Study Notes for Complex Numbers

A complex number is defined as:
- $z = x + iy$ , where:
- $x$ (real part) and $y$ (imaginary part) are both real numbers, denoted as $x, y \in \mathbb{R}$ .
The imaginary unit is denoted by $i$ , where:
- Definition of i: $i^2 = -1$

Real Numbers: Include integers, fractions, and irrational numbers.
Natural Numbers: $1, 2, 3, …$
Whole Numbers: $0, 1, 2, 3, …$
Integers: $…, -1, 0, 1, …$
Rational Numbers: Numbers that can be expressed as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ .
Irrational Numbers: Numbers that cannot be expressed as a fraction. Examples include $\sqrt{10}$ , $\pi$ .
Imaginary Numbers: Numbers of the form $bi$ where $b \in \mathbb{R}$ , such as $-\sqrt{8}$ and $2i$ .
Pure Imaginary Numbers: Numbers that are completely imaginary, such as $11i$ and $-i$ .

Math 204: Complex Numbers
Weekly Content Schedule:
- 1 (23/2/2026): Complex Numbers
- 2 (2/3/2026): Vector and Matrix Algebra
- 3 (9/3/2026): Vector and Matrix Algebra
- 4 (16/3/2026): Vector and Matrix Algebra
- 5 (24/3/2026): System of Linear Equations
- 6 (31/3/2026): System of Linear Equations
- 7 (7/4/2026): Midterm Exam (30%)
- 8 (14/4/2026): Vector Space
- 9 (21/4/2026): Vector Space
- 10 (28/4/2026): Four Fundamental Subspaces
- 11 (5/5/2026): Linear Transformation
- 12 (12/5/2026): Eigenvalues and Eigenvectors
- 13 (19/5/2026): Cayley-Hamilton Theorem
- 14 (1/6/2026): Final Exam (40%)

Argand Diagram: A graphical representation where:
- The horizontal axis (x-axis) represents the real part of the complex number.
- The vertical axis (y-axis) represents the imaginary part of the complex number.

The addition of two complex numbers is defined as:
- $z1 + z2 = (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2)$
Properties:
- Commutative: $z1 + z2 = z2 + z1$
- Associative: $z1 + (z2 + z3) = (z1 + z2) + z3$
- The complex number $z = x + iy$ is zero if and only if $x = 0$ and $y = 0$ .
- Equality: $z1 = z2$ if and only if $x1 = x2$ and $y1 = y2$ .

If $\alpha$ is a scalar, then:
- $\alpha z1 = \alpha (x1 + iy1) = \alpha x1 + i(\alpha y_1)$
Example: If $Z = 1 + 2i$ , then:
- For $2z$ , $2Z = 2 + 4i$ .
- For $\frac{1}{2} Z$ , it's $\frac{1}{2} + i$ .

The subtraction of two complex numbers is defined as:
- $z1 - z2 = (x1 + iy1) - (x2 + iy2) = (x1 - x2) + i(y1 - y2)$

The product of two complex numbers is:
- $z1 z2 = (x1 + iy1)(x2 + iy2) = x1x2 - y1y2 + i(x1y2 + y2x1)$
Properties:
- Commutative: $z1 z2 = z2 z1$
- Associative: $z1(z2 z3) = (z1 z2)z3$
- Distributive: $z1(z2 + z3) = z1 z2 + z1 z_3$

The complex conjugate of $z = x + iy$ is defined as:
- $\overline{z} = x - iy$
Properties:
- $\text{Re}(z) = \frac{z + \overline{z}}{2}$
- $\text{Im}(z) = \frac{z - \overline{z}}{2i}$

The modulus of a complex number, which indicates its distance from the origin, is defined as:
- $|z| = \sqrt{x^2 + y^2}$
Properties:
- $|z| \geq 0$ ; $|z| = 0$ if and only if $z = 0$ .
- The geometrical interpretation of $z1 - z2$ is the distance between the two complex numbers.

Given: $z = \frac{1 + 2i}{4 + 3i}$
Step 1: Multiply by the conjugate:
- $\frac{(1 + 2i)(4 - 3i)}{(4 + 3i)(4 - 3i)} = \frac{4 + 3 + 5i}{16 + 9}$
Step 2: Simplify:
- $\frac{7 + 5i}{25}$
- $\text{Re}(z) = \frac{7}{25}$ and $\text{Im}(z) = \frac{5}{25} = \frac{1}{5}$
- $|z|^2 = \left(\frac{7}{25}\right)^2 + \left(\frac{1}{5}\right)^2$

Definition: It is the angle from the positive real axis to the radius line, calculated as:
- $\theta = \text{arg}(z)$
Calculation Using Tangent:
- $\tan(\theta) = \frac{y}{x}$
The principal value of $\text{arg}(z)$ , denoted as $\text{Arg}(z)$ , satisfies:
- -\pi < \text{Arg}(z) \leq \pi .
For negative $z$ , $\text{Arg}(z)$ has the value $\pi$ , not $-\pi$ .

For a complex number $z = x + iy$ , we can express it in polar coordinates as:
- $z = r(cos \theta + i sin \theta)$ , where:
- $r = |z| = \sqrt{x^2 + y^2}$
- $\theta = \text{arg}(z)$
Euler’s Formula:
- $e^{i\theta} = \cos(\theta) + i \sin(\theta)$
Exponential Form:
- $z = re^{i\theta}$

Multiplication:
- $z1 z2 = r1e^{i\theta1} r2e^{i\theta2} = r1 r2 e^{i(\theta1 + \theta2)}$
Division:
- $\frac{z1}{z2} = \frac{r1}{r2}e^{i(\theta1 - \theta2)}$
Powers:
- $z^n = r^n e^{in\theta}$
De Moivre’s Theorem:
- For a complex number where $r = 1$ , then:
- $z^n = \cos(n\theta) + i\sin(n\theta)$

Definition: A number $w$ is called an $n$ th root of a complex number $z$ if $w^n = z$ .
Written as:
- $w = z^{1/n}$
Using De Moivre's theorem:
- If $n$ is a positive integer:
- $z^{1/n} = r^{1/n}e^{i(\theta + 2k\pi)/n}, k = 0, 1, …, n-1$
- There are $n$ different values for $z^{1/n}$ if $z \neq 0$ .
Example of solving for $z^3 - 1 = 0$ :
- $z^3 = e^{i2k\pi}$
- $z = e^{i(2k\pi)/3}, k = 0, 1, 2$
- Specifically:
- For $k=0$ : $z_0 = 1$
- For $k=1$ : $z_1 = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$
- For $k=2$ : $z_2 = -\frac{1}{2} - \frac{\sqrt{3}}{2}i$