Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02
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

For z=x+iyz=x+iy, set Re⁡z=x\operatorname{Re}z=x, Im⁡z=y\operatorname{Im}z=y, z‾=x−iy\overline z=x-iy, and ∣z∣=x2+y2≥0|z|=\sqrt{x^2+y^2}\ge0. The conventions and prerequisite facts used below are recorded in The complex numbers as R2\mathbb R^2, with their arithmetic, real embedding, and imaginary unit, Square roots exist: a unique a≥0\sqrt{a} \ge 0 with (a)2=a(\sqrt{a})^2 = a; the positives are {x2:x≠0}\{x^2 : x \neq 0\}.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 7 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