🌀 Complex Numbers Introduction

Definition

$$i = \sqrt{-1}$$

This means: $i^2 = -1$ ✨

Powers of $i$

• $i^1 = i$
• $i^2 = -1$
• $i^3 = i^2 \cdot i = -i$
• $i^4 = i^2 \cdot i^2 = 1$
• $i^5 = i$ (the pattern repeats!)

Standard Form

$$a + bi$$

• $a$ is the real part
• $b$ is the imaginary part (coefficient of $i$)

Examples:

• $3 + 4i$ (real part: 3, imaginary part: 4)
• $-2 + 5i$ (real part: -2, imaginary part: 5)
• $6i$ (real part: 0, imaginary part: 6)
• $7$ (real part: 7, imaginary part: 0) - purely real!

Example 1: Addition

$(3 + 2i) + (4 + 5i)$

Combine real parts: $3 + 4 = 7$

Combine imaginary parts: $2i + 5i = 7i$

$$= 7 + 7i$$

Example 2: Subtraction

$(6 + 8i) - (2 + 3i)$

$= (6 - 2) + (8i - 3i)$

$$= 4 + 5i$$

Example 1: $(2 + 3i)(1 + 4i)$

First: $2 \cdot 1 = 2$

Outer: $2 \cdot 4i = 8i$

Inner: $3i \cdot 1 = 3i$

Last: $3i \cdot 4i = 12i^2 = -12$

$= 2 + 8i + 3i - 12$

$$= -10 + 11i$$

Example 2: Simpler Case

$i(3 + 2i) = 3i + 2i^2 = 3i - 2 = -2 + 3i$

Why Conjugates Matter

When you multiply a complex number by its conjugate, you get a real number!

$(a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2i^2 = a^2 + b^2$

Example

$(3 + 4i)(3 - 4i)$

$= 9 - 12i + 12i - 16i^2$

$$= 9 + 16 = 25$$

Example: $\frac{2 + 3i}{1 + i}$

Step 1: Multiply by conjugate $\frac{1 - i}{1 - i}$

$$\frac{(2 + 3i)(1 - i)}{(1 + i)(1 - i)}$$

Step 2: Expand

Numerator: $2 - 2i + 3i - 3i^2 = 2 + i + 3 = 5 + i$

Denominator: $1 - i^2 = 1 + 1 = 2$

$$= \frac{5 + i}{2} = \frac{5}{2} + \frac{1}{2}i$$