HSSLIVE Plus One Maths Chapter 5: Complex Numbers and Quadratic Equations Notes

Chapter 5: Complex Numbers and Quadratic Equations

Complex Numbers:

A complex number is expressed in the form a + bi, where:

• a is the real part
• b is the imaginary part
• i is the imaginary unit where i² = -1

Operations on Complex Numbers:

1. Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
2. Subtraction: (a + bi) – (c + di) = (a – c) + (b – d)i
3. Multiplication: (a + bi)(c + di) = (ac – bd) + (ad + bc)i
4. Division: (a + bi)/(c + di) = [(ac + bd)/(c² + d²)] + [(bc – ad)/(c² + d²)]i

Properties:

• Complex Conjugate: z = a + bi, then z̄ = a – bi
• Modulus: |z| = |a + bi| = √(a² + b²)
• Properties of Conjugates:
  • z + z̄ = 2Re(z)
  • z · z̄ = |z|²
  • z/w = z̄/w̄

Polar Form:

A complex number can be written as z = r(cos θ + i sin θ) = r∠θ where:

• r = |z| = √(a² + b²) (modulus)
• θ = tan⁻¹(b/a) (argument)

De Moivre’s Theorem:

For any complex number z = r(cos θ + i sin θ) and integer n: z^n = r^n(cos nθ + i sin nθ)

Quadratic Equations with Complex Roots:

For a quadratic equation ax² + bx + c = 0:

• The roots are x = [-b ± √(b² – 4ac)]/2a
• If b² – 4ac < 0, the roots are complex conjugates:
  • x₁ = -b/(2a) + i√(4ac – b²)/(2a)
  • x₂ = -b/(2a) – i√(4ac – b²)/(2a)

Applications:

1. Solving equations where real solutions don’t exist
2. Electrical engineering for analyzing AC circuits
3. Control systems and signal processing
4. Fractal geometry and complex dynamics