AHL 1.13 (HL)—Polar and Euler forms

Convert complex numbers between Cartesian and polar form.

Polar form z=r(cosθ+i sinθ) separates magnitude r from direction θ; products multiply magnitudes and add arguments.

Worked example
(2 cis 30°)(3 cis 20°)=6 cis 50°.

Worked example
What happens to arguments when multiplying? add them, accounting for the chosen branch.

Common boundary
Do not add Cartesian coordinates when multiplying complex numbers.

Cartesian-to-polar/Euler example: for $z=1+i$, $r=\sqrt{1^2+1^2}=\sqrt2$ and $\theta=\pi/4$, so $z=\sqrt2\,\mathrm{cis}(\pi/4)=\sqrt2e^{i\pi/4}$. Products multiply moduli and add arguments; quotients divide moduli and subtract arguments. Vector addition is performed most directly in Cartesian form.