Complex Root Calculator

Understanding N-th Roots of Complex Numbers

In real-number arithmetic, every positive number has two square roots, and negative numbers have no real roots. However, in complex analysis, the Fundamental Theorem of Algebra ensures that every non-zero complex number has exactly $n$ distinct $n$-th roots. For example, any complex number has 2 square roots, 3 cube roots, 4 fourth roots, and so on.

Our Complex Root Calculator computes all $n$ roots of any complex number provided in rectangular ($a + bi$) or polar ($r \angle \theta$) form using de Moivre's Formula. You can explore other complex number representations with our a+bi Form Calculator and perform arithmetic with our Complex Calculator.

De Moivre's Formula for Roots

To find the $n$-th roots of a complex number $z = a + bi$, we first express $z$ in polar (or trigonometric) form:

where:

• Modulus ($r$): The distance from the origin on the complex plane, given by $r = |z| = \sqrt{a^2 + b^2}$.
• Argument ($\theta$): The angle in radians or degrees counter-clockwise from the positive real axis, given by $\theta = \operatorname{atan2}(b, a)$.

According to de Moivre's theorem, the $n$ distinct $n$-th roots $w_0, w_1, \dots, w_{n-1}$ are given by:

In exponential notation, this is written concisely as:

Geometric Interpretation on the Argand Plane

The geometry of complex roots reveals one of the most elegant symmetries in mathematics:

1. Equal Magnitude: All $n$ roots lie on a circle centered at the origin with radius $R = \sqrt[n]{r}$.
2. Equal Angular Spacing: The roots are separated by equal angles of $\Delta\theta = \frac{2\pi}{n}$ radians (or $\frac{360^\circ}{n}$).
3. Regular Polygon: Connecting adjacent roots with straight line segments forms a regular $n$-sided polygon (an equilateral triangle for cube roots, a square for 4th roots, a regular pentagon for 5th roots, etc.) inscribed in the circle.

Step-by-Step Example: Finding the Cube Roots of $8i$

Let us determine the 3 cube roots of $z = 0 + 8i$ ($n = 3$):

1. Calculate Modulus and Argument: $$r = \sqrt{0^2 + 8^2} = 8, \quad \theta = \frac{\pi}{2} = 90^\circ$$
2. Calculate Root Radius: $$R = \sqrt[3]{8} = 2$$
3. Calculate Each Root for $k \in \{0, 1, 2\}$:
   • Principal Root ($k = 0$): $\theta_0 = \frac{90^\circ}{3} = 30^\circ \implies w_0 = 2(\cos 30^\circ + i\sin 30^\circ) = \sqrt{3} + i \approx 1.7321 + i$
   • Second Root ($k = 1$): $\theta_1 = \frac{90^\circ + 360^\circ}{3} = 150^\circ \implies w_1 = 2(\cos 150^\circ + i\sin 150^\circ) = -\sqrt{3} + i \approx -1.7321 + i$
   • Third Root ($k = 2$): $\theta_2 = \frac{90^\circ + 720^\circ}{3} = 270^\circ \implies w_2 = 2(\cos 270^\circ + i\sin 270^\circ) = -2i$