nth Roots of a Complex Number

Every non-zero complex number z = r·cis θ has exactly n distinct nth roots. They lie on a circle of radius r^(1/n) centred at the origin, spaced equally at angular intervals of 2π/n — forming a regular n-gon. In Euler form: r^(1/n) · e^(i(θ+2kπ)/n).

Finding the $n$th roots of a complex number is one of the most geometric results in the chapter. The algebra gives a formula; the geometry reveals why there are exactly $n$ roots and where they sit.

Setup

Let $z = re^{i\theta}$ (with $r = |z| > 0$ and $\theta = \arg(z)$). We want all $w$ such that $w^n = z$.

Write $w = \rho e^{i\phi}$. Then:

$$w^n = \rho^n e^{in\phi} = re^{i\theta}$$

Equating moduli and arguments separately:

$$\rho^n = r \implies \rho = r^{1/n} \quad \text{(unique positive real root)}$$ $$n\phi = \theta + 2k\pi, \quad k \in \mathbb{Z} \implies \phi = \frac{\theta + 2k\pi}{n}$$

The n Distinct Roots

$$w_k = r^{1/n} \left(\cos\frac{\theta + 2k\pi}{n} + i\sin\frac{\theta + 2k\pi}{n}\right), \quad k = 0, 1, \ldots, n-1$$

For $k \geq n$, the angle $(\theta + 2k\pi)/n$ differs from some $w_j$ with $0 \leq j \leq n-1$ by a multiple of $2\pi$ — giving the same point. So there are exactly $n$ distinct roots.

The Geometric Picture

All $n$ roots:

• lie on a circle of radius $r^{1/n}$ centred at the origin
• are separated by equal angular gaps of $\mathbf{2\pi/n}$
• form the vertices of a regular $n$-gon

The first root $w_0 = r^{1/n} e^{i\theta/n}$ anchors the polygon. Every subsequent root is obtained by rotating $w_0$ by $2\pi/n$.

Special Case: Roots of Unity

When $z = 1$ ($r = 1$, $\theta = 0$):

$$w_k = e^{2\pi ik/n} = \cos\frac{2k\pi}{n} + i\sin\frac{2k\pi}{n}, \quad k = 0, 1, \ldots, n-1$$

A regular $n$-gon inscribed in the unit circle with one vertex always at $(1, 0)$.

Why the Modulus Has a Unique Value

The equation $\rho^n = r$ with $\rho, r > 0$ has exactly one positive real solution $\rho = r^{1/n}$. All $n$ roots share this modulus — they lie on a single circle. The only freedom is in the argument, which produces the $n$ distinct angular positions.