Polar and Exponential Form

Multiplication of complex numbers is rotation plus scaling. That sentence is hidden in the Cartesian form $a + bi$, but it becomes obvious the moment you switch to polar form. A complex number is a point in the plane; describe that point by its distance from the origin and its angle from the positive real axis, and suddenly multiplication of complex numbers is just multiplication of distances together with addition of angles. De Moivre's theorem and the roots of unity drop out for free.

Polar form: r and theta

Any complex number $z = a + bi$ can be written in polar form as

$z = r (\cos\theta + i \sin\theta),$

where $r = |z| = \sqrt{a^2 + b^2}$ is the modulus (distance from the origin) and $\theta = \arg(z)$ is the argument (angle measured counterclockwise from the positive real axis). The conversion formulas are $a = r\cos\theta$ and $b = r\sin\theta$.

The argument is determined only up to multiples of $2\pi$: angles $\theta$ and $\theta + 2\pi$ describe the same direction. The principal value conventionally lies in (-\pi, \pi] or $[0, 2\pi)$, depending on the textbook.

Euler's formula and exponential form

Euler proved one of the most stunning identities in mathematics:

$e^{i\theta} = \cos\theta + i \sin\theta.$

So polar form has a compact rewrite: $z = r e^{i\theta}$. From this, multiplication of complex numbers is

$z_1 z_2 = (r_1 e^{i\theta_1})(r_2 e^{i\theta_2}) = r_1 r_2 \, e^{i(\theta_1 + \theta_2)}.$

Moduli multiply, arguments add. If you ever wondered why multiplication mysteriously rotated points in the complex plane, that is the answer: the exponent, an angle, just adds when you multiply.

De Moivre's theorem

Powers follow instantly from the multiplication rule:

$(r e^{i\theta})^n = r^n e^{in\theta} = r^n (\cos(n\theta) + i \sin(n\theta)).$

This is De Moivre's theorem. To raise a complex number to a high power, write it in polar form, raise the modulus to that power, multiply the argument by $n$, and convert back if needed. Computing $(1 + i)^8$ in Cartesian form is painful; in polar form $1 + i = \sqrt{2} e^{i\pi/4}$, so $(1 + i)^8 = (\sqrt{2})^8 e^{i \cdot 8 \cdot \pi/4} = 16 e^{i 2\pi} = 16$.

Roots of unity

The $n$th roots of unity are the $n$ solutions to $z^n = 1$. Writing $z = e^{i\theta}$ (modulus must equal 1), the equation $e^{in\theta} = 1$ forces $n\theta$ to be a multiple of $2\pi$, so

$\omega_k = e^{2\pi i k / n} = \cos\dfrac{2\pi k}{n} + i \sin\dfrac{2\pi k}{n}, \qquad k = 0, 1, \ldots, n - 1.$