Roots of Complex Number Calculator

Calculator

Input Mode

Real Part a

Imaginary Part b

Polar Modulus r

Polar Angle

Angle Unit

Root Degree n

Decimal Precision

Output Options

Formula Used

First convert the complex number into polar form:

z = a + bi = r(cos θ + i sin θ)

Where:

r = √(a² + b²)
θ = atan2(b, a)

The nth roots are calculated using De Moivre's theorem:

w_k = r^1/n[cos((θ + 2πk) / n) + i sin((θ + 2πk) / n)]

Here, k = 0, 1, 2, ..., n - 1. These values create all branches of the complex root.

How to Use This Calculator

Choose rectangular input when your number is written as a + bi.
Choose polar input when you already know modulus and angle.
Enter the root degree. For square roots, use 2.
Select the angle unit for polar input.
Choose decimal precision for rounded output.
Press Calculate Roots.
Review the result table shown above the form.
Use CSV or PDF download for saved records.

Example Data Table

Input Type	Value	Root Degree	Expected Meaning
Rectangular	3 + 4i	2	Find both square roots of the complex number.
Rectangular	-8 + 0i	3	Find three cube-root branches.
Polar	16 ∠ 120°	4	Find four fourth-root branches.
Rectangular	0 + 0i	5	Confirm that zero has one unique root.

Understanding Complex Roots

Complex roots extend ordinary square roots into the complex plane. A complex number has a real part and an imaginary part. Written as a + bi, it can also be expressed in polar form. Polar form uses a modulus and an angle. This makes roots easier to calculate, because powers split into size and rotation.

Why Polar Form Helps

For any nonzero complex number, the modulus gives distance from the origin. The angle gives direction from the positive real axis. When taking an nth root, the modulus becomes the nth root of the original modulus. The angle is divided by n. Extra full rotations also matter. Those rotations create n distinct roots, spaced evenly around a circle.

How the Calculator Works

This calculator accepts rectangular input or polar input. Rectangular input uses real and imaginary values. Polar input uses modulus and angle values. The tool converts the number into modulus and argument. It then applies De Moivre's theorem to generate every branch. Each row gives the root index, root modulus, radians, degrees, real part, imaginary part, rectangular form, and polar form.

Interpreting the Answer

The first row is commonly called the principal root. It uses k = 0. Other rows use k = 1 through n - 1. These roots are not errors. They are valid answers caused by periodic angle rotation. If the original number is zero, the only unique root is zero. Repeated branches then represent the same value.

Practical Uses

Complex roots appear in algebra, signal processing, control systems, alternating current, geometry, and numerical methods. They help solve polynomial equations. They also describe rotations and periodic patterns. Students can compare polar and rectangular answers to build intuition. Engineers can check phase changes and magnitudes.

Accuracy Notes

Decimal answers depend on the selected precision. Higher precision gives longer values. Lower precision is easier to read. Rounding can make tiny values appear as zero. Always check the formula section when learning the method. Use the example table to confirm input style. Export results when you need records for homework, reports, or later review. Use positive root degrees only. Start with small degrees, then increase them. Compare every branch visually by noting its angle and equal spacing around the origin plane.

FAQs

What is a complex root?

A complex root is a number that gives the original complex number when raised to a chosen power. For example, a square root becomes the original value after being squared.

Why are there many roots?

Complex angles repeat after full rotations. When the angle is divided for roots, those rotations create separate branches. An nth root usually has n distinct roots.

What is the principal root?

The principal root is usually the first branch, where k equals zero. It is often used as the main representative answer, but other branches are also valid.

Can I enter polar values?

Yes. Select polar input, enter the modulus, enter the angle, and choose degrees or radians. The calculator converts the values and finds every branch.

What happens when the input is zero?

Zero has one unique complex root, which is zero. Multiple listed branches may repeat the same value, because every branch collapses to zero.

Which formula is used?

The calculator uses De Moivre's theorem. It converts the input to polar form, takes the nth root of the modulus, and divides adjusted angles by n.

Why do tiny values show as zero?

Rounding removes very small decimal noise. This often happens near axes, where cosine or sine should be zero but floating-point calculation creates tiny values.

Can I export the results?

Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable report containing inputs, formulas, and calculated roots.