Polar form expresses a complex number by its modulus and argument, and De Moivre's theorem raises it to integer powers.

What this dot point is asking

Cartesian form is convenient for adding, but clumsy for multiplying and taking powers. Polar form describes the same number by how far it is from the origin and in what direction.

Modulus and argument

For $z=a+bi$ the modulus is $r=|z|=\sqrt{a^2+b^2}$ and the argument $\theta=\arg(z)$ is the angle the vector makes with the positive real axis, measured anticlockwise. Then $a=r\cos\theta$ and $b=r\sin\theta$, so

often abbreviated $z=r\operatorname{cis}\theta$.

Multiplication and division in polar form

If $z_1=r_1\operatorname{cis}\theta_1$ and $z_2=r_2\operatorname{cis}\theta_2$, then

So multiplication is a scaling and rotation: the new vector is $r_2$ times as long and rotated by $\theta_2$. This follows from the compound-angle formulae applied to $\cos$ and $\sin$.

De Moivre's theorem

Applying the multiplication rule $n$ times gives De Moivre's theorem: for any integer $n$,

This makes large powers, which would be hopeless to expand in Cartesian form, immediate.

The theorem also derives multiple-angle identities. Expanding $(\cos\theta+i\sin\theta)^3$ by the binomial theorem and equating real parts with $\cos 3\theta$ gives $\cos 3\theta=4\cos^3\theta-3\cos\theta$.

Common errors

Why it matters

Polar form turns multiplication into rotation and division into the reverse, and De Moivre's theorem is the gateway to finding the roots of complex numbers. The conversion technique builds directly on the modulus and conjugate ideas from complex arithmetic.