Math
De Moivre’s Theorem Calculator
Compute complex number powers and n-th roots using De Moivre’s Theorem. Enter rectangular or polar form, choose degrees or radians, and get results in both forms with a roots table and history export.
Complex Powers & Roots Solver
Choose an operation, enter z in rectangular (a+bi) or polar (r, θ), then calculate. Output includes both forms and a roots table when applicable.
Tip: For fast checks, try z=3+4i (rectangular) or r=5, θ≈53.130102° (polar). For roots, use n=3 or n=4 to see evenly spaced angles.
| Task | Use | Formula | What you get |
|---|---|---|---|
| Convert a+bi → polar | Convert | r = √(a²+b²), θ = atan2(b, a) | Magnitude and angle for z |
| Convert polar → a+bi | Convert | a = r cos θ, b = r sin θ | Rectangular components |
| Compute z^n | Power | z^n = r^n(cos(nθ)+i sin(nθ)) | Power result in both forms |
| Compute n-th roots | Roots | z^(1/n) = r^(1/n)(cos((θ+2kπ)/n)+i sin((θ+2kπ)/n)) | All n roots listed by k |
| SSA-like ambiguity? | Roots | Angles wrap by 2π | Multiple valid roots are expected |
Quick Steps
- Pick an operation: Power, Roots, or Convert.
- Choose your input form (rectangular or polar).
- Select degrees or radians to match your source.
- Press Calculate to get rectangular and polar outputs.
- If roots are shown, use k to compare angles spaced by 2π/n.
Reliable check: converting rectangular → polar → rectangular should return the original z (up to rounding). If your angle differs by 360° (or 2π), it’s still equivalent.
Your history will appear here after you run calculations.