Key Idea: This topic writes a complex number by its distance r and angle θ. Polar form is r cisθ and exponential form is r eⁱθ. Multiplying and dividing become easy. It is on Paper 1 and Paper 2.
Polar (modulus-argument) form
- z = r cisθ = r(cosθ + i sinθ).
- Cartesian to polar: r = √(a² + b²), θ = arctan(b/a), then fix the quadrant.
- Polar to Cartesian: a = r cosθ, b = r sinθ.
- Sketch the point first. Add ±π for quadrants 2 and 3.
| Operation | Moduli | Arguments |
|---|---|---|
| Multiply | r₁r₂ | θ₁ + θ₂ |
| Divide | r₁/r₂ | θ₁ − θ₂ |
Exponential (Euler) form
- z = r eⁱθ, with the same r and θ, since eⁱθ = cosθ + i sinθ.
- Index laws: add exponents to multiply, subtract them to divide.
- θ is in radians.
Multiplying a complex number scales and rotates it.
2 cis(π/6) in Cartesian form? √3 + i.
(√2 cis(π/4))²? 2 cis(π/2) = 2i.
A product's argument is outside the range. What do you do? Add or subtract 2π.
3 eⁱπ/² as a + bi? 3i.
arctan(b/a) gives the wrong angle for −1 − i. Why? The point is in quadrant 3, so you must adjust θ by π.