Complex Root Calculator
Calculate nth roots of complex numbers
What are Complex Roots?
Complex roots are the solutions to equations of the form z^n = a + bi, where z is the unknown complex number we're solving for, n is a positive integer (the root index), and a + bi is a given complex number. When finding the nth root of a complex number, there are always exactly n distinct solutions, evenly spaced around a circle in the complex plane.
De Moivre's Formula
De Moivre's formula is essential for finding complex roots. For a complex number in polar form r∠θ, its nth roots are given by:
z_k = r^(1/n) × (cos((θ + 2πk)/n) + i×sin((θ + 2πk)/n))
where:
- r is the modulus (magnitude) of the complex number
- θ is the argument (angle) in radians
- k ranges from 0 to n-1
- n is the root index
How to Find Complex Roots
- Convert the complex number to polar form (r∠θ)
- Calculate the nth root of the modulus (r^(1/n))
- Divide the argument by n and add 2πk/n for k = 0, 1, ..., n-1
- Convert each result back to rectangular form (a + bi)
Applications
Complex roots have various applications in:
- Solving polynomial equations
- Signal processing and analysis
- Control systems engineering
- Electrical circuit analysis
- Quantum mechanics calculations
- Computer graphics and animation
Examples
Example 1: Square Root
- Find √(1 + i)
- r = √2, θ = π/4
- Two roots:
- z₁ = 1.0897 + 0.4551i
- z₂ = -0.4551 - 1.0897i
Example 2: Cube Root
- Find ∛(-8)
- r = 8, θ = π
- Three roots:
- z₁ = -2
- z₂ = 1 + 1.732i
- z₃ = 1 - 1.732i
Properties of Complex Roots
- Every nonzero complex number has exactly n nth roots
- The roots form a regular polygon in the complex plane
- The roots are equally spaced around a circle with radius r^(1/n)
- The angle between consecutive roots is 2π/n radians
- The product of all nth roots equals the original number raised to the power of 1
Common Mistakes to Avoid
- Forgetting that complex numbers have multiple roots
- Using incorrect angle measurements (degrees vs. radians)
- Not considering all possible values of k
- Incorrectly converting between rectangular and polar forms
- Making sign errors in trigonometric calculations