Lecture Notes: Pure Math 3 - Complex Numbers Part 3

Introduction

• Topic: Modulus and Argument of Complex Numbers
• Focus: Trigonometric and Exponential forms of complex numbers
• Notable Formula: De Moivre's Theorem (useful in examples)

Trigonometric Aspect of Complex Numbers

• Transition from Cartesian (x, y) to Polar Coordinates (r, θ)
  • Polar Coordinates:
    • r = distance from origin
    • θ = angle of rotation
• Relationship to Right Triangle:
  • Sine: opposite/hypotenuse
  • Cosine: adjacent/hypotenuse
  • Conversion to Complex Numbers:
    • Real axis = x-axis
    • Imaginary axis = y-axis
• Trigonometric Form of Complex Number:
  • r(cosθ + i sinθ) or r cis θ

Example 1: Finding Modulus and Argument

• Given: u = -√3 + i
• Find Modulus (r):
  • r = √((-√3)² + 1²) = √(3 + 1) = 2
• Find Argument (θ):
  • θ = arctan(opposite/adjacent) = arctan(1/(-√3))
  • θ = -π/6 (Check quadrant: second quadrant)
  • Final θ = 5π/6
• Result: u in modulus-argument form = 2 cis (5π/6)

Example 2: Converting from Trigonometric to Algebraic Form

• Given: Modulus = 1, Argument = π/3
• w = 1(cos(π/3) + i sin(π/3))
  • Cos(π/3) = 1/2, Sin(π/3) = √3/2
• Result: w = 1/2 + (√3/2)i

Example 3: Exponential Form and Power

• Given: u = √3 + i
• Convert to Trigonometric Form:
  • r = √(√3² + 1²) = 2, θ = arctan(1/√3) = π/6
  • Trigonometric Form: 2 cis (π/6)
• Convert to Exponential Form:
  • Exponential Form: u = 2 e^(i π/6)
• Find u⁴:
  • Using De Moivre's Theorem: (2² = 4) → 2⁴ = 16, (4 * π/6 = 2π/3)
  • Result: u⁴ = 16 cis (2π/3)

Example 4: Finding a Value for Argument

• Given: Argument of conjugate u* = π/3
• u* = (6/(a² + 4)) - (3a/(a² + 4))i
• Find a:
  • tan(θ) = opposite/adjacent
  • tan(π/3) = √3 = - (3a/6)
  • Solve for a: a = -2√3

Final Illustrations of De Moivre's Theorem

• Concept of roots in complex numbers.
  • Example: x³ = 8 → Find all roots.
  • Roots occur in complex number field, spaced 120 degrees apart.
  • Resulting roots in algebraic, trigonometric, and exponential forms discussed.

Conclusion

• Flexibility across platforms (Algebraic, Trigonometric, Exponential)
• Use appropriate method based on the problem type.
• Next Lecture: Graphing Complex Numbers on the Argand Diagram.