Complex Numbers and Quadratic Equations
4.1 Introduction
Complex numbers extend the real number system to solve equations that have no real solutions (e.g., x² + 1 = 0).
Key Concepts:
- Definition of imaginary unit: i = √(-1) where i² = -1
- Standard form: z = a + ib (a = real part, b = imaginary part)
4.2 Complex Numbers
Definition: A complex number is an ordered pair (a, b) where a, b ∈ ℝ.
Representations:
- Rectangular form: z = a + ib
- Geometric interpretation: Point in complex plane
Example:
3 + 4i is a complex number with Re(z) = 3, Im(z) = 4
4.3 Algebra of Complex Numbers
Operations:
| Operation | Formula |
|---|---|
| Addition | (a + ib) + (c + id) = (a + c) + i(b + d) |
| Subtraction | (a + ib) - (c + id) = (a - c) + i(b - d) |
| Multiplication | (a + ib)(c + id) = (ac - bd) + i(ad + bc) |
| Division | (a + ib)/(c + id) = [(ac + bd) + i(bc - ad)]/(c² + d²) |
Properties:
- Commutative, Associative, and Distributive laws hold
- Additive identity: 0 + i0
- Multiplicative identity: 1 + i0
4.4 The Modulus and the Conjugate of a Complex Number
Modulus (Absolute Value):
|z| = |a + ib| = √(a² + b²)
Conjugate:
z̄ = a - ib
Properties:
- z + z̄ = 2Re(z)
- z - z̄ = 2iIm(z)
- z·z̄ = |z|²
- |z₁z₂| = |z₁||z₂|
4.5 Argand Plane and Polar Representation
Argand Plane: Geometric representation where:
- X-axis represents real part
- Y-axis represents imaginary part
[Argand Diagram: Point (a,b) representing z = a + ib]
Polar Form:
z = r(cosθ + isinθ) = re^(iθ)
Where:
- r = |z| = √(a² + b²) (modulus)
- θ = arg(z) = tan⁻¹(b/a) (argument)
Euler's Formula:
e^(iθ) = cosθ + isinθ
Example:
Convert 1 + i to polar form:
r = √(1² + 1²) = √2
θ = tan⁻¹(1/1) = π/4
Polar form: √2(cos(π/4) + isin(π/4)) or √2e^(iπ/4)
Key Applications:
- Solving quadratic equations with negative discriminants
- Electrical engineering (AC circuit analysis)
- Signal processing
- Quantum mechanics