Calculator Inputs
Example Data Table
| Case | Input | n | Example output |
|---|---|---|---|
| Power | 2 + 3i | 3 | -46 + 9i |
| Roots | 1 + 0i | 4 | 1, i, -1, -i |
| Power | 3(cos 45° + i sin 45°) | 2 | 9i |
Formula Used
1) Convert to polar form
For z = a + bi:
r = √(a² + b²)
θ = atan2(b, a)
2) De Moivre theorem for powers
If z = r(cos θ + i sin θ), then:
zⁿ = rⁿ [cos(nθ) + i sin(nθ)]
3) De Moivre theorem for roots
The nth roots of z are:
zk = r1/n [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], where k = 0, 1, 2, ..., n-1.
4) Rectangular reconstruction
After finding modulus and angle:
x = r cos θ and y = r sin θ
How to Use This Calculator
- Select rectangular form if your number is a + bi.
- Select polar form if your number is r and θ.
- Choose whether you want a power or all nth roots.
- Enter the integer n. Use a positive value for roots.
- For roots, choose the highlighted branch index k.
- Set your preferred precision and angle unit.
- Click Calculate to show the result above the form.
- Use the CSV or PDF buttons to export the result tables.
Frequently Asked Questions
1) What does De Moivre theorem do?
It turns complex powers and roots into angle and modulus operations. After converting to polar form, powers multiply the angle and raise the modulus, while roots divide the angle and take the nth root of the modulus.
2) Why is polar form useful here?
Polar form separates size and direction. That makes exponent and root calculations easier because the modulus and angle follow clean formulas, unlike repeated expansion in rectangular form.
3) Can the calculator handle negative exponents?
Yes. A negative exponent computes the reciprocal power, provided the input is not zero. Zero cannot be raised to a negative exponent because division by zero would occur.
4) What does branch k mean for roots?
Branch k identifies one specific nth root from the full root set. Each branch adds another full-turn offset before the angle is divided by n, producing evenly spaced roots.
5) Why do nth roots appear evenly spaced?
Because each branch changes the angle by 2π/n. On the complex plane, that creates equal angular spacing, so the roots lie on a circle with the same modulus.
6) What happens when the input is zero?
For positive powers, the result stays zero. For roots, every branch collapses to zero. For negative powers, the value is undefined because the reciprocal of zero does not exist.
7) Should I use degrees or radians?
Use whichever matches your source data. The calculator converts internally and reports both degree and radian outputs, so either input style is fine for the computation.
8) Why do I sometimes see tiny decimal leftovers?
Trigonometric calculations use floating-point arithmetic. Values that should be zero may appear as extremely small decimals, and the calculator cleans near-zero noise for clearer output.