← Complex numbers

4 Argand diagram & polar form

Plot z = a + bi as the point (a, b). Its distance from the origin is the modulus r = |z|, and the angle from the positive real axis is the argument θ = arg(z). Polar (mod-arg) form z = r(cos θ + i sin θ) = r cis θ r = √(a²+b²), and tan θ = b/a (check the quadrant!). Multiply in polar r₁cisθ₁ × r₂cisθ₂ = r₁r₂ cis(θ₁+θ₂) Multiply moduli, ADD arguments. Divide in polar \frac{r_1\operatorname{cis}\theta_1}{r_2\operatorname{cis}\theta_2}=\frac{r_1}{r_2}\operatorname{cis}(\theta_1-\theta_2) Divide moduli, SUBTRACT arguments.

Getting the argument right

• Always sketch z on the Argand plane first to see which quadrant it’s in.
• tan θ = b/a gives a reference angle; adjust for the quadrant (add/subtract π).
• By convention, arg(z) is given in −π < θ ≤ π (principal argument).

Worked example Convert to polar form Write z = −1 + i in polar form. r = √((−1)² + 1²) = √2. z is in the 2nd quadrant (real −, imaginary +). Reference angle: tan⁻¹(1/1) = π/4, so θ = π − π/4 = 3π/4. ✓ z = √2 cis(3π/4).

Practise just this