Complex Root Calculator
Results
How to Use
- Enter the real part of your complex number
- Enter the imaginary part of your complex number
- Enter the root number (n)
- Click "Calculate Roots" to find all nth roots
About Complex Roots
The nth roots of a complex number z = r(cos θ + i sin θ) are given by:
z^(1/n) = r^(1/n)[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]
where k = 0, 1, 2, ..., n-1
Properties:
- Every non-zero complex number has exactly n distinct nth roots
- The roots are equally spaced around a circle in the complex plane
- The radius of the circle is the nth root of the magnitude
- The angle between consecutive roots is 2π/n
Common Examples
| Number | Root (n) | Results |
|---|---|---|
| 1 | 2 | 1, -1 |
| 1 | 3 | 1, -0.5 + 0.866i, -0.5 - 0.866i |
| 1 | 4 | 1, i, -1, -i |
| i | 2 | 0.707 + 0.707i, -0.707 - 0.707i |
Applications
- Solving polynomial equations
- Signal processing
- Electrical engineering
- Quantum mechanics
- Fractal geometry
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