Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-07-24
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.

Arithmetic on the rationals

Definition

On Q\mathbb{Q} (The rationals as equivalence classes of pairs of integers) define, on representatives,

[(a,b)]+[(c,d)]:=[(ad+cb,  bd)],[(a,b)]⋅[(c,d)]:=[(ac,  bd)],[(a,b)] + [(c,d)] := [(ad + cb,\; bd)], \qquad [(a,b)] \cdot [(c,d)] := [(ac,\; bd)], −[(a,b)]:=[(−a,b)],0:=[(0,1)],1:=[(1,1)],-[(a,b)] := [(-a, b)], \qquad 0 := [(0,1)], \qquad 1 := [(1,1)],

and, for [(a,b)]≠0[(a,b)] \ne 0 (equivalently a≠0a \ne 0), the inverse [(a,b)]−1:=[(b,a)][(a,b)]^{-1} := [(b,a)].

Remarks

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 10 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