Complex Root Calculator

Find all nth roots of complex numbers using De Moivre's theorem with detailed step-by-step solutions.

Enter Complex Number & Root

Result

Number of Roots
3
cube roots of 1
Original z1 + 0i
Modulus (r)1
Argument (θ)0°
r^(1/n)1

Step-by-Step Solution

z^(1/n) = r^(1/n) [cos((θ+2kπ)/n) + i·sin((θ+2kπ)/n)]

Understanding Complex Roots

Every nonzero complex number has exactly n distinct nth roots. These roots are evenly spaced around a circle in the complex plane, centered at the origin with radius r1/n. Finding these roots is made systematic by De Moivre's theorem, which provides a straightforward formula for computing each root.

The concept of complex roots extends the idea of square roots and cube roots to the complex plane, where every number (except zero) has exactly n nth roots. This is a fundamental theorem of algebra that shows the completeness of the complex number system.

De Moivre's Theorem for Roots

If z = r(cos θ + i·sin θ) is a complex number in trigonometric form, then the n distinct nth roots of z are given by:

zk = r1/n [cos((θ + 2kπ)/n) + i·sin((θ + 2kπ)/n)]

where k = 0, 1, 2, ..., n-1.

Step-by-Step Process

  1. Convert to polar form: Find the modulus r = |z| and argument θ.
  2. Compute r1/n: Take the nth root of the modulus.
  3. Find each root: For k = 0, 1, ..., n-1, compute the angle (θ + 2kπ)/n.
  4. Convert back: Optionally convert each root back to rectangular form a + bi.

Key Formulas

De Moivre's Theorem (Powers)

Raise a complex number in polar form to any integer power.

z^n = r^n(cos nθ + i·sin nθ)

De Moivre's Theorem (Roots)

Find all nth roots of a complex number.

z_k = r^(1/n) cis((θ+2kπ)/n)

Roots of Unity

The nth roots of 1, evenly spaced on the unit circle.

ω_k = cis(2kπ/n), k=0..n-1

Angular Spacing

The angle between consecutive roots is always the same.

Δθ = 360°/n = 2π/n

Modulus of Each Root

All nth roots share the same modulus.

|z_k| = r^(1/n) for all k

Product of All Roots

The product of all n roots equals the original number z.

z_0 · z_1 · ... · z_(n-1) = z

Geometric Interpretation

The n nth roots of a complex number form the vertices of a regular n-gon (polygon with n sides) inscribed in a circle of radius r1/n. For example, the three cube roots of a number form an equilateral triangle, and the four fourth roots form a square.

This geometric property makes complex roots visually elegant and helps in understanding their symmetric distribution around the complex plane.

Common Examples

Cube Roots of Unity

The three cube roots of 1 are: 1, -1/2 + (√3/2)i, and -1/2 - (√3/2)i. These form an equilateral triangle inscribed in the unit circle, and they satisfy x³ - 1 = 0.

Square Roots of -1

The two square roots of -1 are i and -i. These are the solutions to x² + 1 = 0, and they are diametrically opposite on the unit circle.

Fourth Roots of 1

The four fourth roots of 1 are: 1, i, -1, and -i. These form a square inscribed in the unit circle, and they satisfy x4 - 1 = 0.

Applications

Complex roots are fundamental in many areas of mathematics and engineering. They appear in solving polynomial equations, in Fourier analysis (where roots of unity are the basis for the discrete Fourier transform), in signal processing, in quantum mechanics, and in number theory. The Fast Fourier Transform (FFT) algorithm, which is central to modern computing, relies heavily on properties of roots of unity.