Complex Numbers: Introduction

• Complex numbers expand the real number system using the imaginary unit $i$, where $i^2 = -1$.
• Complex numbers are expressed as $a + bi$, with $a$ and $b$ being real numbers.

Key Concepts

• A complex number $z$ is defined as $z = a + bi$, where $a$ and $b$ are real numbers.
• $\text{Re}(z)$ denotes $a$, the real part of $z$.
• $\text{Im}(z)$ denotes $b$, the imaginary part of $z$.

Arithmetic Operations

• Given two complex numbers $z_1 = a + bi$ and $z_2 = c + di$:
• Addition: $z_1 + z_2 = (a + c) + (b + d)i$
• Subtraction: $z_1 - z_2 = (a - c) + (b - d)i$
• Multiplication: $z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$
• Division: $\frac{z_1}{z_2} = \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

Complex Conjugate

• The complex conjugate of $z = a + bi$ is written as $\bar{z} = a - bi$.

Modulus (Absolute Value)

• The modulus of $z = a + bi$ or absolute value, is $|z| = \sqrt{a^2 + b^2}$.

Argument

• The argument of $z = a + bi$ is the angle $\theta$, where $a = r \cos(\theta)$ and $b = r \sin(\theta)$, with $r = |z|$.
• $\theta$ can be found using the equation $\theta = \arctan\left(\frac{b}{a}\right)$.

Polar Form

• $z = a + bi$ in polar form is $z = r(\cos(\theta) + i\sin(\theta))$.
• Using Euler's formula, $z = re^{i\theta}$ where $r = |z|$ and $\theta$ is the argument of $z$.

Euler's Formula

• Connects complex exponentials to trigonometric functions via $e^{i\theta} = \cos(\theta) + i\sin(\theta)$.

De Moivre's Theorem

• For any complex number $z = r(\cos(\theta) + i\sin(\theta))$ and integer $n$, $z^n = r^n(\cos(n\theta) + i\sin(n\theta))$.

Roots of Complex Numbers

• To find the $n$-th roots of $z = r(\cos(\theta) + i\sin(\theta))$:
• $\sqrt[n]{z} = \sqrt[n]{r} \left( \cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right) \right)$ for $k = 0, 1, 2, \dots, n-1$.

Applications

• Complex numbers are used in electrical engineering.
• Complex numbers are used in quantum mechanics.
• Complex numbers are used in fluid dynamics.
• Complex numbers are used in signal processing.
• Complex numbers are used in control theory.

Examples

• For $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$:
• $z_1 + z_2 = 4 + 2i$
• $z_1 \cdot z_2 = 11 - 2i$
• $\bar{z_1} = 3 - 4i$
• $|z_1| = 5$
• To express $z = 1 + i$ in polar form:
• $r = |z| = \sqrt{2}$
• $\theta = \arctan\left(\frac{1}{1}\right) = \frac{\pi}{4}$
• $z = \sqrt{2}e^{i\frac{\pi}{4}} = \sqrt{2}\left(\cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right)\right)$