AlgebraVery High Yield (2-3 Qs in JEE)

Complex Numbers Formulas

Argand plane, polar & Euler forms, triangle inequalities, cube roots of unity, De Moivre's theorem, and geometric loci.

8 Formulas•658+ Questions in Bank

Core Formulas & Mathematical Identities

z = x + iy = r(\cos\theta + i\sin\theta) = r e^{i\theta} \quad (r = |z| = \sqrt{x^2 + y^2}, \; \theta = \arg(z))

Exponential Euler representation for complex numbers.

Examiner Tip: e^(iπ) + 1 = 0 (Euler's identity).

||z_1| - |z_2|| \le |z_1 \pm z_2| \le |z_1| + |z_2|

Bounds for modulus of sum and difference of complex numbers.

Examiner Tip: Equality holds when origin, z₁, and z₂ are collinear.

1 + \omega + \omega^2 = 0, \quad \omega^3 = 1, \quad \omega = \frac{-1 + i\sqrt{3}}{2} = e^{i 2\pi/3}

Roots of equation z³ - 1 = 0.

Examiner Tip: 1 + ω^n + ω^(2n) = 3 if n is multiple of 3, else 0.

(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) = e^{in\theta}

Powers and roots of trigonometric complex forms for integer and rational n.

Examiner Tip: Always convert to polar form before taking high powers like z²⁰²⁴.

|z - z_0| = r \; (\text{Circle}), \quad |z - z_1| = |z - z_2| \; (\text{Perpendicular Bisector}), \quad \arg\left(\frac{z - z_1}{z - z_2}\right) = \theta \; (\text{Circular Arc})

Translating complex equations into 2D Cartesian geometric figures.

Examiner Tip: If θ = ±π/2, the locus is a circle with z₁ and z₂ as diametric endpoints.

\frac{z_3 - z_1}{z_2 - z_1} = \frac{|z_3 - z_1|}{|z_2 - z_1|} e^{i\alpha}

Rotates directed line segment (z₂ - z₁) by an angle α counter-clockwise around pivot point z₁ to obtain direction of (z₃ - z₁).

Examiner Tip: For an equilateral triangle with vertices z₁, z₂, z₃: z₁² + z₂² + z₃² = z₁z₂ + z₂z₃ + z₃z₁.

\frac{|z - z_1|}{|z - z_2|} = k \; (k \ne 1 \implies \text{Circle}), \quad |z - z_1| + |z - z_2| = 2a \; (2a > |z_1 - z_2| \implies \text{Ellipse})

Modulus ratio represents an Apollonius circle, while sum and difference of distances represent ellipses and hyperbolas.

Examiner Tip: If |z - z₁| + |z - z₂| = |z₁ - z₂|, locus is the line segment joining z₁ and z₂.

z^n = 1 \implies z_k = e^{i \frac{2k\pi}{n}} \quad (k = 0, 1, \dots, n-1), \quad \sum_{k=0}^{n-1} z_k = 0, \quad \prod_{k=0}^{n-1} z_k = (-1)^{n-1}

The n-th roots of unity form vertices of a regular n-gon inscribed in unit circle centered at origin.

Examiner Tip: Factorization identity: x^n - 1 = (x - 1)(x - α)(x - α²) ... (x - α^(n-1)).

Frequently tested misconceptions in Complex Numbers

•Forgetting that √(a)*√(b) = √(ab) is valid ONLY when at least one of a or b is non-negative.
•Principal argument must lie in (-π, π].

4 Topics

›A complex number is defined as z = a + ib where a, b ∈ ℝ and i = √(-1). a is the Real part Re(z) and b is the Imaginary part Im(z).
›Integral powers of i: i⁰ = 1, i¹ = i, i² = -1, i³ = -i, i⁴ = 1. In general, i^(4k) = 1, i^(4k+1) = i, i^(4k+2) = -1, i^(4k+3) = -i for any integer k.
›Crucial identity: The sum of any four consecutive integral powers of i is always zero: i^n + i^(n+1) + i^(n+2) + i^(n+3) = 0.
›Equality of complex numbers: z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂ are equal if and only if a₁ = a₂ AND b₁ = b₂. Complex numbers cannot be compared using inequalities (> or <) unless their imaginary parts are zero.

›Every complex number z = x + iy corresponds to a unique point P(x, y) in the Argand plane. The modulus |z| = √(x² + y²) represents the Euclidean distance of P from the origin.
›Conjugate: z̄ = x - iy is the mirror reflection of z about the real axis (x-axis). Important properties: z · z̄ = |z|², z + z̄ = 2 Re(z), z - z̄ = 2i Im(z).
›Conjugate distributes over all basic operations: (z₁ ± z₂)̄ = z̄₁ ± z̄₂, (z₁ · z₂)̄ = z̄₁ · z̄₂, and (z₁/z₂)̄ = z̄₁/z̄₂ (z₂ ≠ 0).
›Triangle Inequalities: ||z₁| - |z₂|| ≤ |z₁ + z₂| ≤ |z₁| + |z₂|. Maximum value of |z₁ + z₂| is |z₁| + |z₂| (occurs when origin, z₁, and z₂ are collinear with z₁ and z₂ on the same side of origin: arg(z₁) = arg(z₂)).

›Argument θ is the angle made by vector OP with positive real axis: tan α = |y/x| (where α is acute angle).
›Principal Argument Arg(z) strictly lies in (-π, π]: Quad I: θ = α; Quad II: θ = π - α; Quad III: θ = -(π - α); Quad IV: θ = -α.
›Logarithmic properties of arguments: arg(z₁ · z₂) = arg(z₁) + arg(z₂) + 2kπ; arg(z₁/z₂) = arg(z₁) - arg(z₂) + 2kπ (where k ∈ {-1, 0, 1} adjusts into principal range).

›Roots of z³ = 1 are z = 1, ω = (-1 + i√3)/2 = e^(i 2π/3), and ω² = (-1 - i√3)/2 = e^(i 4π/3).
›Fundamental properties: 1 + ω + ω² = 0, and ω³ = 1 (ω^(3k) = 1, ω^(3k+1) = ω, ω^(3k+2) = ω²).
›Conjugate relationship: ω̄ = ω² and (ω²)̄ = ω.
›Factorization identities: a³ - b³ = (a - b)(a - ωb)(a - ω²b); a³ + b³ = (a + b)(a + ωb)(a + ω²b); a³ + b³ + c³ - 3abc = (a + b + c)(a + ωb + ω²c)(a + ω²b + ωc).