Complex Numbers: Modulus, Argument & Loci JEE Main Guide

Complex Numbers is the most versatile chapter in JEE Main Algebra — it connects to Trigonometry (argument and De Moivre's theorem), Coordinate Geometry (loci in the Argand plane), and Polynomial theory (roots of unity). Students who understand complex numbers geometrically — as points or vectors in the plane — find most JEE questions become intuitive rather than computational. This guide builds both the algebraic and geometric perspectives you need.

Modulus, Argument, and Polar Form

For a complex number z = x+iy: modulus |z| = √(x²+y²), argument arg(z) = θ = tan⁻¹(y/x) (adjusted for quadrant). The principal argument θ ∈ (−π, π]. Polar form: z = |z|(cosθ + i sinθ) = |z|e^(iθ). Euler's formula: e^(iθ) = cosθ + i sinθ — the most beautiful equation in mathematics and one of the most useful in JEE. De Moivre's theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) = e^(inθ). This is used to find powers of complex numbers in polar form and to derive multiple-angle trigonometric identities.

Modulus properties: |z₁z₂| = |z₁||z₂|; |z₁/z₂| = |z₁|/|z₂|; |z₁+z₂| ≤ |z₁|+|z₂| (triangle inequality); ||z₁|−|z₂|| ≤ |z₁−z₂|. Argument properties: arg(z₁z₂) = arg(z₁)+arg(z₂) (mod 2π); arg(z₁/z₂) = arg(z₁)−arg(z₂). Conjugate: if z = x+iy, then z̄ = x−iy; |z|² = zz̄; z+z̄ = 2x (real part); z−z̄ = 2iy (imaginary part). Take a free Complex Numbers mock to test your modulus-argument fluency. For deeper complex number coverage, see our complex numbers guide.

Loci in the Argand Plane

The Argand plane represents z = x+iy as the point (x,y). Key loci: |z−a| = r represents a circle of radius r centred at a (where a is a fixed complex number = point in the plane). |z−a| = |z−b| represents the perpendicular bisector of the segment joining a and b. arg(z−a) = α represents the ray from a making angle α with the positive real axis. |z−a|/|z−b| = k (constant): if k ≠ 1, this is a circle (Apollonius circle); if k = 1, the perpendicular bisector. JEE tests all four locus types.

The condition for four points z₁, z₂, z₃, z₄ to be concyclic: Im[(z₃−z₁)/(z₃−z₂) · (z₄−z₂)/(z₄−z₁)] = 0 (the cross-ratio is real). Condition for three points to be collinear: (z₃−z₁)/(z₂−z₁) is real. Rotation by angle α: new position z' = z·e^(iα) = z(cosα + i sinα). If P represents z₁ and Q represents z₂, the vector from P to Q rotated by angle α about P gives z' = z₁ + (z₂−z₁)e^(iα). These rotation results are used in geometry problems. For related Argand plane geometry, see our complex plane Argand guide.

nth Roots of Unity and Their Properties

The n-th roots of unity are the n solutions to zⁿ = 1: z_k = e^(2πik/n) = cos(2πk/n) + i sin(2πk/n), for k = 0, 1, ..., n−1. They are equally spaced on the unit circle. Let ω = e^(2πi/n) be the primitive root; then all roots are 1, ω, ω², ..., ωⁿ⁻¹. Key properties: sum of all roots = 0 (for n > 1); product of all roots = (−1)ⁿ⁺¹. For the cube roots of unity (n=3): ω = e^(2πi/3), and 1+ω+ω² = 0, ω³ = 1. These properties — particularly the cube root sum — are tested as JEE shortcuts in polynomial evaluation questions.

JEE application: evaluate 1+ω+ω²+...+ωⁿ⁻¹ = 0. Use this to evaluate expressions like Σ(ωᵏ) or polynomial values at ω. If P(x) = Σaₖxᵏ, then P(1)+P(ω)+P(ω²) = 3·(a₀+a₃+a₆+...) — useful for finding sums of coefficients at positions that are multiples of 3. Roots of unity appear in several JEE integer questions. For related polynomial and algebra theory, see our De Moivre's theorem guide.

Miscellaneous Identities and JEE Shortcuts

The identity |z|² = zz̄ converts modulus-squared expressions to algebra: |z₁+z₂|² = |z₁|²+|z₂|²+2Re(z₁z̄₂) and |z₁−z₂|² = |z₁|²+|z₂|²−2Re(z₁z̄₂). These convert geometric-looking JEE questions into algebraic ones. The triangle inequality tightens to equality when z₁ and z₂ have the same argument — i.e., they point in the same direction. Geometric interpretation: this is the collinearity condition. For a complex number on the unit circle, z = e^(iθ) and z̄ = e^(−iθ) = 1/z — useful for simplifying |z| = 1 problems. For the full complex numbers topic, see our complex numbers geometry guide.