Principal Root Calculator

Find principal nth roots for real and complex numbers. Control precision, output format, and significant digits easily. Handle negative radicands, polar forms, and scientific notation. Export CSV or PDF with batch inputs and examples. Reliable. Compute confidently with validated methods and clean design everywhere.

Enter Value

Use a+bi form, or a,b for real and imaginary parts.

Results

Root Rectangular form (a + bi) Polar form (r ∠ θ) Verification: result^n
No results yet. Enter a value and click Calculate.

CSV/PDF buttons enable after a calculation.

Formula Used

Let z = a + bi with magnitude r = √(a² + b²) and principal argument θ = atan2(b, a) ∈ (−π, π].

The principal n-th root is: w₀ = r^(1/n) [cos(θ/n) + i sin(θ/n)].

All n roots are given by: w_k = r^(1/n) [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], for k = 0,…,n−1. The principal root corresponds to k = 0.

How to Use

  1. Enter a real value (e.g., -16) or complex value (e.g., 3+4i).
  2. Set the index n (positive integer).
  3. Choose fixed decimals or significant digits, and rounding mode.
  4. Optional: display angles in degrees and show all n roots.
  5. Click Calculate to populate the results table.
  6. Use CSV or PDF buttons to export the results.

Example Data Table

Input n Principal root (a + bi) Polar form (r ∠ θ, radians)
16 4 2.000000 + 0.000000i 2.000000 ∠ 0.000000 rad
-16 2 0.000000 + 4.000000i 4.000000 ∠ 1.570796 rad
-8 3 1.000000 + 1.732051i 2.000000 ∠ 1.047198 rad
3+4i 2 2.000000 + 1.000000i 2.236068 ∠ 0.463648 rad
1+1i 3 1.084215 + 0.290515i 1.122462 ∠ 0.261799 rad
0 5 0.000000 + 0.000000i 0.000000 ∠ 0.000000 rad

FAQs

For a given number and index n, the principal root is the canonical root obtained using the principal argument θ ∈ (−π, π]. It corresponds to k = 0 in the n distinct roots.

Yes. For even n, negative reals yield purely imaginary principal roots (e.g., √(-16) = 4i). For odd n, a real negative root exists but is not the principal value; for example, the principal ∛(-8) = 1 + √3 i.

Enter real numbers like 12.5 or -16, complex numbers like 3+4i or 3-4i, or a pair a,b meaning a+bi. Scientific notation is supported.

Set either fixed decimals or significant digits. Rounding modes include half up, half even, floor, and ceil. Internally, computations use double precision floating point.

Every nonzero complex number has n distinct n-th roots arranged evenly on a circle in the complex plane. Checking the option lists them alongside the principal root.

Finite precision arithmetic and rounding can introduce small numerical differences when raising a computed root to the n-th power. Increasing precision reduces visible error.

Yes. Use the CSV or PDF export buttons above the results table. A separate button downloads the example table as CSV.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.