# Question 6939: Fraction Product and Two Reciprocal Guards [Return to catalogue page 35](https://ocme.evemisslab.com/en/questions/page/35/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261011-advance-20261015-r01-fraction-foundations/#q-20261011-advance-20261015-r01-fraction-0064) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0012-06939 · q-20261011-advance-20261015-r01-fraction-0064 ## Original question and givens Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/3, 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=1/3, 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. - [mko-fraction](https://ocme.evemisslab.com/en/mko/mko-fraction/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: fraction_representation_value_scope · Uses: G001 S002 · calculationBefore fraction product := mul(1/3, -2/3) After fraction product = -2/9 Conditions: none Multiply the values of the two valid fractions;zero values may still be multiplied. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore fraction reciprocal0 := div(1, 1/3) After fraction reciprocal0 = 3 Conditions: none A multiplicative reciprocal exists only when the fraction's value is nonzero. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore reciprocal reverse product0 := mul(1/3, 3) After reciprocal reverse product0 = 1 Conditions: none Multiply the value by its reciprocal to check that the result is1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore value=1/3;nonzero=Yes After reciprocal=3/1 Conditions: none 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. Step identifiers and dependenciesRule: fraction_reciprocal_value_guard · Uses: G001, S004 S006 · calculationBefore fraction reciprocal1 := div(1, -2/3) After fraction reciprocal1 = -3/2 Conditions: none A multiplicative reciprocal exists only when the fraction's value is nonzero. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore reciprocal reverse product1 := mul(-2/3, -3/2) After reciprocal reverse product1 = 1 Conditions: none Multiply the value by its reciprocal to check that the result is1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore value=-2/3;nonzero=Yes After reciprocal=-3/2 Conditions: none 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. Step identifiers and dependenciesRule: fraction_reciprocal_value_guard · Uses: G001, S007 S009 · calculationBefore product of two reciprocals := mul(3, -3/2) After product of two reciprocals = -9/2 Conditions: none Multiply the reciprocals only when both are defined. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 ## Reverse checks - 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 (-2/9, [Yes, Yes], [3/1, -3/2], -9/2) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 3c2fd6142878644828d9334b76e439b1d05c394af7fabec21fcd8b636541869f - candidate record SHA-256: 3627a2f50ac6c850985f7cac23d12dcd485f4dda657f0b360b8e2fefe12d82e0 - semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00 - question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261011-advance-20261015-r01-fraction-0064.json)