Complex Numbers

Complex Numbers: Basics

• A complex number is of the form z = a + ib, where a, b ∈ R and i² = −1.
• Set of complex numbers: C = {a + ib | a, b ∈ R}
• Real part: Re(z) = a, Imaginary part: Im(z) = b
• Equality: z₁ = z₂ ⇔ Re(z₁) = Re(z₂) and Im(z₁) = Im(z₂)

Algebra of Complex Numbers

Addition & Subtraction

• (a₁ + ib₁) + (a₂ + ib₂) = (a₁ + a₂) + i(b₁ + b₂)
• (a₁ + ib₁) − (a₂ + ib₂) = (a₁ − a₂) + i(b₁ − b₂)
• Additive identity: 0 = 0 + i0
• Additive inverse of z is −z

Multiplication

• (a₁ + ib₁)(a₂ + ib₂) = (a₁a₂ − b₁b₂) + i(a₁b₂ + a₂b₁)
• Multiplicative identity: 1 = 1 + i0
• Multiplication is commutative, associative, and distributive

Division

• For z ≠ 0: 1/z = (a − ib)/(a² + b²)
• Division is multiplication by the reciprocal

Conjugate of a Complex Number

• If z = a + ib, then z̄ = a − ib
• z + z̄ = 2Re(z)
• z − z̄ = 2i Im(z)
• z·z̄ = |z|²
• z is purely real ⇔ z = z̄
• z is purely imaginary ⇔ z + z̄ = 0

Modulus of a Complex Number

• |z| = √(a² + b²) = √[{Re(z)}² + {Im(z)}²]
• |z₁z₂| = |z₁||z₂|
• |z₁ / z₂| = |z₁| / |z₂|
• |z̄| = |z|
• |z₁ ± z₂| ≤ |z₁| + |z₂|

Argument of a Complex Number

• Argument (arg z) is the angle made by the line OP with positive x-axis.
• If z = x + iy and x > 0: arg z = tan⁻¹(y/x)
• Quadrant-wise adjustment is required for x < 0.
• arg(z₁z₂) = arg z₁ + arg z₂
• arg(z₁/z₂) = arg z₁ − arg z₂
• arg(zⁿ) = n arg z
• arg z = 0 ⇒ z purely real
• arg z = ±π/2 ⇒ z purely imaginary

Polar & Euler Form

• Polar form: z = r(cosθ + i sinθ), where r = |z|, θ = arg z
• Euler form: z = r e^iθ
• e^iθ = cosθ + i sinθ
• e^−iθ = cosθ − i sinθ

De Moivre’s Theorem

• (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ), n ∈ Z
• Used to find powers and roots of complex numbers

Roots of Unity

• n^th roots of unity: ω_k = e^2kπi/n, k = 0,1,2,…,n−1
• They lie on the unit circle and form a regular polygon
• Sum of all n^th roots of unity = 0
• Product of all n^th roots of unity = (−1)ⁿ⁻¹

Cube Roots of Unity

• 1, ω, ω² where ω = −1/2 + i√3/2
• 1 + ω + ω² = 0
• ω³ = 1

Geometry in Argand Plane

• Distance between points z₁, z₂: |z₁ − z₂|
• Midpoint: (z₁ + z₂)/2
• Section formula (internal): (mz₂ + nz₁)/(m + n)
• Centroid of triangle: (z₁ + z₂ + z₃)/3

Equation of Circle & Line

• Circle with centre z₀ and radius R: |z − z₀| = R
• Circle with diameter z₁z₂: |z − z₁|² + |z − z₂|² = |z₁ − z₂|²
• General line: az + āz̄ + b = 0, where b ∈ R
• Two lines parallel ⇔ slopes equal
• Two lines perpendicular ⇔ w₁ + w̄₂ = 0

JEE Main Focus Points

• Roots of unity (especially cube roots)
• Modulus–argument inequalities
• De Moivre applications
• Geometry-based loci questions
• Conversion between algebraic and polar forms