Formula Used
Rectangular to polar: r = √(a² + b²), θ = atan2(b, a)
All n-th roots: wk = r1/n [cos((θ + 2πk) / n) + i sin((θ + 2πk) / n)]
Index range: k = 0, 1, 2, ..., n - 1
How to Use This Calculator
- Select rectangular input for a + bi values.
- Select polar input for magnitude and argument values.
- Enter the root degree n.
- Choose degree or radian angle units.
- Select the angle range for final polar roots.
- Press calculate to show all roots above the form.
- Use CSV or PDF buttons to save the result.
Example Data Table
| Input | n | Root magnitude | Root angles | Polar roots |
|---|---|---|---|---|
| 1 + 0i | 4 | 1 | 0°, 90°, 180°, 270° | 1∠0°, 1∠90°, 1∠180°, 1∠270° |
| 0 - 8i | 3 | 2 | 330°, 90°, 210° | 2∠330°, 2∠90°, 2∠210° |
| 16∠120° | 4 | 2 | 30°, 120°, 210°, 300° | 2∠30°, 2∠120°, 2∠210°, 2∠300° |
About Complex Roots in Polar Form
Why polar form is useful
Complex roots are easier to study in polar form. A complex number has a distance from the origin. It also has an angle from the positive real axis. These two values are called magnitude and argument. Rectangular form is useful for addition. Polar form is better for powers and roots. This calculator supports both views, so you can enter the value in the form you already have.
How the roots are spaced
Every nonzero complex number has n distinct n-th roots. They share one common magnitude. That magnitude is the original magnitude raised to the power one over n. Their angles are evenly spaced around a circle. The spacing is 360 divided by n degrees, or 2π divided by n radians. This makes the roots form a regular pattern on the complex plane.
Handling the argument
The argument is not unique. Adding a full turn gives the same complex number. De Moivre’s theorem uses this fact. The formula adds 2πk to the original argument. Then it divides the total angle by n. Each value of k gives another root. The calculator lists k from zero through n minus one.
Rectangular and polar results
The final answer is shown in polar form first. This is the most natural form for roots. Rectangular form is also shown for checking. Small rounding differences can appear, especially with trigonometric values. Increase the precision when you need more detailed output. You can also switch between degrees and radians to match your lesson or workbook.
Exporting your work
The download buttons help with records and assignments. The CSV file opens in spreadsheet tools. The PDF file gives a simple report. Both include the main input, root degree, root magnitude, and every calculated root. Use the note field to describe your problem, class example, or verification task.
FAQs
What does this calculator find?
It finds all n-th roots of a complex number and presents them in polar form. It also gives rectangular form for checking and comparison.
Can I enter rectangular form?
Yes. Choose rectangular mode, then enter the real part and imaginary part. The calculator converts the number into polar form before finding roots.
Can I enter polar form directly?
Yes. Choose polar mode, then enter magnitude and argument. You can enter the angle in degrees or radians.
Why are there n roots?
A nonzero complex number has n different n-th roots. The roots have equal angle spacing around the origin.
What happens when the input is zero?
Zero has zero as its n-th root. The calculator lists zero roots for each index to match the selected root degree.
Why do angles repeat in different forms?
Complex angles repeat after a full turn. For example, 30 degrees and 390 degrees describe the same direction.
Should I use degrees or radians?
Use degrees for geometry-style work. Use radians for calculus, advanced algebra, and most theoretical formulas.
Why are rectangular values slightly rounded?
Rectangular values use sine and cosine calculations. Rounded decimals may appear. Increase decimal precision for more detailed results.