Complex Root Calculator - nth Roots in Two Forms

A complex root calculator finds every nth root of a complex number a+bi in one step. The result is n roots on a circle of radius |z|^(1/n), evenly spaced by 2*pi/n radians around the origin. Enter a, b, and n, and the tool returns the principal root plus every other root in rectangular and polar form. Students, engineers, and signal-processing practitioners reach for it whenever a polynomial with complex coefficients or a phasor needs to be unwrapped into multiple harmonics.

• • Solving z^n = w: Find every complex solution of z^n=w in one step, including the angles between them.
• • Roots of unity and phasors: Generate the n-th roots of unity (or any complex number on the unit circle) for DFT, FFT, and phasor work.
• • Companion-matrix eigenvalues: A size-n companion matrix has characteristic polynomial lambda^n - c = 0, so its eigenvalues are the n-th roots of c printed here.
• • Signal and control systems: Decompose a complex pole or zero into n evenly spaced points on the unit circle, a common step in digital-filter design.

Every non-zero complex number has exactly n distinct nth roots, all sitting on a circle whose radius is the nth root of the input magnitude and whose angles are spaced by 2*pi/n. The tool prints each root in a+bi form and as a magnitude/argument pair.

If the next step is to add or multiply two of the roots, the Complex Number Calculator handles the arithmetic in rectangular form. The principal argument atan2(b, a) sits inside this calculation, and a standalone inverse-tangent tool is useful whenever the next part of the same problem needs that angle in degrees or radians.

If you are working on a related problem such as converting a complex number to polar form or adding two complex numbers together, the Arctan Calculator is a natural next step whenever the problem needs the angle that this calculator relies on.

The tool reads a, b, and n, converts a+bi to polar form, and applies the De Moivre rotation to print all n roots. The same calculation reports the shared magnitude, the angle between roots, and the principal argument. A complex root calculator does all three steps at once so the user can focus on reading the result.

• a, b: Real and imaginary parts of the input. Accept any finite float, including 0 and negative values.
• n: Root index. Must be a positive integer in [1, 12]. n = 1 returns the original number; n >= 2 returns the n distinct n-th roots.
• r: Magnitude of the input: r = sqrt(a^2 + b^2). The n-th roots share the magnitude r^(1/n).
• theta: Principal argument of the input, returned by atan2(b, a) in radians and wrapped to (-pi, pi].
• z_k: The k-th n-th root of a + bi. Angles are spaced exactly 2*pi / n radians apart starting from theta / n.

After computing the polar form, the calculator applies the De Moivre rotation 2*pi*k/n for each k, producing n evenly spaced directions, and the constant magnitude r^(1/n) keeps all n points on a single circle. The principal root sits at k=0, at the smallest non-negative angle theta/n.

According to Wikipedia: nth root, every non-zero complex number has exactly n distinct n-th roots, which lie on a circle of radius |z|^(1/n) at angles spaced 2*pi / n apart.

For users who want to revisit the underlying conversion, the Cartesian to Polar Calculator is a quick way to confirm the polar form of a + bi before feeding it to the root calculation.

These four concepts are the building blocks for understanding the calculation. Skim them once and the result panel will read like a description of a familiar geometric picture.

If you are comfortable with these four ideas, the result panel will read like a small table of n points on a circle. The calculator essentially prints the polar form, applies De Moivre, and writes the result back into rectangular form, the same operations you would do on paper.

If you need the magnitude r = sqrt(a^2 + b^2) on its own before applying the root formula, the Vector Magnitude Calculator returns the same Euclidean length sqrt(a^2 + b^2) in a single field.

The tool takes only three inputs and prints a full set of n roots. Enter a and b as the real and imaginary parts of the input, choose a positive integer n, and read the result.

1. 1 Enter the real part a: Type the real part of the input. The default 8 gives the cube roots of 8 as a starting point.
2. 2 Enter the imaginary part b: Type the imaginary part. Use 0 for a real number, or 1 to start from i.
3. 3 Pick the root index n: Choose a positive integer between 1 and 12. n=1 returns the original number, n=2 the two square roots, and so on.
4. 4 Read every root: The principal (k=0) root is in the card as a+bi, and all n roots are listed below as a+bi with arguments. Magnitude, angle, and principal argument are reported in between.

Suppose you need the four fourth roots of 1. The complex root calculator handles this directly: enter a=1, b=0, n=4 to get the principal root 1+0i, magnitude 1, angle between roots 90 degrees, and the four roots 1, i, -1, -i on the unit circle. Change n to 6 for the sixth roots of unity at angles 0, 60, 120, 180, 240, and 300 degrees, the canonical example for the discrete Fourier transform.

Once the n roots are in rectangular form, the Complex Number Calculator multiplies, divides, adds, or subtracts them in a + bi form without re-deriving the polar representation.

A tool that returns all n roots in rectangular and polar form, with the shared magnitude and the angle between roots, saves time on homework, design work, and code reviews. A complex root calculator printing both forms at once is the easiest way to read off a result that would otherwise need a polar conversion, a rotation, and a rectangular conversion by hand.

• • All n roots in one step: Enter a, b, and n once and the tool returns every distinct nth root, no solving z^n=w by hand.
• • Rectangular and polar at the same time: Each root is printed in a+bi form and as a magnitude/argument pair.
• • Geometric pattern visible: The shared magnitude r^(1/n) and the angle 2*pi/n between consecutive roots are reported next to the principal root.
• • Covers special cases cleanly: The tool handles n=1, z=0, and large n with clear messages, and caps n at 12 for readability.
• • Sits next to related math tools: The page links to the cartesian-to-polar, complex-number, and vector-magnitude helpers, so the polar, arithmetic, and magnitude steps are one click away.

The biggest win is avoiding re-deriving the polar form and the De Moivre rotation three times. Reading the shared magnitude and the angle between roots next to the principal root is a quick way to internalise how the n roots sit on a circle.

When the problem is the eigenvalues of a companion matrix, those eigenvalues are the n-th roots of one complex number, and the Characteristic Polynomial Calculator expands the same det(A - lambda*I) = 0 form to the roots printed here.

A few factors control what the tool gives you. Knowing them up front prevents the most common mistakes, especially near special cases like the origin or the unit circle.

• • The tool uses the principal argument atan2(b,a) in (-pi, pi]. Inputs on the negative real axis have theta=pi, so the principal root lands at pi/n, not at the geometrically equivalent 2*pi-pi/n.
• • The tool caps the root index at n=12 to keep the result list readable. Inputs above n=12 are rejected, so the result panel always lists every one of the n roots.

To draw the roots on an Argand diagram by hand, plot the principal root at angle theta/n on a circle of radius r^(1/n), then mark off the other n-1 roots in steps of 2*pi/n.

According to Wolfram MathWorld: Root, the n-th roots of a complex number z are r^(1/n) * exp(i*(theta + 2*pi*k) / n) for k = 0, 1, ..., n-1.

If the printed arguments need to be reused in a calculation that expects gradians or turns, the Angle Converter handles the conversion in both directions.