Nth Root Complex Number Calculator

What is the nth root of a complex number?

The nth root of a complex number z is a complex number w such that w^n = z. Unlike real numbers, complex numbers have multiple roots when n > 1. For example, the square roots of -1 are i and -i.

Complex numbers are typically represented in rectangular form (a + bi) or polar form (r(cosθ + i sinθ)). The polar form is particularly useful for finding roots because it separates the magnitude and angle of the number.

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)
2. Calculate the magnitude r = √(a² + b²)
3. Calculate the argument θ = arctan(b/a) (with appropriate quadrant adjustment)
4. Find the nth roots using the formula below

The roots will be equally spaced around a circle in the complex plane, with angles separated by 2π/n radians.

The formula for nth root of complex numbers

Where:

• w_k is the kth root
• r is the magnitude of z
• θ is the argument of z
• n is the root degree

The principal root (k=0) is the one with the smallest positive angle.

Worked example

Let's find the cube roots of z = -8 - 8i.

1. Convert to polar form:
   • r = √((-8)² + (-8)²) = √(64 + 64) = √128 ≈ 11.3137
   • θ = arctan(-8/-8) = arctan(1) = π (225°)
2. Calculate the roots using the formula:
   • w₀ ≈ 11.3137^(1/3) [cos(π/3) + i sin(π/3)] ≈ 2.25 [0.5 + i 0.866] ≈ 1.125 + i 1.952
   • w₁ ≈ 2.25 [cos(π/3 + 2π/3) + i sin(π/3 + 2π/3)] ≈ 2.25 [cos(π) + i sin(π)] ≈ -2.25 + i 0
   • w₂ ≈ 2.25 [cos(π/3 + 4π/3) + i sin(π/3 + 4π/3)] ≈ 2.25 [cos(5π/3) + i sin(5π/3)] ≈ 1.125 - i 1.952

The three cube roots of -8 - 8i are approximately 1.125 + 1.952i, -2.25 + 0i, and 1.125 - 1.952i.