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Instant Calculation
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Every non-zero complex number has exactly n distinct n-th roots, evenly spaced around a circle in the complex plane. This tool applies De Moivre's theorem to find every one of them, in both rectangular (a+bi) and polar form.
How it works
The complex number is first converted to polar form (r, θ). Then each of the n roots is computed as r^(1/n) times cos((θ+360°k)/n) + i·sin((θ+360°k)/n), for k = 0 through n-1.
- Enter real part (a).
- Enter imaginary part (b).
- Enter root degree (n).
- Click Calculate to see your results.
Examples
Cube roots of unity
The three cube roots of 1 are 1, -0.5+0.866i, and -0.5-0.866i — evenly spaced 120° apart on the unit circle.
Square roots of i
The two square roots of i are approximately 0.707+0.707i and -0.707-0.707i.
Who should use it
- Precalculus and complex analysis coursework on De Moivre's theorem.
- Verifying hand-computed complex roots.
Industry applications
- Mathematics and engineering education (signal processing, control theory)
Advantages
- Computes every root, not just the principal one.
- Shows both rectangular and polar form for each root.
Limitations
- Limited to root degrees from 2 to 8.
Common mistakes to avoid
- Only finding one root (usually the "principal" one) and forgetting the other n-1 roots exist.
Best practices
- Plot the roots mentally (or on paper) as points evenly spaced around a circle of radius r^(1/n) to sanity-check the result.
Tips
- The roots of unity (n-th roots of 1) always include 1 itself as one of the roots — a quick way to check your setup is correct.