Complex Numbers Class 11 Notes for IIT JEE

A complex number is a number of the form z = a + ib, where a is the real part, b is the imaginary part, and i = √(−1). Complex numbers extend the real number system to include solutions of equations like x² + 1 = 0. They form the basis for many advanced topics in IIT JEE Mathematics, including geometry in the Argand plane and polynomial theory.

The most important formulas are: modulus |z| = √(a² + b²), argument tan θ = b/a, conjugate z̄ = a − ib, polar form z = r(cosθ + i sinθ), De Moivre's theorem (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ), and the cube root properties 1 + ω + ω² = 0 and ω³ = 1. These appear in 2–3 JEE Main questions annually.

The modulus |z| is the distance of z from the origin on the Argand plane, calculated as √(a² + b²). The argument θ is the angle that the line from origin to z makes with the positive real axis, found using tan θ = b/a. Modulus is always a non-negative real number; argument is an angle, typically expressed in radians in the range (−π, π].

De Moivre's theorem is not explicitly part of the standard CBSE Class 11 board exam syllabus, but it is an important topic for IIT JEE preparation. It is covered under the JEE Advanced syllabus and is essential for questions involving nth roots of complex numbers and proving trigonometric identities. Students targeting JEE Advanced should study it thoroughly alongside the Class 11 notes.

The three cube roots of unity are 1, ω, and ω², where ω = −½ + i(√3/2). They satisfy the equation x³ = 1. Their two key properties are: the sum 1 + ω + ω² = 0, and the product ω³ = 1. These properties are used to simplify expressions in JEE algebra questions, particularly those involving factorisation of sums of cubes.