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R0011-06443 · q-20261009-advance-20261013-r03-rational-number-0043
Exact rational representations have nonzero denominators; a=-19/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.
G001 — Exact rational representations have nonzero denominators; a=-19/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). 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
raw a=[-19, -12];raw b=[1, 3]
After
a=19/12;b=1/3
Conditions: none
Original denominators need only be nonzero;negative denominators are valid.Normalize to reduced positive-denominator fractions for exact arithmetic while retaining the original representations and sources.
Rule: nonzero_signed_rational_representation · Uses: G001
Before
rational comparison difference := sub(19/12, 1/3)
After
rational comparison difference = 5/4
Conditions: none
Compare using the sign of the exact difference,not rounded decimals.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
a−b=5/4
After
a > b
Conditions: none
A negative,zero or positive difference means less,equal or greater;normalization prevents an erroneous order reversal from a negative-denominator representation.
Rule: rational_order_from_difference · Uses: G001, S002
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Recompute rational comparison difference 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.(>, 5/4)
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: c19b6dc2a197ea827e223cb94da9f9958992550a879bfcde1880a666ac67be57candidate record SHA-256: 60b652ed2acc6887deb42b06c9f4f3c7337c787eae2773be7519cf7b6f7a396dsemantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED