Enter Complex Graph Inputs
Use the inputs below to evaluate a complex number, graph its related points, and inspect roots, powers, rotation, scaling, and translation.
Example Data Table
| Input z | Power n | Root degree m | |z| | arg(z) | z^n | Principal root |
|---|---|---|---|---|---|---|
| 3.0000 + 4.0000i | 3 | 4 | 5.0000 | 53.1301° | -117.0000 + 44.0000i | 1.4553 + 0.3436i |
| -2.0000 + 5.0000i | 2 | 3 | 5.3852 | 111.8014° | -21.0000 - 20.0000i | 1.3949 + 1.0614i |
| 1.5000 - 2.5000i | 4 | 5 | 2.9155 | -59.0362° | -40.2500 + 60.0000i | 1.2124 - 0.2534i |
Formula Used
1) Standard form
Let z = a + bi, where a is the real part and b is the imaginary part.
2) Modulus and argument
|z| = √(a² + b²) and arg(z) = atan2(b, a).
3) Conjugate and reciprocal
z̄ = a - bi and, for nonzero z, 1/z = (a - bi) / (a² + b²).
4) Power by De Moivre’s theorem
If z = r(cosθ + i sinθ), then zⁿ = rⁿ(cos(nθ) + i sin(nθ)).
5) nth roots
Each root is r^(1/m) [cos((θ + 2πk)/m) + i sin((θ + 2πk)/m)] for k = 0, 1, ..., m-1.
6) Rotation, scaling, and shifting
A rotated and scaled point uses s · z · (cosφ + i sinφ). A shifted point then adds (u + vi).
How to Use This Calculator
Enter the real and imaginary parts of your complex number first. Choose the power value for exponentiation and the root degree for root generation. Add a rotation angle, scale factor, and translation values if you want transformed points. Set the decimal precision, then press Calculate and Plot.
After submission, the result section appears directly below the header and above the form. It shows the main values, the plotted points on the complex plane, and a detailed result table. Use the CSV and PDF buttons to export the current result set for later reference or reporting.
Frequently Asked Questions
1) What does this calculator plot?
It plots the input complex number, its conjugate, reciprocal when defined, selected power, transformed point, and every nth root on the complex plane.
2) Why are the roots spread evenly around the plane?
Nth roots divide the original argument into equal angular steps. Each new point differs by 360/m degrees, so the roots form a regular circular pattern.
3) What happens when the input is zero?
Zero has modulus zero, argument treated as zero here, and every root is also zero. The reciprocal is undefined because division by zero is not allowed.
4) Why does the power result change so quickly?
Complex powers multiply the angle and raise the modulus. Even moderate inputs can produce much larger coordinates when the exponent increases.
5) What is the principal root?
The principal root is the k=0 root generated from the main argument branch. It is often used as the default representative root in analysis.
6) Does the rotation angle use degrees or radians?
The form accepts degrees for convenience. The script converts that angle internally to radians before applying the rotation formulas.
7) What is the benefit of the shifted transform?
It helps you see a transformed point after rotation and scaling, then moves it by a chosen real and imaginary offset for comparison.
8) Can I export the plotted results?
Yes. The calculator includes CSV export for tabular data and PDF export for a formatted report that also includes the graph image.