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.
Enter values to calculate.
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
- Rectangular ↔ polar conversion of one complex number, and an engine-linked Argand figure.
- 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
- 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
- NIST DLMF — 4.2 Definitions: Logarithm and Exponential
- Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare)
- Evidence
- 6 legacy golden · 4 legacy boundary · legacy regression suite · 11/11 oracle-backed golden · 9/9 invalid · Artifact integrity PASS
- Semantic contract
- PASS
Full verification
Formulas
Core equations used by this calculator.
How to use
Choose direction
Rect → polar needs re and im. Polar → rect needs r and θ in degrees.
Read both forms
Every result includes re, im, r, and θ. The Argand figure updates with z.
Example calculations
Common configurations with formula and result.
3+4i
rect to polar
5 ∠ 53.13°
polar to rect
Complex Polar calculator specification
Version 1.1.0 · Engine tested
- Engine tested 18/18 tests · Production surface contract 5/5
- Named expert review Not performed
- Calculation version 1.1.0
- Definition
- z = re + im i = r (cos θ + i sin θ) with r = |z| and θ = atan2(im, re) in degrees. This page is a discovery surface on the complex engine (rect_to_polar / polar_to_rect). Arithmetic stays on /calc/math/complex-number.
- What it calculates
- Rectangular ↔ polar conversion of one complex number, and an engine-linked Argand figure.
- Inputs
- mode
- re?
- im?
- r?
- arg_deg?
- Outputs
- re
- im
- r
- arg_deg
- Formula
re = r cos θ, im = r sin θ- Assumptions
- atan2 argument in degrees
- r ≥ 0
- Not a CAS
- Units
- dimensionless
- degrees
- Boundary conditions
- r < 0 → VALUE_MUST_BE_NON_NEGATIVE
- unknown mode (including simplify) → INVALID_MODE
- Example
- mode=rect_to_polar re=3 im=4 → r=5
- Validation cases
2 published on this page · 18/18 tests · Production surface contract 5/5 · View evidence
- mode=rect_to_polar re=3 im=4 → r=5
- mode=polar_to_rect r=5 arg_deg=53.1301023542 → re≈3 im≈4
- Sources
- ISO 80000-2:2019 Quantities and units — MathematicsSupports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
- NIST DLMF — 4.2 Definitions: Logarithm and ExponentialSupports: Complex exponential and logarithm definitions, including the exponential’s trigonometric representation and branch conventions. Mathematical reference only; not validation or endorsement of this implementation.
- Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare) — §9.4 Complex NumbersSupports: 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.
- ISO 80000-2:2019 Quantities and units — Mathematics
- Calculation version
- 1.1.0
Background
Interpretation and common distinctions.
What does this complex polar calculator compute?
z = re + im i = r (cos θ + i sin θ) with r = |z| and θ = atan2(im, re) in degrees.
The published formula or method is re = r cos θ, im = r sin θ. 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.
- NIST DLMF — 4.2 Definitions: Logarithm and Exponential: Complex exponential and logarithm definitions, including the exponential’s trigonometric representation and branch conventions. Mathematical reference only; not validation or endorsement of this implementation.
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 rect to polar with r=5. 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 v1.1.0 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.
Related tools
Other calculators in this family: Complex Number Calculator, Complex Power Calculator, Complex Roots Calculator, Euler Form Calculator . Explore all Complex Numbers.
Frequently asked questions
Key distinctions behind the calculation.
Is this a second complex engine?
No. It is rect_to_polar / polar_to_rect on math.complex, the same engine as /calc/math/complex-number.
Can I add two complex numbers here?
Use /calc/math/complex-number for add, subtract, multiply, divide, conjugate, and modulus.