AHL 1.13 (HL)—Complex forms and applications

A complex number can be written in Cartesian form z = a + bi or polar form z = r(cos θ + i sin θ) = re^{iθ}, where r = |z| and θ is an argument. Cartesian form is natural for addition; polar or exponential form makes multiplication, division and powers easier.

For z₁ = 2e^{iπ/3} and z₂ = 3e^{iπ/6}, z₁z₂ = 6e^{iπ/2}. Multiplication multiplies moduli and adds arguments; converting back gives 6i.

Arguments differ by multiples of 2π, and the principal argument depends on the chosen range. When converting from Cartesian form, use the correct quadrant rather than relying on arctan(b/a) alone.

Integer powers follow De Moivre's pattern: $[r\operatorname{cis}\theta]^n=r^n\operatorname{cis}(n\theta)$, while division divides moduli and subtracts arguments. Geometrically, multiplication by $r\operatorname{cis}\theta$ scales by $r$ and rotates by $\theta$. Adding same-frequency sinusoids is vector addition of complex amplitudes, which determines the resultant amplitude and phase; finding complex roots is not required in this objective.