Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • 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.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Real and imaginary parts, complex conjugation, and modulus

Definition

Every z∈C has unique coordinates z=a+bi (C=R[x]/(x2+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a−bi)/(a2+b2)). Define Re⁡z=a,Im⁡z=b,z‾=a−bi, and define its modulus by ∣z∣=a2+b2. The real numbers are least-upper-bound complete (The Cauchy-sequence reals have the least-upper-bound property), so the nonnegative square root exists and is unique by Square roots exist: a unique a≥0 with (a)2=a; the positives are {x2:x≠0}.

Depends on

Dependency tree · 2 levels

Sources