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.

The complex numbers as $\mathbb R^2$ , with their arithmetic, real embedding, and imaginary unit

Definition

Set $\mathbb C=\mathbb R^2$ , write $(x,y)=x+iy$ , embed $x\in\mathbb R$ as $(x,0)$ , and put $i=(0,1)$ . Define $(x+iy)+(u+iv)=(x+u)+i(y+v),\qquad (x+iy)(u+iv)=(xu-yv)+i(xv+yu).$ Then $i^2=-1$ . The conventions and prerequisite facts used below are recorded in The real numbers.

Depends on

The real numbers

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 8 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

J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)

The complex numbers as R2\mathbb R^2R2, with their arithmetic, real embedding, and imaginary unit