By the end of this chapter, you will be able to:
Mastering these skills will help you handle complex calculations with confidence, making your work in engineering tasks more precise and reliable.
Complex numbers are fundamental in Industrial Mechatronics Technology for analyzing and designing systems involving alternating current circuits, control systems, and signal processing. Their use extends to representing electrical impedances, oscillations, and vibrations in machinery and electronic components. Mastery of complex geometry allows mechatronics professionals to solve practical problems involving phase angles, magnitudes, and system stability efficiently.
The real part of a complex number represents its horizontal component on the complex plane, corresponding to the actual numerical value without the imaginary unit. A complex number \(z\) is generally expressed as \(z = x + yi\), where \(x\) is the real part.
$$ \text{Real part of } z = \Re(z) = x $$
Example 1:
Given a complex number \(z = 4 + 3i\), find the real part.
Given: \(z = 4 + 3i\)$$\Re(z) = 4$$
Answer: 4
Example 2:
Given \(z = -7 + 2i\), determine the real part.
Given: \(z = -7 + 2i\)$$\Re(z) = -7$$
Answer: -7
Example 3:
For \(z = 0 + 5i\), find the real part.
Given: \(z = 0 + 5i\)$$\Re(z) = 0$$
Answer: 0
Example 4:
If \(z = 12 - 9i\), what is the real part?
Given: \(z = 12 - 9i\)$$\Re(z) = 12$$
Answer: 12
Example 5:
Find the real part of \(z = -3.5 + 0i\).
Given: \(z = -3.5 + 0i\)$$\Re(z) = -3.5$$
Answer: -3.5
The imaginary part of a complex number is the coefficient of the imaginary unit \(i\), representing the vertical component on the complex plane. For \(z = x + yi\), the imaginary part is \(y\).
$$ \text{Imaginary part of } z = \Im(z) = y $$
Example 1:
Given \(z = 6 + 8i\), find the imaginary part.
Given: \(z = 6 + 8i\)$$\Im(z) = 8$$
Answer: 8
Example 2:
For \(z = 5 - 4i\), determine the imaginary part.
Given: \(z = 5 - 4i\)$$\Im(z) = -4$$
Answer: -4
Example 3:
Find the imaginary part of \(z = -2 + 0i\).
Given: \(z = -2 + 0i\)$$\Im(z) = 0$$
Answer: 0
Example 4:
Given \(z = 0 + 11i\), what is the imaginary part?
Given: \(z = 0 + 11i\)$$\Im(z) = 11$$
Answer: 11
Example 5:
Calculate the imaginary part of \(z = -7.5 - 3.2i\).
Given: \(z = -7.5 - 3.2i\)$$\Im(z) = -3.2$$
Answer: -3.2
An Argand diagram is a graphical representation of complex numbers on a plane, with the horizontal axis representing the real part and the vertical axis the imaginary part. This visualization is essential in mechatronics for understanding system phases and signal components.
Example 1:
Plot the point for \(z = 3 + 4i\).
Given: \(z = 3 + 4i\)
Plot on horizontal axis at 3 and vertical axis at 4.
Example 2:
Plot \(z = -5 + 2i\).
Given: \(z = -5 + 2i\)
Plot at -5 on the real axis and 2 on the imaginary axis.
Example 3:
Plot \(z = 0 - 6i\).
Given: \(z = 0 - 6i\)
Plot at 0 on the real axis and -6 on the imaginary axis.
Example 4:
Plot \(z = -4 - 3i\).
Given: \(z = -4 - 3i\)
Plot at -4 on the real axis and -3 on the imaginary axis.
Example 5:
Plot \(z = 7 + 0i\).
Given: \(z = 7 + 0i\)
Plot at 7 on the real axis and 0 on the imaginary axis.
The modulus or magnitude of a complex number \(z = x + yi\) represents its distance from the origin in the Argand plane, calculated by the Pythagorean theorem.
$$ |z| = \sqrt{x^{2} + y^{2}} $$
Example 1:
Find the modulus of \(z = 3 + 4i\).
Given: \(x=3, y=4\)$$|z| = \sqrt{3^{2} + 4^{2}}$$$$|z| = \sqrt{9 + 16}$$$$|z| = \sqrt{25}$$
Answer: 5
Example 2:
Calculate the modulus of \(z = -6 + 8i\).
Given: \(x=-6, y=8\)$$|z| = \sqrt{(-6)^{2} + 8^{2}}$$$$|z| = \sqrt{36 + 64}$$$$|z| = \sqrt{100}$$
Answer: 10
Example 3:
Find the magnitude of \(z = 5 - 12i\).
