root complex number calculator - Aaron Graves, PhDude Replica

Complex Number Root Finder

Enter a complex number in rectangular form z = a + bi and a root index n. The calculator returns all n-th roots in rectangular and polar form.

What this root complex number calculator does

This tool calculates the n-th roots of any complex number. If your input is z = a + bi, the calculator converts that number to polar form and applies De Moivre’s theorem to produce every valid root. Unlike real-number roots, complex roots usually come in a full set of evenly spaced solutions around the complex plane.

Quick refresher: roots in the complex plane

A complex number can be written in two common forms:

Rectangular form: z = a + bi
Polar form: z = r(cos θ + i sin θ), where r is magnitude and θ is angle

To find n-th roots, polar form is the easiest route because angles divide cleanly and include periodic rotations by 2π.

Formula used by the calculator

r = √(a² + b²) θ = atan2(b, a) For k = 0, 1, 2, ..., n-1: Root magnitude = r^(1/n) Root angle = (θ + 2πk) / n x_k = r^(1/n) cos((θ + 2πk)/n) y_k = r^(1/n) sin((θ + 2πk)/n) So each root is: w_k = x_k + i y_k

How to use it

Step 1: Enter a and b

Type the real part and imaginary part of your complex number. Example: for 3 + 4i, enter a = 3 and b = 4.

Step 2: Enter n

Choose the root index. For square roots use n = 2, cube roots use n = 3, and so on.

Step 3: Click “Calculate Roots”

The output includes:

Input number in both rectangular and polar form
Root magnitude
All n roots listed in rectangular and angle formats (radians and degrees)

Worked example

Suppose you want the cube roots of -8 (that is, -8 + 0i) with n = 3. The calculator gives 3 equally spaced roots:

2 + 0i
-1 + 1.732051i
-1 - 1.732051i

Geometrically, these points are 120° apart on a circle of radius 2.

Why this matters

Complex roots appear in many fields: electrical engineering (phasors and impedance), control theory, signal processing, quantum mechanics, and polynomial solving. A reliable calculator saves time and helps verify hand calculations quickly.

Common mistakes to avoid

Using a non-integer or negative n (root index must be a positive integer)
Forgetting that complex roots produce multiple answers
Using ordinary arctangent instead of atan2, which can place angles in the wrong quadrant
Rounding too aggressively and thinking different roots are the same

Final note

If your input is 0 + 0i, every n-th root collapses to 0, so the calculator reports that special case clearly. For any nonzero input, you’ll get n distinct roots arranged symmetrically around the origin.