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 welding engineering mathematics, particularly when analyzing alternating current (AC) welding circuits, signal processing, and control systems. Understanding complex geometry allows welding professionals to represent and manipulate electrical parameters such as impedance and phase angles effectively. This chapter develops the skills to work with complex numbers graphically and algebraically, enabling precise calculations critical for welding equipment design and troubleshooting.
Complex geometry involves representing complex numbers visually and algebraically to facilitate calculations in welding electrical circuits. This section covers the fundamental components of complex numbers, their graphical representation on the Argand diagram, and key properties such as modulus, argument, and conjugate.
The real part of a complex number corresponds to the horizontal coordinate on the Argand diagram and represents the magnitude of the number along the real axis. For a complex number \( z = x + yi \), the real part is \( x \).
$$ \text{Real part} = \text{Re}(z) = x $$
Example 1: A complex voltage in a welding circuit is \( z = 6 + 8i \) volts. Find the real part.
Given: \( z = 6 + 8i \)
$$ \text{Re}(z) = 6 $$
Answer: 6 \text{ volts}
Example 2: The complex current is \( z = -4 + 3i \) amperes. Calculate the real part.
Given: \( z = -4 + 3i \)
$$ \text{Re}(z) = -4 $$
Answer: -4 \text{ amperes}
Example 3: Determine the real part of the complex impedance \( z = 0 + 10i \) ohms.
Given: \( z = 0 + 10i \)
$$ \text{Re}(z) = 0 $$
Answer: 0 \text{ ohms}
Example 4: Find the real part of \( z = -7 - 5i \).
Given: \( z = -7 - 5i \)
$$ \text{Re}(z) = -7 $$
Answer: -7
Example 5: For \( z = 12 \), a purely real number, find the real part.
Given: \( z = 12 \)
$$ \text{Re}(z) = 12 $$
Answer: 12
The imaginary part of a complex number is the coefficient of \( i \) and corresponds to the vertical coordinate on the Argand diagram. For \( z = x + yi \), the imaginary part is \( y \).
$$ \text{Imaginary part} = \text{Im}(z) = y $$
Example 1: For \( z = 6 + 8i \), find the imaginary part.
Given: \( z = 6 + 8i \)
$$ \text{Im}(z) = 8 $$
Answer: 8
Example 2: Find the imaginary part of \( z = -4 + 3i \).
Given: \( z = -4 + 3i \)
$$ \text{Im}(z) = 3 $$
Answer: 3
Example 3: Determine the imaginary part of \( z = 0 + 10i \).
Given: \( z = 0 + 10i \)
$$ \text{Im}(z) = 10 $$
Answer: 10
Example 4: Calculate the imaginary part of \( z = -7 - 5i \).
Given: \( z = -7 - 5i \)
$$ \text{Im}(z) = -5 $$
Answer: -5
Example 5: For the purely real number \( z = 12 \), find the imaginary part.
Given: \( z = 12 \)
$$ \text{Im}(z) = 0 $$
**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. This visual tool helps welding professionals understand the magnitude and phase relationships in AC circuits.
$$ z = x + yi \quad \Rightarrow \quad \text{point } (x,y) \text{ on the plane} $$
Example 1: Plot \( z = 3 + 4i \) on the Argand diagram and identify coordinates.
Given: \( z = 3 + 4i \)
Coordinates: \( (3,4) \)
Answer: Point at (3,4)
Example 2: Identify the point representing \( z = -5 + 0i \).
Given: \( z = -5 + 0i \)
Coordinates: \( (-5,0) \)
Answer: Point at (-5,0)
Example 3: Represent \( z = 0 - 6i \) on the Argand diagram.
Given: \( z = 0 - 6i \)
Coordinates: \( (0,-6) \)
Answer: Point at (0,-6)
Example 4: Plot \( z = -2 - 3i \).
Given: \( z = -2 - 3i \)
Coordinates: \( (-2,-3) \)
Answer: Point at (-2,-3)
Example 5: For \( z = 0 + 0i \), find the point on the Argand diagram.
Given: \( z = 0 + 0i \)
Coordinates: \( (0,0) \)
Answer: Origin
The modulus (or magnitude) of a complex number is its distance from the origin on the Argand diagram, calculated using the Pythagorean theorem. It represents the size or absolute value of the complex number.
$$ |z| = \sqrt{x^{2} + y^{2}} $$
Example 1: Find the modulus of \( z = 3 + 4i \).
Given: \( z = 3 + 4i \)
$$ |z| = \sqrt{3^{2} + 4^{2}} $$
$$ |z| = \sqrt{9 + 16} $$
$$ |z| = \sqrt{25} $$
Answer: 5
Example 2: Calculate the modulus of \( z = -5 + 12i \).
