Math calculator

Euler Form Calculator

Q: Is this a second complex engine?

No. It is mode=euler on math.complex, the same engine as /calc/math/complex-number.

Q: Is this complex analysis?

No. It writes Euler form. Poles, residues, and contour integrals stay out of scope.

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.

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: Euler / cis form of one complex number, and an engine-linked Argand figure.
Scope: atan2 argument in degrees
Verification: Engine tested · Specification 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
Specification basis: ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
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

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

Formulas

Core equations used by this calculator.

Eulerz = r e^{iθ}

Cise^{iθ} = cos θ + i sin θ

Identitye^{iπ} = −1

iθ in degrees (−180°, 180°]. r ≥ 0. Polar → Euler with r=1, θ=180° is Euler’s identity. Not a second complex engine. Not contour integrals.

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°

e^{iπ} = −1

−1

i

0+1i

e^{i 90°}

1 e^{i 90°}