Third Root of A Complex Number Calculator

The third root of a complex number is a fundamental operation in complex analysis. This calculator provides an accurate way to find the cube roots of any complex number, with clear explanations of the mathematical process.

What is the Third Root of a Complex Number?

The third root of a complex number z = a + bi (where a and b are real numbers) is a complex number w = x + yi such that w³ = z. Every non-zero complex number has exactly three distinct cube roots in the complex plane.

This concept extends the familiar real number cube roots to the complex plane, where solutions exist for all non-zero numbers. The roots are equally spaced around a circle in the complex plane, forming an equilateral triangle when plotted.

How to Calculate the Third Root

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

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.
Find the cube roots of the magnitude: r^(1/3).
Find the cube roots of the argument: θ/3, θ/3 + 2π/3, θ/3 + 4π/3.
Convert each root back to rectangular form using Euler's formula.

This process gives three distinct roots, each separated by 120° in the complex plane.

The Formula Explained

The three cube roots of a complex number z = a + bi are given by:

w_k = r^1/3 [cos((θ + 2πk)/3) + i sin((θ + 2πk)/3)] for k = 0, 1, 2

Where:

r = √(a² + b²) is the magnitude of z
θ = arctan(b/a) is the argument of z
k is the root index (0, 1, or 2)

This formula accounts for the periodic nature of trigonometric functions and the three distinct roots that exist for any non-zero complex number.

Worked Example

Let's find the cube roots of z = 1 + i:

Convert to polar form: r = √(1² + 1²) = √2, θ = arctan(1/1) = π/4 radians.
Calculate the magnitude root: √2^(1/3) ≈ 1.1447.
Calculate the argument roots:
- θ₀ = π/12 ≈ 0.2618 radians
- θ₁ = π/12 + 2π/3 ≈ 2.3562 radians
- θ₂ = π/12 + 4π/3 ≈ 4.4506 radians
Convert back to rectangular form:
- w₀ ≈ 1.1447(cos(0.2618) + i sin(0.2618)) ≈ 1.0966 + 0.3969i
- w₁ ≈ 1.1447(cos(2.3562) + i sin(2.3562)) ≈ -0.5966 + 1.1219i
- w₂ ≈ 1.1447(cos(4.4506) + i sin(4.4506)) ≈ -0.5480 - 0.8256i

These three roots form an equilateral triangle in the complex plane when plotted.

FAQ

How many cube roots does a complex number have?

Every non-zero complex number has exactly three distinct cube roots. The zero complex number has exactly one cube root (itself).

What's the difference between real and complex cube roots?

Real numbers have one real cube root and two complex conjugate roots. Complex numbers have three distinct roots in the complex plane, none of which are real unless the original number is real.

How do I plot the cube roots of a complex number?

Plot each root in the complex plane using its real part as the x-coordinate and imaginary part as the y-coordinate. The three roots will form an equilateral triangle centered at the origin.

Can I find cube roots of negative numbers?

Yes, negative real numbers have one real cube root and two complex conjugate roots. For example, the cube roots of -8 are -2, 2i√2, and -2i√2.