Polar Form

Every complex number can be written in polar form using its modulus r and argument θ. This form makes multiplication and division geometric: multiply moduli and add arguments. The expression cos θ + i sin θ is often abbreviated as cis θ.

The polar form rewrites a complex number using distance and angle instead of horizontal and vertical components. It is not a new object — it is the same complex number, described differently.

From Cartesian to Polar

Let $z = a + ib$ with $z \neq 0$. Define:

$$r = |z| = \sqrt{a^2 + b^2} > 0$$

Since $r > 0$, divide $a$ and $b$ by $r$:

$$\frac{a}{r} = \cos\theta, \qquad \frac{b}{r} = \sin\theta$$

where $\theta = \arg(z)$ is the angle such that $\cos^2\theta + \sin^2\theta = 1$ — consistent since $(a/r)^2 + (b/r)^2 = (a^2+b^2)/r^2 = 1$.

Therefore:

$$a = r\cos\theta, \qquad b = r\sin\theta$$

Substituting back:

$$z = a + ib = r\cos\theta + ir\sin\theta = r(\cos\theta + i\sin\theta)$$ $$\boxed{z = r(\cos\theta + i\sin\theta)}$$

Uniqueness

$r$ is unique: $r = |z|$ is determined by $z$ alone. There is only one positive real number equal to the distance from the origin.

$\theta$ is not unique: Any $\theta + 2k\pi$ ($k \in \mathbb{Z}$) gives the same point. The principal value $\operatorname{Arg}(z) \in (-\pi, \pi]$ is the conventional choice.

Multiplication in Polar Form

$$z_1 z_2 = r_1(\cos\theta_1 + i\sin\theta_1) \cdot r_2(\cos\theta_2 + i\sin\theta_2)$$ $$= r_1 r_2 [(\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2) + i(\cos\theta_1\sin\theta_2 + \sin\theta_1\cos\theta_2)]$$ $$= r_1 r_2 [\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)]$$

Moduli multiply, arguments add. This is why polar form is natural for multiplication and powers — operations that are clumsy in Cartesian form become clean in polar form.

The Euler Connection

Combined with Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$:

$$z = re^{i\theta}$$

The polar form and Euler's form are the same representation written two ways. Euler's form is more compact and makes exponentiation immediate.