Complex Number Calculator

Enter real and imaginary parts for two complex numbers and calculate the result, magnitude, and angle.

Stable numericsSubstitute / reconstructIndependent checkLocal calculation

a (real)

b (imag)

c (real)

d (imag)

Operation

4 - 2iComplex-number result

4.472136Magnitude

-26.565°Argument

Algebra round-trip cluster

Solve once, then verify the same object through another representation

Independent round trips and invariants expose sign, scaling, reconstruction and degeneracy mistakes that a single formatted answer can hide.

Quadratics Matrices Systems

All 28 algebra tools

Matrix InverseA↔A⁻¹ identity round trips MidpointEndpoint reflection round trip Linear EquationScale invariance + degeneracy classes DeterminantTranspose/row-operation invariants Quadratic VertexStandard ↔ vertex coefficient round trip Slope InterceptPoints ↔ line-form equivalence Polynomial DivisionQuotient/remainder reconstruction Standard FormSlope-intercept ↔ standard round trip

Polar form, powers, all nth roots & Argand verification

The primary two-number arithmetic stays first. This optional layer reuses either z₁ or the current arithmetic result, reports rectangular and polar forms, raises the value to an integer power, lists every nth root, plots the source and roots, and checks the largest |wⁿ − z| residual.

SourceOperationExponent / root order

Complex-number verification

Angles use the principal atan2 convention in degrees; floating-point residuals are shown for verification.

Transform, then verify

Equivalent forms should agree. Use discriminants, roots, factoring, vertex form, substitution, matrix checks, or the Remainder Theorem where appropriate to verify the result rather than relying on a single opaque output.

Independent algebra verification

The primary calculation is paired with a substitution, invariant, inverse operation, reconstruction, or equivalent-form check so the displayed answer is easier to audit.

Scope and precision

These are bounded browser calculators, not a universal computer algebra system. WebToolArc keeps exact algebraic identities visible where practical, uses stable floating-point algorithms for numeric work, distinguishes undefined/no-solution/infinite-solution cases, and shows an independent check after the main Calculate action.

Practical guide and verification

Use the tool first, then apply these checks to verify inputs, interpret the result, and hand it off without displacing the primary workflow.

Keep rectangular and polar forms tied to the same complex value

The forms a + bi and r∠θ describe the same number when r is the nonnegative modulus and θ is an argument. Convert using atan2 for the angle so the quadrant is preserved. Because arguments differ by integer multiples of 2π, two correct tools can display different but equivalent angles. Record the angle unit and principal-angle convention when comparing results or handing a phasor to another system.

Use polar form for powers and roots, then verify in rectangular form

De Moivre’s theorem makes integer powers and nth roots transparent in polar coordinates, but every nonzero complex number has n distinct nth roots for integer n greater than one. Listing only the principal root loses valid solutions. After computing a power or every root, convert each result back to rectangular form and raise each root to the requested power; the residual against the original complex number should be near zero within floating-point precision.

Treat division by zero and the zero argument explicitly

Division by 0 + 0i is undefined. The argument of the zero complex number is also not uniquely defined, even though software may display zero degrees as a placeholder. Root formulas need special handling at zero because the only distinct root is zero even though multiplicity remains n. Keep these domain boundaries visible instead of forcing an ordinary polar calculation through cases where its assumptions do not hold.

Use diagrams as orientation checks, not proof

An Argand diagram is valuable for spotting quadrant errors, sign mistakes, and the even angular spacing of nth roots. It is still a scaled visualization, so a point that looks aligned on screen is not a numerical verification. Pair the plot with modulus, argument, rectangular coordinates, and round-trip residuals. For engineering phasors, also confirm the sign convention, frequency-domain convention, and whether the application uses i or j notation.