Complex Numbers for JEE: Key Formulas, Argument Tricks and Solved Problems

Complex Numbers gives 1–2 questions in JEE Main every session and is a favourite for tricky Advanced problems. The theory is small; the tricks matter. Here they are.

Basics

• z = x + iy, i² = −1; i⁴ⁿ = 1, i⁴ⁿ⁺¹ = i, i⁴ⁿ⁺² = −1, i⁴ⁿ⁺³ = −i.
• Conjugate z̄ = x − iy; z z̄ = |z|².
• Modulus |z| = √(x² + y²); argument θ = tan⁻¹(y/x) adjusted for quadrant, principal value in (−π, π].
• Polar: z = r(cos θ + i sin θ) = r e^iθ.

Argument tricks

z	arg z
−1 + i	3π/4 (2nd quadrant)
−1 − i	−3π/4
1 − i	−π/4
i	π/2
−5 (negative real)	π

arg(z₁z₂) = arg z₁ + arg z₂; arg(z₁/z₂) = arg z₁ − arg z₂ (adjust to principal value).

Important results

• |z₁z₂| = |z₁||z₂|; |z₁ + z₂| ≤ |z₁| + |z₂| (triangle inequality).
• |z₁ + z₂|² + |z₁ − z₂|² = 2(|z₁|² + |z₂|²).
• De Moivre: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ.
• nth roots of unity: e^2πik/n, k = 0…n−1; their sum is 0.

Cube roots of unity (ω)

ω = (−1 + i√3)/2, ω² = (−1 − i√3)/2. Key: 1 + ω + ω² = 0 and ω³ = 1. Any power: reduce mod 3. Example: 1 + ω⁵⁰ + ω¹⁰⁰ = 1 + ω² + ω = 0.

Factorisation: a³ + b³ = (a + b)(a + bω)(a + bω²); a² + ab + b² = (a − bω)(a − bω²).

Geometry in the Argand plane

• |z − z₀| = r → circle centre z₀, radius r.
• |z − z₁| = |z − z₂| → perpendicular bisector.
• arg((z − z₁)/(z − z₂)) = α → arc of a circle (α = π/2 gives a semicircle on z₁z₂ as diameter).
• Rotation: to rotate z₁ about z₀ by angle θ: z = z₀ + (z₁ − z₀)e^iθ.

Solved problems

1. Find (1 + i)¹⁰. 1 + i = √2 e^iπ/4 → (√2)¹⁰ e^i10π/4 = 32 e^i5π/2 = 32(cos π/2 + i sin π/2) = 32i.

2. If |z − 3i| = 2, find max |z|. Circle centre (0, 3) radius 2 → max distance from origin = 3 + 2 = 5.

3. If z + 1/z = 1, find z¹⁰⁰ + 1/z¹⁰⁰. z² − z + 1 = 0 → z = −ω or −ω². z¹⁰⁰ = ω¹⁰⁰ = ω; 1/z¹⁰⁰ = ω². Sum = ω + ω² = −1.

Common traps

• √(−4) × √(−9) = 2i × 3i = −6, not +6.
• arg is not simply tan⁻¹(y/x) in the 2nd and 3rd quadrants.
• Inequalities like z₁ > z₂ have no meaning for non-real complex numbers.

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Frequently asked questions

What is the principal argument?

The value of arg z in the interval (−π, π].

How many questions come from complex numbers in JEE Main?

Usually 1 to 2 questions per paper.

What is the sum of the cube roots of unity?

Zero: 1 + ω + ω² = 0.