Complex Number Roots Calculator
Verified Calculation Engine
The online Complex Number Roots Calculator helps you calculate instantly and solve problems related to Complex Numbers. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Math Numbers calculations.
Finds all nth roots of a complex number using De Moivre's theorem: z^(1/n) = r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)].
z^(1/n) = r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)]
where k = 0, 1, 2, ..., n-1
There are exactly n distinct nth roots
A complex number has n distinct nth roots
Inputs
Results
Worked Examples
Example 1: Cube Roots
Find all cube roots of 8
- modulus: 8
- argument: 0
- root: 3
- roots: 3
- rootsPolar: 2(cos(0°) + i sin(0°)), 2(cos(120°) + i sin(120°)), 2(cos(240°) + i sin(240°))
- rootsRectangular: 2, -1 + 1.732i, -1 - 1.732i
Example 2: Board Level
Find all fourth roots of 16
- modulus: 16
- argument: 0
- root: 4
- roots: 4
- rootsPolar: 2(cos(0°) + i sin(0°)), 2(cos(90°) + i sin(90°)), 2(cos(180°) + i sin(180°)), 2(cos(270°) + i sin(270°))
- rootsRectangular: 2, 2i, -2, -2i
About this calculator
Overview
Complex Number Roots Calculator evaluates Complex Number Roots for the math / complex-numbers library. Primary input cue: Modulus r [required]: Modulus of complex number. Method anchor: z^(1/n) = r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] where k = 0, 1, 2, ..., n-1 There are exactly n distinct nth roots. A complex number has n distinct nth.
When to use
Use this page when you need complex number roots from known inputs and want a transparent arithmetic or symbolic check.
Inputs explained
- Modulus r [required]: Modulus of complex number
- Argument θ (Degrees) [required]: Argument in degrees
- Root n [required]: nth root to find
Formula / method
z^(1/n) = r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] where k = 0, 1, 2, ..., n-1 There are exactly n distinct nth roots. A complex number has n distinct nth roots (calculator id: complex-number-roots-calculator).
Worked example
Worked values: modulus = 8, argument = 0, root = 3 → roots = 3, rootsPolar = ['2(cos(0°) + i sin(0°))', '2(cos(120°) + i sin(120°))', '2(cos(240°) + i sin(240°))'], rootsRectangular = ['2', '-1 + 1.732i', '-1 - 1.732i']
Interpreting results
If complex number roots looks implausible, re-check units and which field is optional vs required.
Assumptions
- Units (when shown) must be consistent across inputs for complex number roots. [complex-number-roots-calculator]
- Empty required inputs are not silently replaced with zeros for complex number roots.
- Real-number arithmetic is used unless complex number roots explicitly needs integers.
Limitations
- This page does not provide a full textbook proof for every identity related to complex number roots.
- Complex Number Roots Calculator cannot invent missing inputs; incomplete forms stop short of a forged complex number roots value.
- Complex/branch cuts and undefined points are not always interactively annotated for complex number roots.
Important note
Educational mathematics tool. Check edge cases and domain restrictions when results look unexpected.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
z^(1/n) = r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] where k = 0, 1, 2, ..., n-1 There are exactly n distinct nth roots. A complex number has n distinct nth roots (calculator id: complex-number-roots-calculator).
Example Calculation
Worked values: modulus = 8, argument = 0, root = 3 → roots = 3, rootsPolar = ['2(cos(0°) + i sin(0°))', '2(cos(120°) + i sin(120°))', '2(cos(240°) + i sin(240°))'], rootsRectangular = ['2', '-1 + 1.732i', '-1 - 1.732i']
Frequently Asked Questions
What is Complex Number Roots Calculator?
- Complex Number Roots Calculator evaluates Complex Number Roots for the math / complex-numbers library. Primary input cue: Modulus r [required]: Modulus of complex number. Method anchor: z^(1/n) = r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] where k = 0, 1, 2, ..., n-1 There are exactly n distinct nth roots. A complex number has n distinct nth.
How does Complex Number Roots Calculator work?
- z^(1/n) = r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] where k = 0, 1, 2, ..., n-1 There are exactly n distinct nth roots. A complex number has n distinct nth roots (calculator id: complex-number-roots-calculator).
Why use this Math Numbers calculator?
- Use this page when you need complex number roots from known inputs and want a transparent arithmetic or symbolic check.