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Math calculator

Complex Number Calculator

Add, subtract, multiply, divide complex numbers; convert rectangular ↔ polar. Not a CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 10/10 tests · Production surface contract 3/3 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Complex arithmetic and rectangular/polar conversion.
Scope
atan2 argument in degrees
Verification
Engine tested · 10/10 tests · Production surface contract 3/3 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 2 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status PASS (0 stale; 164 CURRENT) @ 2026-09-16T06:44:24.909Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Modulusr = √(re² + im²)
Product(re₁re₂ − im₁im₂) + (re₁im₂ + im₁re₂)i
Polarre = r cos θ, im = r sin θ
iθ in degrees, atan2 range (−180°, 180°]. Not symbolic simplification.

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.

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 1.0.0 · 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 and rectangular/polar conversion.
Inputs
  • mode
  • re?
  • im?
  • re2?
  • im2?
  • r?
  • arg_deg?
Outputs
  • re
  • im
  • r
  • arg_deg
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 · 10/10 tests · Production surface contract 3/3 · View evidence

  • mode=add re=1 im=2 re2=3 im2=4 → 4+6i
  • mode=rect_to_polar re=3 im=4 → r=5
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Linear equation, Quadratic, Sine .

ax+b Linear equation › Δ Quadratic › sin Sine ›
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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.