Polar form
z = r(cosθ + i sinθ) = re^(iθ) (Euler's formula).
Polar form replaces Cartesian a + bi with modulus r and argument θ. Euler's formula e^(iθ) = cosθ + i sinθ gives the exponential form.
Worked examples
z = 2(cos π/3 + i sin π/3) = 1 + i√3.
z = re^(iθ); zw = rs · e^(i(θ+φ)).
e^(iπ) + 1 = 0. Euler's identity.
Frequently asked questions
Why is e^(iθ) = cosθ + i sinθ?
Comes from the Maclaurin series of ex with x = iθ.
Polar product?
(r·e^(iθ)) × (s·e^(iφ)) = rs · e^(i(θ+φ)). Multiplication is rotation + scaling.