Given: \(x=5, y=-12\)$$|z| = \sqrt{5^{2} + (-12)^{2}}$$$$|z| = \sqrt{25 + 144}$$$$|z| = \sqrt{169}$$
Answer: 13
Example 4:
Calculate the modulus of \(z = 0 + 7i\).
Given: \(x=0, y=7\)$$|z| = \sqrt{0^{2} + 7^{2}}$$$$|z| = \sqrt{49}$$
**Answer: 7$
Example 5:
Find the magnitude of \(z = -9 - 12i\).
Given: \(x=-9, y=-12\)$$|z| = \sqrt{(-9)^{2} + (-12)^{2}}$$$$|z| = \sqrt{81 + 144}$$$$|z| = \sqrt{225}$$
Answer: 15
The argument of a complex number \(z = x + yi\) is the angle \(\theta\) between the positive real axis and the line joining the origin to the point \((x, y)\) in the Argand plane. It is calculated using the arctangent function.
$$ \theta = \tan^{-1} \frac{y}{x} $$
Example 1:
Find the argument of \(z = 1 + \sqrt{3}i\).
Given: \(x=1, y=\sqrt{3}\)$$\theta = \tan^{-1} \frac{\sqrt{3}}{1}$$$$\theta = \tan^{-1} (\sqrt{3})$$$$\theta = 60^\circ$$
Answer: 60 degrees
Example 2:
Calculate the argument of \(z = -1 + \sqrt{3}i\).
Given: \(x=-1, y=\sqrt{3}\)$$\theta = \tan^{-1} \frac{\sqrt{3}}{-1}$$$$\theta = \tan^{-1} (-\sqrt{3})$$
Since \(x < 0\), add 180° to get the principal value:$$\theta = 180^\circ - 60^\circ = 120^\circ$$
Answer: 120 degrees
Example 3:
Find the argument of \(z = -1 - 1i\).
Given: \(x=-1, y=-1\)$$\theta = \tan^{-1} \frac{-1}{-1} = \tan^{-1} (1) = 45^\circ$$
Since both \(x<0\) and \(y<0\), add 180°:$$\theta = 180^\circ + 45^\circ = 225^\circ$$
Answer: 225 degrees
Example 4:
Calculate the argument of \(z = 0 + 2i\).
Given: \(x=0, y=2\)
Since \(x=0\), \(\theta = 90^\circ\) if \(y > 0\)
Answer: 90 degrees
Example 5:
Find the argument of \(z = 0 - 3i\).
Given: \(x=0, y=-3\)
Since \(x=0\), \(\theta = 270^\circ\) if \(y < 0\)
Answer: 270 degrees
The conjugate of a complex number \(z = x + yi\) is \( \bar{z} = x, yi \), reflecting the number across the real axis on the Argand diagram. Conjugates are widely used in simplifying division and finding magnitudes in mechatronics calculations.
$$ \bar{z} = x, yi $$
Example 1:
Find the conjugate of \(z = 5 + 7i\).
Given: \(z = 5 + 7i\)$$\bar{z} = 5 - 7i$$
Answer: 5 - 7i
Example 2:
Calculate the conjugate of \(z = -3 - 4i\).
Given: \(z = -3 - 4i\)$$\bar{z} = -3 + 4i$$
Answer: -3 + 4i
Example 3:
Find the conjugate of \(z = 0 + 9i\).
Given: \(z = 0 + 9i\)$$\bar{z} = 0 - 9i$$
Answer: -9i
Example 4:
Calculate the conjugate of \(z = 8 + 0i\).
Given: \(z = 8 + 0i\)$$\bar{z} = 8 - 0i = 8$$
Answer: 8
Example 5:
Find the conjugate of \(z = -6 + 2.5i\).
Given: \(z = -6 + 2.5i\)$$\bar{z} = -6 - 2.5i$$
Answer: -6 - 2.5i
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Create a free accountThis chapter introduced the geometric representation of complex numbers, covering their real and imaginary parts and how these are plotted on the Argand diagram. It explained how to determine the modulus, which gives the magnitude of a complex number, and the argument, which represents the angle it makes with the positive real axis. The concept of the complex conjugate was also discussed as a reflection across the real axis. The chapter then explored the fundamental operations on complex numbers, including addition, subtraction, multiplication, and division, emphasizing the importance of proper handling in both rectangular and polar forms. Methods to convert between rectangular and polar forms were detailed to facilitate these operations. De Moivre’s theorem was presented as a powerful tool for expanding complex numbers raised to powers and extracting roots, linking complex numbers to trigonometric identities. This connection highlights the utility of complex numbers in solving engineering problems involving oscillations, waves, and rotations.
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 calculate its modulus. (3 marks)
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