Learning Objectives
- Understand complex numbers and their algebraic operations
- Learn the modulus, argument, and polar form of complex numbers
- Study the Argand plane and geometric representation
- Solve quadratic equations with complex roots
Key Concepts
Complex Numbers
A complex number z = a + ib, where a is the real part (Re(z)) and b is the imaginary part (Im(z)), and i = √(-1), so i2 = -1. Powers of i: i1 = i, i2 = -1, i3 = -i, i4 = 1 (cycle of 4). Two complex numbers are equal iff their real and imaginary parts are equal.
Algebra of Complex Numbers
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) = multiply numerator and denominator by conjugate (c - id).
Conjugate: z̄ = a - ib. Properties: z · z̄ = |z|2 = a2 + b2. z₁ + z₂ = z̄₁ + z̄₂. z₁ · z₂ = z̄₁ · z̄₂.
Modulus: |z| = √(a2 + b2). Properties: |z₁z₂| = |z₁||z₂|; |z₁/z₂| = |z₁|/|z₂|; |z₁ + z₂| ≤ |z₁| + |z₂| (triangle inequality).
Argand Plane and Polar Form
Argand plane: Complex number a + ib represented as point (a, b). x-axis = real axis, y-axis = imaginary axis. Argument (arg z): Angle θ that the line from origin to z makes with positive real axis. tan θ = b/a. Principal argument: θ ∈ (-π, π]. Polar form: z = r(cos θ + i sin θ) where r = |z| and θ = arg(z).
Euler's form (JEE): z = reiθ where eiθ = cos θ + i sin θ.
Quadratic Equations
For ax2 + bx + c = 0 (a ≠ 0): Discriminant D = b2 - 4ac. If D > 0: two distinct real roots. If D = 0: two equal real roots. If D < 0: two complex conjugate roots. Roots: x = (-b ± √D)/(2a). When D < 0, roots are x = (-b ± i√|D|)/(2a). Sum of roots = -b/a; Product of roots = c/a.
Square Root of Complex Number
To find √(a + ib): Let √(a + ib) = x + iy. Then a = x2 - y2, b = 2xy. Solve these equations to find x and y.
Summary
Complex numbers extend the real number system with the imaginary unit i. They can be represented algebraically (a + ib), geometrically (Argand plane), or in polar form r(cosθ + isinθ). Quadratic equations with negative discriminant have complex conjugate roots.
Important Terms
- Imaginary unit: i = √(-1); i2 = -1
- Modulus: |z| = √(a2 + b2) — distance from origin
- Argument: Angle made by z with positive real axis
- Conjugate: z̄ = a - ib; reflection about real axis
- Polar form: z = r(cosθ + isinθ)
- Discriminant: D = b2 - 4ac; determines nature of roots
Quick Revision
- i2 = -1; i3 = -i; i4 = 1; cycle repeats every 4
- z · z̄ = |z|2
- |z₁ + z₂| ≤ |z₁| + |z₂| (triangle inequality)
- Polar form: z = r(cosθ + isinθ); r = |z|, θ = arg(z)
- D < 0: complex conjugate roots; D = 0: equal real roots; D > 0: distinct real roots
- Sum of roots = -b/a; Product = c/a
- Principal argument ∈ (-π, π]