Complex Root Calculator

To find the n-th roots of a complex number z = a + bi, the calculator first converts the input into its polar representation (magnitude r and phase angle φ). The magnitude is determined using the Pythagorean relation r = sqrt(a² + b²), and the phase angle is found via φ = atan2(b, a). According to De Moivre's formula, the n distinct roots are computed by taking the n-th root of the magnitude and dividing the phase angle plus multiples of 2π by the root degree n. Each resulting root is then transformed back into its Cartesian components (real part a and imaginary part b) for easy analysis.

Suppose you want to find the square roots (n = 2) of the complex number z = 3 + 4i. First, compute the magnitude: r = sqrt(3² + 4²) = sqrt(25) = 5. Next, find the phase angle: φ = atan2(4, 3) ≈ 0.9273 radians (or 53.13 degrees). For the first root (k = 0), the magnitude of the root is sqrt(5) ≈ 2.236, and the phase is 0.9273 / 2 = 0.4636 radians. Converting back to Cartesian form yields approximately 1.986 + 1.023i. For the second root (k = 1), add 2π to the phase before dividing by 2, resulting in -1.986 - 1.023i.

When working with complex roots, always verify whether your input angles are in degrees or radians to avoid unexpected calculation shifts. If your complex number lies on the negative real axis, pay close attention to the branch cuts of the arctangent function. Utilizing the polar coordinate input mode can drastically simplify calculations involving high root degrees, as multiplying and raising powers becomes straightforward multiplication and division.