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R0008-04618 · q-20261008-advance-20261011-r02-arithmetic-operations-0068
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-18/2, candidate r=1/19. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
G001 — All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-18/2, candidate r=1/19. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
exact arithmetic in Q;N={0,1,2,...} when membership is askedAfter
do not truncate subtraction or rational division
Conditions: none
Compute exact results in Q before testing N-membership;do not clamp negative differences to0 or replace rational division by integer division.Division by0 remains undefined.
Rule: exact_rational_arithmetic_scope · Uses: G001
Before
candidate inverse product := mul(-9, 1/19)
After
candidate inverse product = -9/19
Conditions: none
A multiplicative inverse must multiply the original number to1.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
inverse signed defect := sub(-9/19, 1)
After
inverse signed defect = -28/19
Conditions: none
Compute the signed defect ar−1 in that order.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
a=-9;r=1/19;ar=-9/19
After
inverse valid=No;ar−1=-28/19
Conditions: none
For a=0 every candidate product is0,so no inverse exists;even with nonzero a,the candidate must actually produce1.
Rule: multiplicative_inverse_zero_boundary · Uses: G001, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute inverse signed defect with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute candidate inverse product with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(No, -28/19)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 35f648039597ca289250b187520e91e80e40ec53533d4ce4bcedde6a115f5f86candidate record SHA-256: fbae920b7931b9fc1c199bc5c53955c79f0a62ecbe97305cf5f36b3d359f2ebcsemantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67Lean: NOT_RUN · human review: NOT_PERFORMED