Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Every nonzero complex number has a unique polar form r(cos⁡θ+isin⁡θ)r(\cos\theta+i\sin\theta) with r>0r>0 and −π<θ≤π-\pi<\theta\le\pi

Statement

Every z≠0z\ne0 has a unique representation z=r(cos⁡θ+isin⁡θ)z=r(\cos\theta+i\sin\theta) with r=∣z∣>0r=|z|>0 and −π<θ≤π-\pi<\theta\le\pi. The conventions and prerequisite facts used below are recorded in Real and imaginary parts, complex conjugation, and modulus, t↦(cos⁡t,sin⁡t)t\mapsto(\cos t,\sin t) is a bijection from [0,2π)[0,2\pi) onto the real unit circle, The zero sets of sine and cosine and the least positive common period 2 pi.

Facts & Assumptions

Given: z=x+iy≠0z=x+iy\ne0.

Proof

technique · direct
1.1

The point (x/∣z∣,y/∣z∣)(x/|z|,y/|z|) lies on the unit circle.

algebra
1.2

The unit-circle parametrization supplies an angle, and its endpoint convention converts it uniquely to (−π,π](-\pi,\pi].

given
2.1

Multiplying by ∣z∣|z| proves existence; the period theorem proves uniqueness.

given∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 50 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources