Complex Root Calculator
Enter a complex number z = a + bi and a root degree n. The calculator finds all n distinct nth roots using De Moivre's theorem, showing each root in Cartesian form (x + yi), polar form (r*(cos theta + i*sin theta)), and exponential form (r*e^(i*theta)). Step-by-step working and a visual showing how the roots sit on a circle in the complex plane are included.
Formula
Worked example
Find the square roots (n = 2) of z = 1 + i. Modulus: r = sqrt(1^2 + 1^2) = sqrt(2) ≈ 1.4142. Argument: theta = atan2(1, 1) = pi/4 ≈ 0.7854 rad. Root modulus: sqrt(2)^(1/2) = 2^(1/4) ≈ 1.1892. Root 0 (k=0): w_0 = 1.1892*(cos(pi/8) + i*sin(pi/8)) ≈ 1.0987 + 0.4551i. Root 1 (k=1): w_1 = 1.1892*(cos(9*pi/8) + i*sin(9*pi/8)) ≈ -1.0987 - 0.4551i. Verification: (1.0987 + 0.4551i)^2 = 1.1892*1.1892*(cos(pi/4) + i*sin(pi/4)) = 1.4142*(0.7071 + 0.7071i) ≈ 1 + i.
What is a complex nth root?
Every nonzero complex number z has exactly n distinct nth roots: complex numbers w such that w^n = z. This is guaranteed by the Fundamental Theorem of Algebra. Unlike real numbers, where a negative number has no real square root, complex arithmetic always yields n complete solutions. For example, the cube roots of 8 are not just 2 but also two complex numbers: 2*(-1/2 + (sqrt(3)/2)*i) and 2*(-1/2 - (sqrt(3)/2)*i). The key insight is that the n roots are evenly spaced around a circle in the complex plane, forming a regular n-gon whose vertices all lie at the same distance from the origin.
De Moivre's theorem: the formula
The nth roots of z are found by converting z to polar form r*(cos(theta) + i*sin(theta)), where r = |z| is the modulus and theta = arg(z) is the argument (principal value in (-pi, pi]). De Moivre's theorem then gives: w_k = r^(1/n) * (cos((theta + 2*pi*k)/n) + i*sin((theta + 2*pi*k)/n)) for k = 0, 1, ..., n-1. The principal root is the k=0 case. Each subsequent root rotates 2*pi/n radians counterclockwise from the previous one. The modulus of every root is the same: r^(1/n). The three equivalent notations - Cartesian (a + bi), polar (r*(cos(theta) + i*sin(theta))), and exponential (r*e^(i*theta)) - all describe exactly the same complex number; which form you use depends on context.
Polar and exponential forms
Euler's formula e^(i*theta) = cos(theta) + i*sin(theta) links polar and exponential notation. The exponential form r*e^(i*theta) is especially compact and makes multiplication and exponentiation simple: (r1*e^(i*theta1)) * (r2*e^(i*theta2)) = (r1*r2)*e^(i*(theta1+theta2)), which geometrically means moduli multiply and arguments add. This is why De Moivre's theorem is so clean: raising r*e^(i*theta) to the power n gives r^n * e^(i*n*theta), so the nth root reverses this - dividing the argument by n and taking the positive real nth root of the modulus. The ambiguity in the argument (any multiple of 2*pi/n can be added) is exactly what produces the n distinct solutions.
Applications in engineering and mathematics
Complex roots appear throughout applied mathematics and engineering. In AC circuit analysis, the three-phase power system uses the cube roots of unity to represent 120-degree-shifted voltages. In signal processing, the Fast Fourier Transform (FFT) relies on the nth roots of unity as twiddle factors that efficiently rotate signal components in the frequency domain. Polynomial factoring uses the fact that z^n - c = 0 has exactly the nth roots of c as its solutions. Control theory uses root locus analysis, where the stability of a feedback system depends on where the roots of the characteristic polynomial lie in the complex plane. Crystallography uses roots of unity to describe the rotational symmetry of crystal lattices.
Common roots of unity
| n | Root type | Arguments (degrees) | Applications |
|---|---|---|---|
| 2 | Square roots of unity | 0 deg, 180 deg | Sign changes, alternating series |
| 3 | Cube roots of unity | 0 deg, 120 deg, 240 deg | Three-phase AC power, DFT |
| 4 | Fourth roots of unity | 0 deg, 90 deg, 180 deg, 270 deg | Rotational symmetry, FFT butterflies |
| 6 | Sixth roots of unity | 0 deg, 60 deg, 120 deg, 180 deg, 240 deg, 300 deg | Hexagonal symmetry, crystal structures |
| 8 | Eighth roots of unity | Multiples of 45 deg | FFT radix-8 algorithms |
The nth roots of unity are the n solutions to z^n = 1. They form a regular n-gon on the unit circle.
Frequently asked questions
How many nth roots does a complex number have?
Every nonzero complex number has exactly n distinct nth roots. The number zero has only one nth root, which is zero. The n roots all share the same modulus (r^(1/n)) and are evenly spaced around a circle in the complex plane, separated by an angle of 2*pi/n radians (360/n degrees).
What is De Moivre's theorem?
De Moivre's theorem states that (cos(theta) + i*sin(theta))^n = cos(n*theta) + i*sin(n*theta) for any integer n. Applied to roots, it means the nth roots of r*(cos(theta) + i*sin(theta)) are r^(1/n) * (cos((theta + 2*pi*k)/n) + i*sin((theta + 2*pi*k)/n)) for k = 0 through n-1. This is the core formula used by this calculator.
What is the principal root?
The principal nth root corresponds to k = 0 in De Moivre's formula. It uses the principal argument theta in the range (-pi, pi]. For positive real numbers the principal root is the familiar positive real nth root. For other complex numbers the principal root has the smallest non-negative argument among the n roots.
What is the argument of a complex number?
The argument (or phase) of a complex number z = a + bi is the angle theta = atan2(b, a) measured counterclockwise from the positive real axis. The principal argument lies in the interval (-pi, pi]. In radians, a full circle is 2*pi ≈ 6.2832 rad; in degrees, it is 360 degrees. This calculator uses radians internally and optionally shows degrees in the root table.
Why do square roots of negative numbers give complex results?
The square root of -1 cannot be a real number because squaring any real number gives a non-negative result. Complex numbers extend the real line to the plane by adding the imaginary unit i, defined so that i^2 = -1. The square roots of -1 are i and -i, and this calculator will find them: enter a = 0, b = 0 for the imaginary part is wrong; instead enter a = -1, b = 0 and n = 2, and the two roots will be 0 + 1i and 0 - 1i.
How do I verify a computed root is correct?
Raise the computed root w to the nth power and check that the result equals the original z. In Cartesian form, multiply w by itself n times using the rule (a + bi)*(c + di) = (ac - bd) + (ad + bc)i. In polar form it's simpler: raise the modulus to n and multiply the argument by n, then convert back to Cartesian. If the result matches z (within rounding error), the root is correct.