Euler Form Calculator
Convert rectangular or polar complex numbers into Euler form with angles in degrees. Explore the Argand diagram, worked examples and numerical limits.
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
- Euler / cis form of one complex number, and an engine-linked Argand figure.
- Scope
- atan2 argument in degrees
- Verification
- Engine tested · Source checked · v1.3.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
- Versions
- Calculation 1.3.0 · CVP protocol 1.0.0-proposed · Evidence 2026-10-29.math-uuce-fields-v1
- Verification revision
- 2026-10-29.math-uuce-fields-v1 · 3/3 property · digest e0f5f79d4366
- 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
- 643/643 golden · 5/5 CVP boundary · 9/9 invalid · 3/3 property · 4/4 metamorphic · 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
- Evidence
- 6 legacy golden · 4 legacy boundary · legacy regression suite · 643/643 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 rect or polar
Rect uses re and im. Polar uses r and θ. Default polar 1 ∠ 180° is −1.
Read r e^{iθ}
Arithmetic stays on /calc/math/complex-number. The Argand figure updates with z.
Example calculations
Common configurations with formula and result.
Euler identity
r=1 θ=180°
i
0+1i
Euler Form calculator specification
Version 1.3.0 · Engine tested
- Engine tested 18/18 tests · Production surface contract 5/5
- Named expert review Not performed
- Calculation version 1.3.0
- Definition
- z = r e^{iθ} with θ in degrees, atan2 convention. This page is mode=euler on math.complex. Polar conversion is /complex-polar. Not complex analysis.
- What it calculates
- Euler / cis form of one complex number, and an engine-linked Argand figure.
- Inputs
- from?
- re?
- im?
- r?
- arg_deg?
- Outputs
- re
- im
- r
- arg_deg
- euler
- cis
- Formula
z = r e^{iθ}- Assumptions
- atan2 argument in degrees
- r ≥ 0
- Not a CAS
- Not complex analysis
- Units
- dimensionless
- degrees
- Boundary conditions
- r < 0 → VALUE_MUST_BE_NON_NEGATIVE
- unknown mode (including log) → INVALID_MODE
- Example
- from=polar r=1 arg_deg=180 → −1
- Validation cases
2 published on this page · 18/18 tests · Production surface contract 5/5 · View evidence
- from=polar r=1 arg_deg=180 → re=-1 im=0
- re=0 im=1 → r=1 arg_deg=90
- 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.
- ISO 80000-2:2019 Quantities and units — Mathematics
- Calculation version
- 1.3.0
Background
Interpretation and common distinctions.
What does this euler form calculator compute?
z = r e^{iθ} with θ in degrees, atan2 convention.
The published formula or method is z = r e^{iθ}. The following references support the stated mathematics or method:
- 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 r=1 θ=180° with e^{iπ} = −1. 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.
Related tools
Other calculators in this family: Complex Number Calculator, Complex Polar Calculator, Complex Power Calculator, Complex Roots Calculator . Explore all Complex Numbers.
Frequently asked questions
Key distinctions behind the calculation.
Is this a second complex engine?
No. It is mode=euler on math.complex, the same engine as /calc/math/complex-number.
Is this complex analysis?
No. It writes Euler form. Poles, residues, and contour integrals stay out of scope.