Polar Form and Applications

Complex Numbers: The Polar Form Adventure 🧭

Welcome to the World of Complex Numbers!

Imagine you’re a treasure hunter with a special map. On this map, every treasure location needs TWO pieces of information:

1. How far is the treasure from you?
2. Which direction should you walk?

That’s exactly what the polar form of complex numbers is all about! Instead of saying “go 3 steps right and 4 steps up” (rectangular form), we say “walk 5 steps in THAT direction” (polar form).

📍 The Modulus: How Far is the Treasure?

What is Modulus?

Think of a complex number like z = a + bi as a point on a map.

The modulus (written as |z| or r) tells you: “How far is this point from where you’re standing (the origin)?”

The Magic Formula

|z| = √(a² + b²)

It’s just the Pythagorean theorem! Remember the triangle with sides a, b, and hypotenuse c? The modulus is that hypotenuse!

Simple Example

Find the modulus of z = 3 + 4i

• a = 3 (horizontal distance)
• b = 4 (vertical distance)
• |z| = √(3² + 4²) = √(9 + 16) = √25 = 5

The treasure is 5 units away from you!

graph TD
    A["Start at Origin 0,0"] --> B["Move 3 units right"]
    B --> C["Move 4 units up"]
    C --> D["Distance = √3²+4² = 5"]

Another Example

Find |z| for z = -5 + 12i

|z| = √((-5)² + 12²) = √(25 + 144) = √169 = 13

🧭 The Argument: Which Direction?

What is Argument?

The argument (written as arg(z) or θ) is the angle you need to turn to face the treasure!

• Start facing right (positive x-axis)
• Rotate counterclockwise until you’re pointing at your complex number
• That angle is the argument!

The Magic Formula

θ = tan⁻¹(b/a)

But wait! This formula gives the basic angle. You need to check which quadrant your point is in:

Simple Example

Find the argument of z = 1 + i

• a = 1, b = 1 (both positive = Quadrant I)
• θ = tan⁻¹(1/1) = tan⁻¹(1) = π/4 (or 45°)

You turn 45 degrees counterclockwise!

Another Example

Find arg(z) for z = -1 + √3i

• a = -1, b = √3 (Quadrant II)
• Basic angle: tan⁻¹(√3/1) = π/3
• Since Quadrant II: θ = π - π/3 = 2π/3 (or 120°)

🌟 Polar Form: The Complete Treasure Map

What is Polar Form?

Now we combine both pieces! Instead of writing z = a + bi, we write:

z = r(cos θ + i sin θ)

Or the super-cool short form:

z = r · cis(θ)

Where “cis” means “cos + i·sin”

Step-by-Step Conversion

Convert z = 1 + √3i to polar form

Step 1: Find r (modulus) r = √(1² + (√3)²) = √(1 + 3) = √4 = 2

Step 2: Find θ (argument) θ = tan⁻¹(√3/1) = π/3 (Quadrant I, so no adjustment)

Step 3: Write polar form z = 2(cos π/3 + i sin π/3) = 2 cis(π/3)

graph TD
    A["z = 1 + √3i"] --> B["Find r = 2"]
    A --> C["Find θ = π/3"]
    B --> D["Polar: 2 cis π/3"]
    C --> D

Why is Polar Form Awesome?

Multiplication becomes super easy!

If z₁ = r₁ cis(θ₁) and z₂ = r₂ cis(θ₂), then:

z₁ × z₂ = r₁r₂ cis(θ₁ + θ₂)

Just multiply the distances and add the angles! Magic! ✨

👑 De Moivre’s Theorem: The Power Move

The Big Idea

Abraham de Moivre discovered something amazing: raising a complex number to a power becomes simple in polar form!

[r cis(θ)]ⁿ = rⁿ cis(nθ)

Translation: Raise the distance to the power, and multiply the angle by n!

Simple Example

Find (1 + i)⁸

Step 1: Convert to polar

• r = √(1² + 1²) = √2
• θ = π/4
• So 1 + i = √2 cis(π/4)

Step 2: Apply De Moivre (√2 cis(π/4))⁸ = (√2)⁸ cis(8 × π/4)

Step 3: Calculate

• (√2)⁸ = (2^(1/2))⁸ = 2⁴ = 16
• 8 × π/4 = 2π

Step 4: Convert back 16 cis(2π) = 16(cos 2π + i sin 2π) = 16(1 + 0i) = 16

Wow! A complex number raised to the 8th power gave us a real number!

Finding Roots with De Moivre

We can also find nth roots! If zⁿ = w, then:

z = ⁿ√r cis((θ + 2πk)/n)

Where k = 0, 1, 2, …, n-1 gives us all n roots!

🔮 Cube Roots of Unity: The Three Musketeers

What are They?

The cube roots of unity are the three solutions to:

z³ = 1

These three special numbers, when cubed, all give 1!

Finding Them

Start with 1 = 1 cis(0) = 1 cis(2π) = 1 cis(4π)…

Using De Moivre in reverse:

For k = 0: z₀ = 1 cis(0/3) = cis(0) = 1

For k = 1: z₁ = 1 cis(2π/3) = cos(2π/3) + i sin(2π/3) = -1/2 + (√3/2)i

For k = 2: z₂ = 1 cis(4π/3) = cos(4π/3) + i sin(4π/3) = -1/2 - (√3/2)i

Meet the Special Symbol: ω (omega)

We call ω = -1/2 + (√3/2)i one of these special roots.

The three cube roots of unity are:

• 1 (the obvious one)
• ω = -1/2 + (√3/2)i
• ω² = -1/2 - (√3/2)i

graph TD
    A["Cube Roots of Unity"] --> B["ω⁰ = 1"]
    A --> C["ω¹ = -½ + √3/2 i"]
    A --> D["ω² = -½ - √3/2 i"]
    B --> E["Forms equilateral triangle!"]
    C --> E
    D --> E

Magic Properties of ω

Property 1: They sum to zero!

1 + ω + ω² = 0

Property 2: ω³ = 1

Property 3: ω² = ω̄ (conjugate)

Property 4: They form an equilateral triangle on the unit circle!

Verification Example

Let’s verify 1 + ω + ω² = 0:

1 + (-1/2 + √3/2 i) + (-1/2 - √3/2 i) = 1 - 1/2 - 1/2 + √3/2 i - √3/2 i = 1 - 1 + 0 = 0 ✓

🎯 Quick Summary

🚀 You Did It!

You’ve just learned how to see complex numbers as arrows instead of just points! Now you can:

• Find how far any complex number is from the origin (modulus)
• Find what direction it points (argument)
• Write it in the cool polar form
• Raise it to ANY power using De Moivre’s theorem
• Understand the three magical cube roots of unity

Remember: Complex numbers aren’t complicated—they’re just points on a treasure map with distance and direction!