Complex numbers in polar form

Math

Definition

A complex number a + bi can be written in polar form as r(cos θ + i sin θ), or r cis θ, where r is the modulus (distance from the origin) and θ is the argument (angle from the positive real axis). This form makes multiplication, division, and finding powers much easier.

Examples

  • 1 + i in polar form is √2 · cis(45°)
  • Multiplying two complex numbers in polar form: multiply the moduli and add the angles
  • Using De Moivre's theorem to find (1 + i)⁸
Key Fact

r = √(a² + b²); θ = tan⁻¹(b/a); De Moivre's theorem: [r cis θ]ⁿ = rⁿ cis(nθ)

Related Terms

Complex numbers Polar coordinates Reference angles

Study This Concept

Practice complex numbers in polar form with free review games in these units:

Trigonometry Unit 8: Polar Coordinates and Complex Numbers