“Complex Numbers: Deep Dive” is available to Standard subscribers.
Beyond the textbook: the polar form, multiplication as a rotation, the cube roots of unity, and loci on the Argand plane, the parts JEE asks about most, with a full timed paper.
3Lessons11Problems2Python Labs1Question Set1Mock Test
The polar form: Every complex number other than zero has a length and a direction. The length is the modulus r=∣z∣, and the direction is an argument θ, so z=r(cosθ+isinθ)
. Arguments differ by whole turns; the principal argument is the one in
(−π,π]
. Find it from the quadrant of the point, never from
tan−1
alone.
Multiplying as rotating: In polar form, multiplication is easy: the moduli multiply and the arguments add. So multiplying by a number of modulus 1 only turns a point, and multiplying by i turns it through a right angle. To turn about a point c, move c to the origin, turn, and move back. Powers are repeated turns.
Cube roots of unity and loci: The equation z3=1 has three roots: 1, ω and ω2, equally spaced on the unit circle. They satisfy ω3=1 and 1+ω+ω2=0, which simplify many expressions. On the Argand plane, ∣z−a∣=R is a circle and ∣z−a∣=∣z−b∣ is a perpendicular bisector.