ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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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.

A negative irrational has a regular continued fraction with positive later digits

Example

For the negative irrational $- \sqrt{2}$ , the continued-fraction algorithm gives $- \sqrt{2} = [- 2; 1, 1, \overline{2}] .$ The only negative digit is the initial one; every later digit is positive.

Facts & Assumptions

Given: The real number $- \sqrt{2}$ .

[F1]

The complete-quotient algorithm chooses the unique integer part $a_{n}$ with $a_{n} \leq α_{n} < a_{n} + 1$ , and whenever the next complete quotient exists it is $α_{n + 1} = 1 / (α_{n} - a_{n})$ (Complete quotients in the continued-fraction algorithm).

[F2]

The continued-fraction algorithm reconstructs the original real number from its digits (The continued-fraction algorithm for real numbers).

Verification

technique · direct

1.1

Since $- 2 < - \sqrt{2} < - 1$ , the first digit is $a_{0} = - 2$ . Then. [F1, given, algebra] $α_{1} = \frac{1}{- \sqrt{2} + 2} = \frac{2 + \sqrt{2}}{2}, α_{2} = \frac{1}{α_{1} - 1} = \sqrt{2},$ so $a_{1} = 1$ and $a_{2} = 1$ .

F1givenalgebra

2.1

One more step gives. [F1, F2, step 1.1, algebra] $α_{3} = \frac{1}{\sqrt{2} - 1} = \sqrt{2} + 1,$ so every later digit is $2$ . Hence the digit string is $[- 2; 1, 1, \overline{2}],$ and [F2] identifies its value with $- \sqrt{2}$ . In particular the negative sign is absorbed entirely into the first digit, while every later digit stays positive as required by [F1].

F1F2step 1.1algebra∎

Depends on

Used by

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Sources

Peter Hackman, Elementary Number Theory (standard reference, not scraped)
William Stein, Elementary Number Theory: Primes, Congruences, and Secrets (standard reference, not scraped)