Complex Numbers in Polar Form

A complex number a + bi is a point on a plane, with a across and b up. That makes polar coordinates available to it.

Trigonometric form

Every complex number can be written r(cos θ + i sin θ), where r is the modulus and θ is the argument.

Finding them

The modulus is r = √(a² + b²), the distance from the origin. The argument is the angle to the point.

Multiplication becomes simple

To multiply, multiply the moduli and add the arguments. Multiplying by a complex number stretches and rotates.

De Moivre’s Theorem

Raising to a power follows the same logic: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).

Roots

The theorem runs backwards too. A number has n distinct nth roots, spaced evenly around a circle.

Complex numbers in polar form

a + bi can be written r(cos θ + i sin θ), where r is the modulus and θ the argument. It is the same number located by distance and direction rather than by real and imaginary parts.

Multiplying becomes rotating

To multiply two complex numbers in polar form, multiply the moduli and add the arguments. Multiplication is a rotation combined with a scaling, which is far clearer in polar form than in a + bi.

De Moivre's theorem

Raising to the nth power multiplies the argument by n and raises the modulus to the n. Computing (1 + i)¹⁰ becomes trivial in polar form and is genuinely painful without it.

Roots of unity

The n complex nth roots of 1 sit evenly spaced around the unit circle. That regular arrangement is why complex numbers describe rotational symmetry, and it underlies the fast Fourier transform.