Complex Numbers
JEE Concepts

Complex numbers: a + bi, where a and b are real, i² = 1.

Modulus or Magnitude: √(a² + b²).

Conjugate: z* = a  bi

Argand Diagram: Complex numbers represented on a plane, real =horizontal, imagnary = vertical.

De Moivre’s theorem: r^(1/n)(cos(θ + 2kπ)/n + isin(θ + 2kπ)/n), k = 0, 1, 2, …, n1.

Polar form: z = r(cosθ + isinθ).

Euler’s formula: e^(iθ) = cosθ + isinθ

Applications: Quadratic equations, trigonometry, electrical engineering.
CBSE Board Concepts

Basics: a + bi, where a and b are real.

Modulus & Magnitude: z = √(a² + b²).

Conjugate: a  bi

Argand Diagram: Complex numbers represented on a plane.

Operations: Add, subtract, multiply like a*(b+ci)+d(bci) = (ab+cd) +i(acdb).

Division: (a + bi)/(c + di) = [(ac + bd)/(c² + d²)] + [(bc  ad)/(c² + d²)]i

Applications: Solve equations, trigonometric.