Calculator
Formula Used
Complex number in polar form:
z = r(cos θ + i sin θ)
Principal n-th root:
w₀ = r^(1/n) [cos(θ / n) + i sin(θ / n)]
All n roots:
wₖ = r^(1/n) [cos((θ + 2πk) / n) + i sin((θ + 2πk) / n)], for k = 0, 1, 2, ..., n - 1
For positive real inputs, the principal root is the familiar positive real answer. For complex or negative inputs, the calculator uses the principal argument and returns the principal branch automatically.
How to Use This Calculator
- Choose the input mode: real, complex rectangular, or complex polar.
- Enter the number and set the root index n.
- Choose your decimal precision.
- Tick the option to list every n-th root when needed.
- Press Calculate Principal Root.
- Review the summary, root table, and Argand chart.
- Use the CSV or PDF buttons to export the result set.
Example Data Table
| Input | n | Principal Root | Notes |
|---|---|---|---|
| 16 | 2 | 4 | Positive real values return the positive real root. |
| -8 | 3 | 1.0000 + 1.7321i | The principal branch is complex, even though -2 is also a cube root. |
| 3 + 4i | 2 | 2.0000 + 1.0000i | This is the principal square root of a standard complex example. |
| 16 ∠ 90° | 4 | 1.8478 + 0.7654i | Polar input is converted and solved with the same formula set. |
FAQs
1) What is a principal root?
A principal root is the single root chosen from all possible n-th roots by using the principal argument of the original number. It gives a consistent branch for calculation, graphing, and software outputs.
2) Why can a negative number return a complex principal root?
In complex analysis, the principal argument of a negative real number is π. Dividing that angle by n can place the principal root off the real axis, even when another real root also exists.
3) Does every nonzero number have exactly n n-th roots?
Yes. Any nonzero complex number has exactly n distinct n-th roots. They are evenly spaced around a circle in the complex plane, and the principal root is the one with k = 0.
4) What happens when the input is zero?
Zero is a special case. Its principal n-th root is zero, and the root set collapses to the same point. The plot will place the result at the origin.
5) Should I use degrees or radians?
Use whichever unit matches your source data. The calculator accepts both for polar input, then converts everything internally to radians before computing the principal root and the full root set.
6) Can I enter a complex number in polar form directly?
Yes. Choose the complex polar mode, enter magnitude and angle, then select degrees or radians. The calculator converts that value to rectangular form and solves it immediately.
7) What does the Plotly chart show?
The graph shows the computed roots on the Argand plane using real values on the x-axis and imaginary values on the y-axis. It helps you see symmetry, spacing, and the principal branch visually.
8) What do the CSV and PDF exports include?
They include the main input details, principal root summary, and the table of computed roots. CSV works well for spreadsheets, while PDF is useful for reports, assignments, and saved calculation records.