How do you find the nth roots of a complex number?

To find the nth roots of a complex number z = a + bi, first convert to polar form: r = sqrt(a^2 + b^2) and theta = atan2(b, a). The n distinct nth roots are given by De Moivre's theorem: z_k = r^(1/n) * (cos((theta + 2*pi*k)/n) + i*sin((theta + 2*pi*k)/n)) for k = 0, 1, 2, ..., n-1. Each root has the same modulus r^(1/n) but their arguments are evenly spaced 2*pi/n radians apart around the origin.

How many nth roots does a complex number have?

Every non-zero complex number has exactly n distinct nth roots in the complex plane. For example, a square root has 2 roots, a cube root has 3 roots, and so on. These roots are equally spaced on a circle of radius r^(1/n) in the Argand diagram. This is a consequence of the Fundamental Theorem of Algebra: the equation z^n = c has exactly n solutions (counting multiplicity) in the complex numbers.

What is De Moivre's theorem?

De Moivre's theorem states that for any complex number in polar form r*(cos(theta) + i*sin(theta)) and any integer n: (r*(cos(theta) + i*sin(theta)))^n = r^n * (cos(n*theta) + i*sin(n*theta)). This theorem is named after Abraham de Moivre (1667-1754) and is fundamental to working with powers and roots of complex numbers. It connects complex numbers to trigonometry and is a direct consequence of Euler's formula e^(i*theta) = cos(theta) + i*sin(theta).

Complex Root Calculator

This calculator finds every nth root of a complex number, applying De Moivre's theorem to do in seconds what otherwise takes several steps of trigonometry by hand. You enter a complex number in rectangular form, its real part (a) and imaginary part (b), then choose the root degree n, from square roots (n = 2) up to an eighth root, or type in a custom degree between 2 and 20, and pick whether angles display in degrees or radians. The calculator first converts your number to polar form, showing its modulus and argument, then works out the modulus every root will share. It returns a full results table listing all n distinct roots, each shown in rectangular form (a + bi) and polar form, alongside a verification column that raises each root back to the power of n to confirm it returns your original number. A worked example panel walks through the maths step by step using the default inputs, and a summary note explains how the roots sit evenly spaced around a circle in the Argand plane, forming a regular polygon with n vertices. Use it to check homework, verify hand calculations, or explore how complex roots behave for different values of n. Because every non-zero complex number has exactly n distinct nth roots, expect n rows in the table every time you change the degree.

Calculate.co.nz is proud to be partnered with realtor.co.nz, a trusted resource for navigating the New Zealand property market. Their Helpful Articles section offers clear, well-structured insights across buying, selling, and building, making complex real estate topics more accessible. With a focus on up-to-date guidance and practical knowledge, they empower Kiwis to move forward with clarity and confidence in a constantly evolving property landscape.

Method verified De Moivre's theorem. Standard result from complex analysis.

1. Complex Number

Enter the complex number z = a + bi whose nth roots you want to find.

Real part (a) the part without i

Imaginary part (b) coefficient of i

z = 8 + 0i

2. Root Degree

Choose which root to compute. n = 2 gives square roots, n = 3 gives cube roots, and so on.

Root degree (n)

Custom n integer, 2 to 20

Angle unit for output

Results Summary

Modulus |z|

Distance from origin

Argument arg(z)

Angle in chosen units

Root modulus |z|^(1/n)
-
Modulus of each root

All nth Roots

Root k	Rectangular form (a + bi)	Modulus r	Argument θ	Verification z_k^n

Input in Polar Form

Real part (a)-

Imaginary part (b)-

Modulus r = sqrt(a^2 + b^2)-

Argument theta = atan2(b, a)-

Root degree n-

Root modulus r^(1/n)-

Angle spacing 2*pi/n-

Worked Example (Default)

Inputz = 8 + 0i

Root degreen = 3 (cube root)

Modulusr = 8

Argumenttheta = 0 deg

Root modulus8^(1/3) = 2

Root k=02 + 0i

Root k=1-1 + 1.7321i

Root k=2-1 - 1.7321i

Note: All roots lie on a circle of radius |z|^(1/n) in the Argand plane, equally spaced by 2pi/n radians.

How to Calculate nth Roots of a Complex Number

Any non-zero complex number has exactly n distinct nth roots in the complex plane. The method relies on expressing the number in polar form and applying De Moivre's theorem in reverse.

The Formula

Given a complex number z = a + bi, first convert to polar form:

Modulus: r = sqrt(a² + b²)
Argument: θ = atan2(b, a) (in radians)

Then the n distinct nth roots are:

z_k = r^(1/n) × [ cos((θ + 2πk) / n) + i × sin((θ + 2πk) / n) ]

where k = 0, 1, 2, ..., n-1. Each root has the same modulus r^(1/n) but a different argument, equally spaced by 2π/n radians around the circle.

Worked Example: Cube Roots of 8

Find all cube roots of z = 8 (i.e. a = 8, b = 0, n = 3).

Convert to polar form: r = 8, θ = 0
Root modulus: 8^(1/3) = 2
Angle spacing: 2π/3 = 120 degrees

k	Angle	Rectangular form	Check: z_k³
0	0 deg	2 + 0i	8 + 0i = 8 ✓
1	120 deg	-1 + 1.7321i	8 + 0i = 8 ✓
2	240 deg	-1 - 1.7321i	8 + 0i = 8 ✓

The three cube roots of 8 are 2, -1+√3 i, and -1-√3 i. Only z_0 = 2 is the familiar real cube root; the other two are complex conjugates of each other.

De Moivre's Theorem

De Moivre's theorem states that for a complex number in polar form z = r(cos θ + i sin θ) and any integer n:

z^n = r^n × (cos(nθ) + i sin(nθ))

This theorem, named after Abraham de Moivre (1667-1754), is the basis for calculating roots. To find the nth roots, you reverse the process: instead of multiplying the angle by n, you divide it by n (and account for all n distinct values by adding multiples of 2π/n).

Geometrical Interpretation

The nth roots of a complex number always form a regular polygon with n vertices inscribed in a circle of radius |z|^(1/n) centred at the origin in the Argand diagram. For example, the cube roots of any number form an equilateral triangle, the fourth roots form a square, and the sixth roots form a regular hexagon. The vertices are equally spaced by 360/n degrees.

Special Cases

Square roots (n = 2): Any complex number has exactly 2 square roots that are negatives of each other: z_0 and -z_0.
Roots of unity: The nth roots of 1 (z = 1 + 0i) are evenly spaced on the unit circle. They include the real roots 1 (always) and -1 (when n is even).
Real positive numbers: The real nth root is always z_0 (k = 0). The remaining roots are complex.
Zero: z = 0 has only one nth root, which is 0 itself (zero is a repeated root).

Related Calculators

Maths and Stats Calculators: all maths tools in one place.
Quadratic Formula Solver: find the roots of ax² + bx + c = 0, including complex roots.
Cubic Equation Solver: solve cubic polynomials, which can yield one or three real roots.
Combination (nCr) Calculator: count combinations without repetition.

Method: De Moivre's theorem for nth roots of complex numbers. Standard result from complex analysis; see any undergraduate complex analysis or algebra textbook (e.g. Stewart, Complex Analysis; Brown and Churchill, Complex Variables and Applications).

Results are calculated in double-precision floating point. Rounding to 6 significant figures may cause small discrepancies in verification. For exact symbolic results, use a computer algebra system.