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R0011-06410 · q-20261009-advance-20261013-r03-rational-number-0060
Exact rational representations have nonzero denominators; a=10/-3, b=11/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.
G001 — Exact rational representations have nonzero denominators; a=10/-3, b=11/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). 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=[10, -3];raw b=[11, -4]
After
a=-10/3;b=-11/4
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
guarded rational quotient := div(-10/3, -11/4)
After
guarded rational quotient = 40/33
Conditions: none
Use the exact quotient only after checking b≠0.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
rational reverse multiplication := mul(40/33, -11/4)
After
rational reverse multiplication = -10/3
Conditions: none
Multiply back by the same divisor to recover a.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
b=-11/4≠0;quotient=40/33
After
quotient·b=-10/3=a
Conditions: none
This division/reverse-multiplication chain requires a nonzero divisor;it is not a formal proof of the entire rational-number structure.
Rule: rational_nonzero_division_reversal · 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 rational reverse multiplication with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute guarded rational quotient 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.(Yes, 40/33, -10/3)
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.
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