Complex Numbers

Concept overview

A complex number z = x + iy has real part x and imaginary part y, where i² = −1. The Argand plane represents z as a point (x, y). Modulus |z| = √(x² + y²) is the distance from origin; argument arg(z) = arctan(y/x) is the angle from positive x-axis (with sign convention for the correct quadrant).

Polar form z = r(cos θ + i sin θ) = re^(iθ) makes multiplication effortless: multiplying multiplies moduli and adds arguments. Division divides moduli and subtracts arguments. This is why complex numbers work well for rotational geometry.

De Moivre's theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). Applied in reverse, it gives the n-th roots of any complex number — n equally spaced points on a circle around the origin. The n-th roots of unity are e^(2πik/n) for k = 0, 1, ..., n − 1, sitting at vertices of a regular n-gon.

Complex-number problems often reduce to geometric arguments: |z − a| = constant is a circle; |z − a| = |z − b| is a perpendicular bisector; arg(z − a)/(z − b) = θ is an arc of a circle.

Key formulas & concepts

• Modulus: |z| = √(x² + y²); |z|² = z · z̄
• Conjugate: z̄ = x − iy; z + z̄ = 2x; z − z̄ = 2iy
• Polar / Euler form: z = re^(iθ) with r = |z|, θ = arg z
• De Moivre: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
• n-th roots of unity: ω_k = e^(2πik/n), k = 0..n−1
• Sum of n-th roots: 1 + ω + ω² + ... + ω^(n−1) = 0 (for n ≥ 2)
• |z₁ z₂| = |z₁||z₂|: and arg(z₁z₂) = arg z₁ + arg z₂

Common mistakes to avoid

• Applying arctan without checking the quadrant — arg(z) needs the correct signed angle.
• Assuming arg(z_1/z_2) = arg z_1 − arg z_2 always — modular arithmetic modulo 2π matters.
• Using z² instead of |z|² — they differ; z² = (x² − y²) + 2ixy, but |z|² = x² + y².
• Confusing De Moivre for real n vs complex n — the formula is guaranteed only for integer n.

JEE Main relevance

JEE Main asks 1–2 complex-number questions every paper — typically one modulus/argument calculation and one roots-of-unity or De Moivre application.

JEE Advanced relevance

Advanced problems disguise complex numbers as geometric locus problems: 'find the region in the Argand plane satisfying two given inequalities' — combining |z − a| < r with arg constraints.

Previous-year questions (PYQ)

Locus and roots-of-unity problems are the most repeated PYQ archetypes for this chapter.