Enter a complex number in rectangular form (real part + imaginary part) and choose the root degree (n) to find all n-th roots using De Moivre's formula. You get back every distinct complex root displayed in both rectangular (a + bi) and polar form, along with a unit-circle chart showing their geometric positions.
Modulus of Each Root (r^(1/n))
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Base Argument (θ₀) in Degrees
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Number of Distinct Roots
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Modulus of Input |z|
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A complex number w is an n-th root of another complex number z if w^n = z. Every complex number has exactly n distinct n-th roots. For example, every complex number has two square roots, three cube roots, and so on. These roots are evenly spaced around a circle in the complex plane.
Convert the complex number to polar form r(cos θ + i sin θ), where r is the modulus and θ is the argument. Each n-th root has modulus r^(1/n) and argument (θ + 2πk)/n for k = 0, 1, …, n−1. This is De Moivre's Formula, the standard method for finding all complex roots.
De Moivre's Formula states that all n-th roots of r(cos θ + i sin θ) are given by r^(1/n) · (cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)) for k = 0, 1, …, n−1. It provides a systematic way to compute every distinct root of a complex number.
The n-th roots of unity are the n complex numbers that satisfy w^n = 1. They are evenly distributed on the unit circle in the complex plane, forming the vertices of a regular n-gon. For example, the square roots of unity are 1 and −1, while the cube roots are 1, (−1+i√3)/2, and (−1−i√3)/2.
Every non-zero complex number has exactly n distinct n-th roots. So every complex number has 2 square roots, 3 cube roots, 4 fourth roots, and so on. These roots are equally spaced in angle by 2π/n radians (360°/n) on a circle of radius r^(1/n) in the complex plane.
The polar form expresses a complex number z = a + bi as r(cos θ + i sin θ), where r = √(a² + b²) is the modulus (distance from the origin) and θ = atan2(b, a) is the argument (angle from the positive real axis). Polar form is essential for applying De Moivre's Formula.
The complex square root is the special case of n = 2. Every complex number z = a + bi has exactly two square roots, which are negatives of each other (w and −w). For a purely real positive number, these are the familiar ±√a. For other complex numbers, both roots have the same modulus but arguments differing by 180°.
The n-th roots of a complex number all share the same modulus r^(1/n) but have arguments separated by exactly 2π/n radians. Since they lie on a circle of the same radius and are equally spaced in angle, they form the vertices of a regular n-gon inscribed in that circle in the complex plane.