Complex Roots Calculator

What does this complex roots calculator compute?

The n nth-roots of z are r^{1/n} cis((θ+360°k)/n) for k=0…n−1.

The published formula or method is r^{1/n} cis((θ+360k)/n). The following references support the stated mathematics or method:

• Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare): Complex arithmetic, conjugates, modulus and argument, polar/exponential multiplication and division, de Moivre’s formula and complex roots. Noninteger powers require a specified branch; those branch and numerical policies remain CalculatorX’s own contract. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Keep Cartesian and polar representations consistent. Angle units and the selected root or power branch affect the interpretation of a complex result.

The worked example uses re=1 im=0 n=3 with 1, ω, ω². Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.complex v1.3.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical policy and branch conventions

All angles use degrees. The principal argument lies in (−180°,180°]; a negative real number uses +180°, including a signed zero imaginary part. At the origin, the displayed argument0° is a convention, not a defined mathematical direction. Nonzero inputs and output components/modulus must be finite normal binary64 values (at least2⁻¹⁰²² in magnitude); scaling or intermediate range loss returns NUMERIC_RANGE with rescaling advice. Nonzero decimal text that converts to zero is rejected as NUMERIC_UNDERFLOW.

Power accepts integer |n|≤1000, including negative powers of nonzero numbers;0⁰ is undefined. Roots accept integer2≤n≤64 and return every distinct root in k order; zero has one distinct root. Principal root is k=0 with the argument convention above. Integer powers use scaled rectangular arithmetic; division scales both operands to avoid squared-denominator overflow/underflow. Polar angles are reduced modulo360° before trigonometry, with exact cardinal directions.

The verification budget is2e−11 times the result modulus for rectangular coordinates and modulus, and2e−9 degrees for argument on well-conditioned examples. Components near cancellation may have large relative error even with small vector error. Direction is sensitive near zero and discontinuous at the negative real axis; rounded angles alone cannot reconstruct tiny components. Values remain binary64 approximations, not arbitrary precision or symbolic results.

Interactive calculation runs in your browser. Opening a shared URL sends its query to the site for server rendering; REST/MCP also receive their inputs. The diagram shows all roots; the text work list and CSV/JSON export retain all64 possible roots without truncation.