Given: \( z = -5 + 12i \)
$$ |z| = \sqrt{(-5)^{2} + 12^{2}} $$
$$ |z| = \sqrt{25 + 144} $$
$$ |z| = \sqrt{169} $$
Answer: 13
Example 3: Find the modulus of \( z = 7 - 24i \).
Given: \( z = 7 - 24i \)
$$ |z| = \sqrt{7^{2} + (-24)^{2}} $$
$$ |z| = \sqrt{49 + 576} $$
$$ |z| = \sqrt{625} $$
Answer: 25
Example 4: Calculate the modulus of \( z = 0 + 0i \).
Given: \( z = 0 + 0i \)
$$ |z| = \sqrt{0^{2} + 0^{2}} $$
$$ |z| = 0 $$
Answer: 0
Example 5: Find the modulus of \( z = -8 - 15i \).
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 formed between the positive real axis and the line segment joining the origin to the point representing the complex number on the Argand diagram. It is measured in radians or degrees.
$$ \theta = \tan^{-1} \frac{y}{x} $$
Example 1: Find the argument of \( z = 1 + \sqrt{3}i \).
Given: \( z = 1 + \sqrt{3}i \)
$$ \theta = \tan^{-1} \frac{\sqrt{3}}{1} $$
$$ \theta = \tan^{-1} (\sqrt{3}) $$
$$ \theta = 60^\circ $$
Answer: 60^\circ
Example 2: Calculate the argument of \( z = -1 + \sqrt{3}i \).
Given: \( z = -1 + \sqrt{3}i \)
$$ \theta = \tan^{-1} \frac{\sqrt{3}}{-1} $$
$$ \theta = \tan^{-1} (-\sqrt{3}) $$
Since \( x < 0 \), add 180°:
$$ \theta = 180^\circ - 60^\circ = 120^\circ $$
Answer: 120^\circ
Example 3: Find the argument of \( z = -1 - 1i \).
Given: \( z = -1 - 1i \)
$$ \theta = \tan^{-1} \frac{-1}{-1} = \tan^{-1} 1 = 45^\circ $$
Since both \( x < 0 \) and \( y < 0 \), add 180°:
$$ \theta = 180^\circ + 45^\circ = 225^\circ $$
Answer: 225^\circ
Example 4: Calculate the argument of \( z = 0 + 5i \).
Given: \( z = 0 + 5i \)
Since \( x = 0 \), the argument is:
$$ \theta = 90^\circ $$
Answer: 90^\circ
Example 5: Find the argument of \( z = 5 + 0i \).
Given: \( z = 5 + 0i \)
Since \( y = 0 \), the argument is:
$$ \theta = 0^\circ $$
Answer: 0^\circ
The conjugate of a complex number \( z = x + yi \) is \( \overline{z} = x, yi \). It reflects the point across the real axis on the Argand diagram and is used in division and simplifying expressions involving complex numbers.
$$ \overline{z} = x, yi $$
Example 1: Find the conjugate of \( z = 3 + 4i \).
Given: \( z = 3 + 4i \)
$$ \overline{z} = 3 - 4i $$
Answer: 3 - 4i
Example 2: Determine the conjugate of \( z = -5 + 12i \).
Given: \( z = -5 + 12i \)
$$ \overline{z} = -5 - 12i $$
Answer: -5 - 12i
Example 3: Find the conjugate of \( z = 7 - 24i \).
Given: \( z = 7 - 24i \)
$$ \overline{z} = 7 + 24i $$
Answer: 7 + 24i
Example 4: Calculate the conjugate of \( z = 0 + 10i \).
Given: \( z = 0 + 10i \)
$$ \overline{z} = 0 - 10i $$
Answer: -10i
Example 5: Find the conjugate of \( z = -8 - 15i \).
Given: \( z = -8 - 15i \)
$$ \overline{z} = -8 + 15i $$
Answer: -8 + 15i
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Create a free accountThis chapter explored the fundamental concepts of complex numbers starting with their geometric representation, including the real and imaginary parts, the Argand diagram, and the calculation of modulus and argument. The concept of the complex conjugate was also examined as a key tool in complex number manipulation. Operations on complex numbers such as addition, subtraction, multiplication, and division were demonstrated with step-by-step calculations. The chapter then covered the conversion between rectangular and polar forms, enabling easier computation and interpretation of complex quantities. De Moivre's theorem was introduced as a powerful method for expanding powers of complex numbers and finding their roots. Finally, the use of complex numbers to derive trigonometric identities was presented, linking algebraic and geometric interpretations in engineering mathematics. This comprehensive treatment equips learners with the skills to apply complex numbers effectively in various engineering contexts.
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