Complex Numbers

1. Introduction

Complex numbers extend the real numbers by introducing the imaginary unit \( i \), defined by: \[ i^2 = -1 \] A complex number \( z \) is written as: \[ z = a + i b \] where \( a, b \in \mathbb{R} \), \( a \) is the real part \( \Re(z) \), and \( b \) is the imaginary part \( \Im(z) \).

2. Set of Complex Numbers

The set of all complex numbers is denoted by \( \mathbb{C} \): \[ \mathbb{C} = \{ a + i b \mid a, b \in \mathbb{R} \} \] Real numbers are a subset of \( \mathbb{C} \) with \( b = 0 \), and purely imaginary numbers have \( a = 0 \).

3. Equality of Complex Numbers

Two complex numbers \( z_1 = a_1 + i b_1 \) and \( z_2 = a_2 + i b_2 \) are equal if and only if: \[ a_1 = a_2 \quad \text{and} \quad b_1 = b_2 \]

4. Algebra of Complex Numbers

4.1 Addition and Subtraction

\[ (a + i b) \pm (c + i d) = (a \pm c) + i (b \pm d) \]

4.2 Multiplication

\[ (a + i b)(c + i d) = (ac - bd) + i (ad + bc) \]

4.3 Conjugate

The conjugate of \( z = a + i b \) is: \[ \overline{z} = a - i b \] Properties: \[ z \overline{z} = a^2 + b^2 = |z|^2 \]

4.4 Division

For \( z_2 \neq 0 \): \[ \frac{z_1}{z_2} = \frac{z_1 \overline{z_2}}{|z_2|^2} \]

5. Modulus and Argument

The modulus (magnitude) of \( z = a + i b \) is: \[ |z| = \sqrt{a^2 + b^2} \] The argument (angle) \( \theta \) is: \[ \theta = \arg(z) = \tan^{-1}\left(\frac{b}{a}\right) \] (adjusted for the correct quadrant).

6. Polar and Exponential Form

Using Euler’s formula: \[ e^{i\theta} = \cos\theta + i\sin\theta \] we can write: \[ z = r(\cos\theta + i\sin\theta) = r e^{i\theta} \] where \( r = |z| \) and \( \theta = \arg(z) \).

7. Powers and Roots

7.1 De Moivre’s Theorem

For integer \( n \): \[ [r(\cos\theta + i\sin\theta)]^n = r^n [\cos(n\theta) + i\sin(n\theta)] \]

7.2 \(n\)-th Roots

The \( n \)-th roots of \( z = r e^{i\theta} \) are: \[ z_k = r^{1/n} e^{i(\theta + 2k\pi)/n}, \quad k = 0, 1, \dots, n-1 \]

8. Geometric Interpretation

• Complex numbers correspond to points in the complex plane: \( a \) on the real axis, \( b \) on the imaginary axis.
• Addition corresponds to vector addition.
• Multiplication corresponds to scaling by \( |w| \) and rotating by \( \arg(w) \).
• Conjugation reflects a point across the real axis.

9. Special Complex Numbers

• Imaginary unit: \( i = \sqrt{-1} \)
• Unit circle: \( |z| = 1 \) implies \( z = e^{i\theta} \)
• Roots of unity: Solutions to \( z^n = 1 \) are \( e^{2\pi i k / n} \), \( k = 0, \dots, n-1 \)

10. Applications in Engineering

• AC circuit analysis: Phasor representation of voltages and currents.
• Signal processing: Fourier transforms use \( e^{i\omega t} \) basis functions.
• Control systems: Pole-zero plots in the complex plane.
• Fluid dynamics: Complex potentials for 2D flow.

11. Worked Example

Let \( z_1 = 3 + 4i \), \( z_2 = 1 - 2i \).

Addition: \[ z_1 + z_2 = (3+1) + i(4-2) = 4 + 2i \] Multiplication: \[ z_1 z_2 = (3)(1) - (4)(-2) + i[(3)(-2) + (4)(1)] = 3 + 8 + i(-6 + 4) = 11 - 2i \] Modulus of \( z_1 \): \[ |z_1| = \sqrt{3^2 + 4^2} = 5 \] Argument of \( z_1 \): \[ \arg(z_1) = \tan^{-1}\left(\frac{4}{3}\right) \]

12. Summary

Complex numbers unify algebra and geometry, enabling elegant solutions to engineering problems involving oscillations, waves, and rotations. Mastery of their algebraic, polar, and geometric forms is essential for advanced mathematical modeling.