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Math calculator

Complex Number Calculator

Add, subtract, multiply, divide complex numbers; convert rectangular ↔ polar; integer powers, nth roots, and Euler form. Not a CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 12/12 declared partitions (add, sub, mul, div, rect-to-polar, polar-to-rect, conjugate, modulus, power, root, euler, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Complex arithmetic, rectangular/polar conversion, and an engine-linked Argand figure of the result z.
Scope
atan2 argument in degrees
Verification
Engine tested · Source checked · v1.1.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.vector-la-cvp
Verification revision
2026-09-22.vector-la-cvp · 3/3 property · digest 403fe3cb25d5
Legacy regression
18/18 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
11/11 golden · 5/5 CVP boundary · 9/9 invalid · 3/3 property · 5/5 cross-interface · 2/2 cross-calculator · 5/5 CVP contract · Manifest
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • CRC Standard Mathematical Tables — De Moivre and Euler form
Sources
Evidence
6 legacy golden · 4 legacy boundary · legacy regression suite · 11/11 oracle-backed golden · 9/9 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Modulusr = √(re² + im²)
Product(re₁re₂ − im₁im₂) + (re₁im₂ + im₁re₂)i
De Moivrez^n = r^n (cos nθ + i sin nθ)
Eulerz = r e^{iθ}
iθ in degrees, atan2 range (−180°, 180°]. Integer n for powers and roots. Not symbolic simplification. Not i^i.

How to use

1

Choose arithmetic or conversion

Arithmetic needs z₁ (re, im) and z₂ (re2, im2). Polar conversion uses r and arg_deg.

2

Read rectangular and polar together

Every result includes re, im, r, and θ. The Argand figure updates with the result.

Example calculations

Common configurations with formula and result.

ϟ

Sum

(1+2i) + (3+4i)

4 + 6i
4 + 6i
ϟ

3+4i

rect to polar

r=5
5 ∠ 53.13°

Complex Number calculator specification

Version 1.1.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A complex number z = re + im i has modulus r = |z| and argument θ = atan2(im, re).
What it calculates
Complex arithmetic, rectangular/polar conversion, and an engine-linked Argand figure of the result z.
Inputs
  • mode
  • re?
  • im?
  • re2?
  • im2?
  • r?
  • arg_deg?
  • n?
Outputs
  • re
  • im
  • r
  • arg_deg
  • euler
  • roots?
Formula
z = re + im i
Assumptions
  • atan2 argument in degrees
  • Not a CAS
Units
  • dimensionless
  • degrees
Boundary conditions
  • divide by 0+0i → DIVISION_BY_ZERO
  • unknown mode (including simplify) → INVALID_MODE
Example
mode=add re=1 im=2 re2=3 im2=4 → 4+6i
Validation cases

2 published on this page · 18/18 tests · Production surface contract 5/5 · View evidence

  • mode=add re=1 im=2 re2=3 im2=4 → 4+6i
  • mode=rect_to_polar re=3 im=4 → r=5
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Complex number z = re + im i, modulus, and argument
  • CRC Standard Mathematical Tables — De Moivre and Euler form
    Supports: z^n = r^n (cos nθ + i sin nθ); z = r e^{iθ}; nth roots equally spaced on a circle
Calculation version
1.1.0

Other calculators in this family: Complex Polar Calculator, Complex Power Calculator, Complex Roots Calculator, Euler Form Calculator . Explore all Complex Numbers.

→ Complex Polar Calculator Convert a complex number between rectangular (re, im) and polar (r, θ°) form. Same engine as Complex Number. Not a CAS. Runs locally. › → Complex Power Calculator Integer power of a complex number via De Moivre. Same engine as Complex Number. Not a CAS. Not i^i. Runs locally. › → Complex Roots Calculator All n nth-roots of a complex number via De Moivre. Same engine as Complex Number. Not a CAS. Not the real root page. Runs locally. › → Euler Form Calculator Euler form of a complex number: z = r e^{iθ} = r (cos θ + i sin θ). Same engine as Complex Number. Not a CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a computer-algebra system?

No. It does arithmetic and rectangular/polar conversion. It does not simplify expressions.

Where does this run?

Locally in the browser by default. REST and MCP call the same complex engine.