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 Engine tested · Specification checked · 18/18 tests · Production surface contract 3/3 · v1.1.0

Input interpretation: Enter values to calculate.
Result: —
Model: Euler / cis form of one complex number.
Scope: atan2 argument in degrees
Verification: Engine tested · 18/18 tests · Production surface contract 3/3 · Specification checked · v1.1.0
Named expert review: Optional · Not performed
Specification basis: ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence: 6 golden · 4 boundary · 8 property · Production surface contract 3/3 · Artifact integrity PASS
Production: Embedded snapshot: STALE · Last attested schema matched 1.1.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract: PASS

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.

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°}