Nth Root of A Complex Number Calculator

What is the nth root of a complex number?

The nth root of a complex number is a complex number that, when raised to the nth power, equals the original complex number. Unlike real numbers, complex numbers have multiple roots due to their polar form representation.

Complex numbers are expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (√-1). The roots are equally spaced around a circle in the complex plane.

How to calculate the nth root of a complex number

To find the nth roots of a complex number z = a + bi:

1. Convert the complex number to polar form: z = r(cosθ + i sinθ), where r = √(a² + b²) is the magnitude and θ = arctan(b/a) is the argument.
2. Calculate the nth roots using De Moivre's Theorem: z^(1/n) = r^(1/n) [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], where k = 0, 1, 2, ..., n-1.
3. Convert the roots back to rectangular form if needed.

This process yields n distinct roots, each separated by 2π/n radians in the complex plane.

Formula for the nth root of a complex number

The formula uses De Moivre's Theorem to find all possible roots by considering the principal root and its rotations around the origin.

Example calculation

Let's find the cube roots of the complex number 1 + i.

1. Convert to polar form: r = √(1² + 1²) = √2, θ = arctan(1/1) = π/4 radians.
2. Calculate the roots using k = 0, 1, 2:
   • First root (k=0): √2^(1/3) [cos(π/12) + i sin(π/12)] ≈ 1.122 + 0.588i
   • Second root (k=1): √2^(1/3) [cos(9π/12) + i sin(9π/12)] ≈ -0.588 + 1.122i
   • Third root (k=2): √2^(1/3) [cos(17π/12) + i sin(17π/12)] ≈ -1.122 - 0.588i

These roots are equally spaced at 120° intervals in the complex plane.