Complex Root Calculator
Calculate all nth roots of complex numbers in Cartesian and polar forms
Calculate Complex Roots
The degree of the root (e.g., 2 for square root, 3 for cube root)
Complex Roots Results
Enter a complex number to calculate its roots
Mathematical Properties
Example Calculation
Square Roots of 1 + i
Complex number: z = 1 + i
Magnitude: r = √(1² + 1²) = √2 ≈ 1.414
Argument: θ = arctan(1/1) = π/4 = 45°
Root degree: n = 2
Calculation
Root magnitude: ²√(√2) = 2^(1/4) ≈ 1.189
w₀ = 1.189 × e^(i×22.5°) ≈ 1.098 + 0.455i
w₁ = 1.189 × e^(i×202.5°) ≈ -1.098 - 0.455i
Common Root Types
Square Roots
Two roots forming a line
Cube Roots
Three roots forming a triangle
Fourth Roots
Four roots forming a square
Complex Root Tips
Every complex number has exactly n distinct nth roots
Roots are equally spaced on a circle
All roots have the same magnitude
Angle between consecutive roots is 2π/n
Understanding Complex Roots
What are Complex Roots?
A complex number w is an nth root of another complex number z if w^n = z. Every complex number has exactly n distinct nth roots, which form the vertices of a regular n-sided polygon on the complex plane.
Geometric Interpretation
- •All roots lie on a circle with radius ⁿ√r
- •Roots are separated by angles of 2π/n radians
- •First root has argument θ/n
- •Forms regular polygons for different values of n
Complex Root Formula
wₖ = ⁿ√r × e^(i(θ + 2kπ)/n)
= ⁿ√r × (cos((θ + 2kπ)/n) + i·sin((θ + 2kπ)/n))
- wₖ: The kth root (k = 0, 1, ..., n-1)
- r: Magnitude of the complex number
- θ: Argument (angle) of the complex number
- n: Degree of the root
Special Case: Roots of unity are the nth roots of 1, forming regular polygons centered at the origin on the unit circle.
Roots of Unity
The nth roots of unity are the complex nth roots of 1. They have special significance in mathematics and form the vertices of regular n-gons on the unit circle.
Examples:
- • Square roots of unity: 1, -1
- • Cube roots of unity: 1, ω, ω² where ω = e^(2πi/3)
- • Fourth roots of unity: 1, i, -1, -i
Formula:
e^(2πki/n) = cos(2πk/n) + i·sin(2πk/n)
where k = 0, 1, ..., n-1