.5. Write the complex number and state the imaginary part.
Given: \( \Re(z) = 0 \), \( \Im(z) = 7.5 \)
Complex number: \( z = 0 + 7.5i \)
$$\Im(z) = 7.5$$
Answer: 7.5
An Argand diagram is a graphical representation of complex numbers on a two-dimensional plane, where the horizontal axis is the real part and the vertical axis is the imaginary part.
$$z = x + yi \quad \Rightarrow \quad \text{point }(x,y) \text{ on Argand plane}$$
Example 1: Plot \( z = 3 + 4i \) on the Argand diagram and state its coordinates.
Given: \( z = 3 + 4i \)
Coordinates: (3, 4)
Answer: Point at (3,4) on Argand diagram
Example 2: Represent \( z = -2 + 5i \) on the Argand plane and give its coordinates.
Given: \( z = -2 + 5i \)
Coordinates: (-2, 5)
Answer: Point at (-2, 5)
Example 3: A complex number \( z = -6 - 3i \) is given. Plot its position on the Argand diagram.
Given: \( z = -6 - 3i \)
Coordinates: (-6, -3)
Answer: Point at (-6, -3)
The modulus or magnitude of a complex number measures its distance from the origin on the Argand diagram.
$$|z| = \sqrt{x^2 + y^2}$$
Example 1: Find the modulus of \( z = 3 + 4i \).
Given: \( z = 3 + 4i \)
$$|z| = \sqrt{3^2 + 4^2}$$
$$= \sqrt{9 + 16}$$
$$= \sqrt{25}$$
Answer: 5
Example 2: Calculate the magnitude of \( z = -5 + 12i \).
Given: \( z = -5 + 12i \)
$$|z| = \sqrt{(-5)^2 + 12^2}$$
$$= \sqrt{25 + 144}$$
$$= \sqrt{169}$$
Answer: 13
Example 3: Determine the modulus of \( z = -7 - 24i \).
Given: \( z = -7 - 24i \)
$$|z| = \sqrt{(-7)^2 + (-24)^2}$$
$$= \sqrt{49 + 576}$$
$$= \sqrt{625}$$
Answer: 25
The argument of a complex number is the angle \( \theta \) it makes with the positive real axis in the Argand diagram.
$$\theta = \tan^{-1}(\frac{y}{x})$$
Example 1: Find the argument of \( z = 1 + \sqrt{3}i \).
Given: \( z = 1 + \sqrt{3}i \)
$$\theta = \tan^{-1}(\frac{\sqrt{3}}{1})$$
$$= \tan^{-1}(\sqrt{3})$$
$$= 60^\circ$$
Answer: 60^\circ
Example 2: Calculate the argument of \( z = -1 + \sqrt{3}i \) (second quadrant).
Given: \( z = -1 + \sqrt{3}i \)
$$\theta = 180^\circ - \tan^{-1}(\frac{\sqrt{3}}{1})$$
$$= 180^\circ - 60^\circ$$
$$= 120^\circ$$
Answer: 120^\circ
Example 3: Find the argument of \( z = -1 - \sqrt{3}i \) (third quadrant).
Given: \( z = -1 - \sqrt{3}i \)
$$\theta = 180^\circ + \tan^{-1}(\frac{\sqrt{3}}{1})$$
$$= 180^\circ + 60^\circ$$
$$= 240^\circ$$
Answer: 240^\circ
The conjugate of a complex number \( z = x + yi \) is \( \bar{z} = x, yi \). It reflects the point across the real axis.
$$\bar{z} = x, yi$$
Example 1: Find the conjugate of \( z = 5 + 2i \).
Given: \( z = 5 + 2i \)
$$\bar{z} = 5 - 2i$$
Answer: 5 - 2i
Example 2: Determine the conjugate of \( z = -3 - 7i \).
Given: \( z = -3 - 7i \)
$$\bar{z} = -3 + 7i$$
Answer: -3 + 7i
Example 3: If \( z = 0 + 9i \), find its conjugate.
Given: \( z = 0 + 9i \)
$$\bar{z} = 0 - 9i$$
Answer: -9i
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Create a free accountThis chapter explored the fundamental concepts of complex geometry, beginning with the identification of the real and imaginary parts of complex numbers. It introduced the Argand diagram as a graphical representation of complex numbers, highlighting how to determine the modulus or magnitude and the argument or angle of a complex number. The concept of the complex conjugate was also explained as an essential tool in complex number operations. The chapter then covered the basic operations of complex numbers, including addition, subtraction, multiplication, and division, while also detailing the conversions between polar and rectangular forms. De Moivre's theorem was presented as a powerful method for expanding complex numbers raised to powers and finding their roots. Finally, the chapter demonstrated how trigonometric identities can be derived and utilized using complex numbers, linking algebraic and geometric interpretations in engineering mathematics.
Find the real part and imaginary part of the complex number \( z = 7 - 4i \). (2 marks)
Plot the complex number \( z = -3 + 4i \) on an Argand diagram and determine its modulus. (3 marks)
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