Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Signal Addition
Add two phasor signals in complex form to find the resultant.
Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition
Circuit Impedance
Divide two impedances to compute the transfer ratio.
Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division
De Moivre Power
Raise a complex number to the 10th power using De Moivre's theorem.
Key values: z₁ = 1 + i · n = 10 · Power
Cube Roots of −8
Find all three cube roots of −8, visualized as equidistant points on a circle.
Key values: z₁ = −8 + 0i · n = 3 · nth Root
Complex Number Calculator
Enter complex numbers and choose an operation to see the result.
This calculator is also known as Complex Roots Calculator.
Read the complete guidenth Root Formula
The n distinct nth roots of z = r·cis(θ) are z_k = r^(1/n)·cis((θ + 2πk)/n) for k = 0, 1, ..., n−1. All roots share the same modulus r^(1/n) and are separated by an angle of 2π/n radians.
Examples
Cube Roots of −8
Find all three cube roots of −8.
−8 = 8∠180°. The cube root modulus is 8^(1/3) = 2. The three roots are at angles 60°, 180°, and 300°, giving 1 + √3·i, −2, and 1 − √3·i.
Key takeaway: The "obvious" real cube root −2 is just one of the three complex cube roots. The other two are equally valid and equally spaced on a circle of radius 2.
Finding Complex Roots
Steps for computing nth roots:
- Convert to polar form: find r and θ
- Compute the root modulus: r^(1/n)
- Compute the k-th root angle: (θ + 2πk)/n for k = 0, 1, ..., n−1
Frequently Asked Questions about Complex Roots Calculator
How many nth roots does a complex number have?
Every nonzero complex number has exactly n distinct nth roots. The number 0 technically has only one nth root: 0 itself.
What are roots of unity?
The nth roots of unity are the nth roots of 1 + 0i. They are equally spaced on the unit circle and form a cyclic group under multiplication.
Specialized Calculators
Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
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