Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=3/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
G001 — Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=3/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).
Definition and scope
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.
original integer numerator/denominator representations;original denominators nonzero
After
representation and rational value are distinct data
Conditions: none
Negative denominators are valid,but reduced values do not replace original representations.Candidate algorithms about raw numerators/denominators must use those raw data.This finite audit is not a general proof of quotient well-definedness.
Test whether the value is0,not the sign of the original denominator.A zero numerator over a negative nonzero denominator is valid zero,but has no reciprocal.
Test whether the value is0,not the sign of the original denominator.A zero numerator over a negative nonzero denominator is valid zero,but has no reciprocal.
S009 · premise_and_result_audit — Recompute product of two reciprocals with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Recompute reciprocal reverse product1 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute fraction reciprocal1 with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Recompute reciprocal reverse product0 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute fraction reciprocal0 with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute fraction 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.
Result
(-2/3, [Yes, Yes], [1/1, -3/2], -3/2)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
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.