Complex Numbers – Basics
A complex number z = a + ib (a, b real)
Re(z) = a, Im(z) = b
✓ If Re(z) = 0 → purely imaginary
✓ If Im(z) = 0 → real number
Re(z) = a, Im(z) = b
✓ If Re(z) = 0 → purely imaginary
✓ If Im(z) = 0 → real number
Polar (Trigonometric) Form
z = r (cos θ + i sin θ)
where r = |z| = √(a² + b²), θ = arg(z)
where r = |z| = √(a² + b²), θ = arg(z)
General form: z = r [cos(2nπ + θ) + i sin(2nπ + θ)], n ∈ Z
Properties of Conjugate (¯z)
Basic
- z = ¯z ⇔ z is real
- z = –¯z ⇔ z is purely imaginary
- ¯¯z = z
Real & Imaginary Part
- Re(z) = (z + ¯z)/2
- Im(z) = (z – ¯z)/(2i)
Operations
- ¯(z₁ + z₂) = ¯z₁ + ¯z₂
- ¯(z₁ – z₂) = ¯z₁ – ¯z₂
- ¯(z₁z₂) = ¯z₁ ¯z₂
- ¯(z₁/z₂) = ¯z₁/¯z₂ (z₂ ≠ 0)
Properties of Modulus |z|
Key Properties
- |z| ≥ 0 ⇒ |z| = 0 iff z = 0
- –|z| ≤ Re(z) ≤ |z|, –|z| ≤ Im(z) ≤ |z|
- |z| = |¯z| = |–z|
- |z|² = z ¯z
- |z₁z₂| = |z₁||z₂| → |z₁z₂…zn| = |z₁||z₂|…|zn|
- |z₁/z₂| = |z₁|/|z₂| (z₂ ≠ 0)
- |z₁ ± z₂| ≤ |z₁| + |z₂| (Triangle inequality)
Argument of Complex Number
Quadrant-wise arg(z)
| z | Point | arg(z) |
|---|---|---|
| 1 + i | (1,1) | π/4 |
| 1 – i | (1,–1) | –π/4 |
| –1 + i | (–1,1) | 3π/4 |
| –1 – i | (–1,–1) | –3π/4 |
Properties of Argument
- arg(z₁z₂) = arg(z₁) + arg(z₂) + 2kπ (k = 0, ±1)
- arg(z₁/z₂) = arg(z₁) – arg(z₂) + 2kπ
- arg(zⁿ) = n arg(z) + 2kπ
- arg(¯z) = –arg(z)
- arg(z) = 0 ⇒ z is positive real
De Moivre’s Theorem & Cube Roots of Unity
[cos θ + i sin θ]ⁿ = cos(nθ) + i sin(nθ) (for any rational n)
Also valid for negative exponents with conjugate forms.
Cube Roots of Unity (Roots of x³ = 1)
Roots
1, ω, ω² where ω = –½ + i√3/2
ω² = –½ – i√3/2
1 + ω + ω² = 0
ω³ = 1, (ω²)³ = 1
Properties
- ω³ = 1, 1 + ω + ω² = 0
- ω³ᵏ = 1, ω³ᵏ⁺¹ = ω, ω³ᵏ⁺² = ω²
Quadratic Equation ax² + bx + c = 0 (a ≠ 0)
Discriminant D = b² – 4ac
Roots: α, β = [–b ± √D] / (2a)
Roots: α, β = [–b ± √D] / (2a)
Nature of Roots (a, b, c real)
| Condition | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots |
| D < 0 | Complex conjugate roots |
Sum & Product of Roots
α + β = –b/a
αβ = c/a
αβ = c/a
Both roots positive → sum > 0, product > 0
Both roots negative → sum < 0, product > 0
Graph & Inequations
y = ax² + bx + c = a(x + b/(2a))² + (D/(4a))
Vertex: (–b/(2a), –D/(4a))
Vertex: (–b/(2a), –D/(4a))
a > 0 → opens upwards (concave up)
a < 0 → opens downwards (concave down)
Quadratic Inequations
ax² + bx + c ≥ 0 or ≤ 0 are solved using sign chart and nature of roots.
Important Theorems
- Factor Theorem: α root ⇒ (x – α) divides f(x)
- Every odd degree polynomial has at least one real root
- Rolle’s Theorem: f(a) = f(b) ⇒ ∃ c ∈ (a,b) where f'(c) = 0
JEE Main Tips & Weightage
One of the highest weightage chapters (4–6 questions every year).
Must Remember
- All properties of conjugate, modulus, argument
- Triangle inequality |z₁ + z₂| ≤ |z₁| + |z₂|
- Cube roots of unity: 1 + ω + ω² = 0, ω³ = 1
- Nature of roots based on discriminant
- Vertex form and condition for common roots
- Argument addition/subtraction with 2kπ