Lecture Notes: Imaginary Numbers and Complex Numbers

Introduction to Imaginary Numbers

• Imaginary Numbers: Complex numbers with the imaginary unit 'i'
• Imaginary Unit (i):
  • i = √(-1)
  • i² = -1
  • i³ = -i
  • i⁴ = 1

Simplifying Powers of i

• i to the Third Power (i³):
  • i³ = i² × i = (-1) × i = -i
• i to the Fourth Power (i⁴):
  • i⁴ = i² × i² = (-1) × (-1) = 1

Simplifying Large Exponents of i

• Method: Break exponents using the highest multiple of 4
• Examples:
  • i⁷ = i⁴ × i³ = 1 × (-i) = -i
  • i²⁶ = i²⁴ × i² = 1 × (-1) = -1
  • i³³ = i³² × i = 1 × i = i
  • i⁴³ = i⁴⁰ × i³ = 1 × (-i) = -i

Arithmetic with Imaginary Numbers

Addition and Subtraction

• Example Problem: 5(2 + 3i) - 4(7 - 2i)
  • Distribute and combine like terms
  • Result: -18 + 23i

Multiplication

• Example: (5 - 2i) × (8 + 3i)
  • Use FOIL method:
    • 5 × 8 = 40
    • 5 × 3i = 15i
    • (-2i) × 8 = -16i
    • (-2i) × 3i = -6i²
  • Simplify using i² = -1:
    • Result: 46 - i

Division

• Example: (3 + 2i) / (4 - 3i)
  • Multiply by the conjugate of the denominator (4 + 3i)
  • Simplify:
    • Result: (6/25) + (17/25)i

Solving Equations with Imaginary Numbers

Solving Complex Equations

• Example: 4x + 3i = 12 - 15yi
  • Equate real and imaginary parts separately
  • Solve for x and y:
    • x = 3
    • y = -1/5

Solving Algebraic Equations

• Example: x² + 36 = 0
  • Solve for x:
    • x = ±6i

Plotting Complex Numbers and Calculating Absolute Value

Plotting

• Real axis: x-axis
• Imaginary axis: y-axis
• Example: Plot 4 + 3i
  • Move 4 units along the real axis and 3 units along the imaginary axis

Calculating Absolute Value

• Formula: |a + bi| = √(a² + b²)
• Example: |4 + 3i| = √(16 + 9) = 5

Conclusion

• Introduction to complex number operations: addition, subtraction, multiplication, division
• Solving equations and plotting complex numbers
• Finding absolute values of complex numbers