Complex Numbers is a foundational chapter for JEE Main, contributing 2-3 questions annually. The chapter extends the real number system to solve equations like $x^2$ + 1 = 0 by introducing i = sqrt(-1).

Every complex number z = a + ib has three representations: algebraic (a+ib), polar (re^(itheta)), and geometric (point in the Argand plane). The choice of representation depends on the operation: algebraic for addition/subtraction, polar for multiplication/division/powers, and geometric for locus and distance problems.

Core operations include conjugation (z-bar = a-ib), modulus (|z| = sqrt(a^{2+b}^2)), and argument (angle with positive real axis). The identity zz-bar = |z|^2 is used in virtually every problem. Euler's formula e^(itheta) = cos theta + i sin theta connects exponential and trigonometric forms.

De Moivre's theorem (cos theta + i sin theta)^n = cos(ntheta) + i sin(ntheta) handles powers and roots. The nth roots of unity form a regular n-gon on the unit circle and sum to zero. Cube roots of unity (1, w, $w^2$) with 1+w+$w^2$=0 appear in 30% of JEE complex number problems.

Geometric applications include: |z-z0| = r (circle), |z-z1| = |z-z2| (perpendicular bisector), arg(z-z0) = theta (ray), and the triangle inequality |z1+z2| <= |z1|+|z2|. Maximum and minimum modulus problems use geometric interpretation of circles and distances.

Key exam strategies: switch to polar form for powers, use 1+w+$w^2$=0 immediately for cube roots, check quadrant when computing argument, and use |z|^2 = z*z-bar to rationalize denominators.