Explore related knowledge and questions →
← Question derivation index · Original question batch
R0011-06403 · q-20261009-advance-20261013-r03-rational-number-0053
Exact rational representations have nonzero denominators; a=3/-3, b=4/5. 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=3/-3, b=4/5. 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=[3, -3];raw b=[4, 5]
After
a=-1/1;b=4/5
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(-1, 4/5)
After
guarded rational quotient = -5/4
Conditions: none
Use the exact quotient only after checking b≠0.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
rational reverse multiplication := mul(-5/4, 4/5)
After
rational reverse multiplication = -1
Conditions: none
Multiply back by the same divisor to recover a.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
b=4/5≠0;quotient=-5/4
After
quotient·b=-1/1=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, -5/4, -1/1)
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: f8ef5d23b7ba6167ae820ed90a31ceeaa62f144e48b122eba2106f1d43b50c85candidate record SHA-256: 288267478292111118e99b71d5e9ce74c30e11704df4cac9b7ca556cbdc26ce6semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED