polar form

Polar form describes a complex number not by its horizontal and vertical coordinates, but by its distance and direction: how far it is from the origin, and which way it points. It is the difference between giving someone an address as “3 blocks east, 4 blocks north” versus “5 blocks away, heading northeast.” Both pinpoint the same spot.

If r is the modulus and θ is the argument, then z = r(cos θ + i sin θ). The conversion back to rectangular form a + bi is a = r cos θ and b = r sin θ. The shorthand r cis θ means exactly cos θ + i sin θ scaled by r, and Euler's formula packs it even tighter as z = r e^(iθ).

Polar form shines for multiplication, division, and powers, where the rectangular a + bi form gets messy. To multiply two complex numbers you multiply their moduli and add their arguments — a rule that is almost effortless in polar form and is the foundation of De Moivre's theorem.