Samacheer Kalvi Class 12 Mathematics
Chapter 2: Complex Numbers (Overview)

Chapter Scope: This chapter introduces the imaginary unit i, operations on complex numbers, geometric representation in the Argand plane, modulus and argument properties, polar forms, and applications of De Moivre's Theorem.

1. Definition & Basic Form

A complex number z is written in rectangular form as:

z = x + iy   (where x, y ∈ &mathbb;R and i = √-1)

• Re(z) = x (Real part)
• Im(z) = y (Imaginary part)
• Powers of i: i¹ = i, i² = -1, i³ = -i, i⁴ = 1.

2. Key Concepts & Formulas

Concept	Formula / Relation	Key Property
Conjugate (z̄)	z̄ = x - iy	z · z̄ = |z|2
Modulus (|z|)	|z| = √(x2 + y2)	|z1 z2| = |z1||z2|
Triangle Inequality	||z1| - |z2|| ≤ |z1 + z2| ≤ |z1| + |z2|	Crucial for 5-mark proofs
Square Root	√(a + ib) = ± [ √((|z|+a)/2) + i·sgn(b) √((|z|-a)/2) ]	Useful for 2-mark & 3-mark problems

3. Polar Form and Euler Form

For any complex number z = x + iy, its polar form is:

z = r (cos θ + i sin θ) = r e^iθ

• Modulus: r = |z| = √(x² + y²)
• Principal Argument (θ): θ ∈ (-π, π]
• Let α = tan^-1(|y / x|):
  • 1st Quadrant: θ = α
  • 2nd Quadrant: θ = π - α
  • 3rd Quadrant: θ = -(π - α)
  • 4th Quadrant: θ = -α

4. De Moivre's Theorem & Its Applications

For any real number n:

(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

N-th Roots of Unity

The roots of zⁿ = 1 are given by:

ω_k = e^{i (2kπ / n)} = cos(2kπ / n) + i sin(2kπ / n)   (for k = 0, 1, 2, ..., n-1)

• Sum of all n-th roots of unity = 0
• Product of all n-th roots of unity = (-1)^n-1

5. Exercise-wise Chapter Blueprint

Exercise	Core Topic	Importance for Exams
Ex 2.1 - 2.3	Powers of i, Basic Operations, Complex Properties	2-mark questions
Ex 2.4 - 2.5	Conjugates, Modulus & Distance, Square Root	3-mark & 5-mark proofs (Triangle Inequality)
Ex 2.6	Locus of a Complex Number	3-mark & 5-mark geometric loci questions
Ex 2.7	Polar Form & Principal Argument	3-mark conversion problems
Ex 2.8	De Moivre's Theorem & n-th Roots of Unity	High chance of 5-mark long questions