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
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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