Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-02
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • 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) with r>0 and −π<θ≤π

Statement

Every z≠0 has a unique representation z=r(cos⁡θ+isin⁡θ) with r=∣z∣>0 and −π<θ≤π. The conventions and prerequisite facts used below are recorded in Real and imaginary parts, complex conjugation, and modulus, t↦(cos⁡t,sin⁡t) is a bijection from [0,2π) 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≠0.

Proof

technique · direct
1.1algebra

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

1.2given

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

2.1given∎

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

Depends on

Dependency tree · 2 levels

Sources