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R0008-04632 · q-20261008-advance-20261011-r02-arithmetic-operations-0007
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-6/1, b=7/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
G001 — All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-6/1, b=7/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. 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
exact arithmetic in Q;N={0,1,2,...} when membership is askedAfter
do not truncate subtraction or rational division
Conditions: none
Compute exact results in Q before testing N-membership;do not clamp negative differences to0 or replace rational division by integer division.Division by0 remains undefined.
Rule: exact_rational_arithmetic_scope · Uses: G001
Before
a+b := sum(-6, 7/4)
After
a+b = -17/4
Conditions: none
Compute addition as exact fractions.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
a−b := sub(-6, 7/4)
After
a−b = -31/4
Conditions: none
Retain operand order in subtraction.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
ab := mul(-6, 7/4)
After
ab = -21/2
Conditions: none
Multiply in Q without approximation.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
a/b := div(-6, 7/4)
After
a/b = -24/7
Conditions: none
Only a nonzero b permits division by its reciprocal.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
b=7/4
After
division defined=Yes;quotient=-24/7
Conditions: none
A zero divisor is not a small nonzero denominator;neither0/0 nor a/0 may be assigned0.
Rule: division_zero_guard · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute a/b with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute ab with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute a−b with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute a+b 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.(-17/4, -31/4, -21/2, -24/7)
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: 38fec3f16613c415fcb0c8e6aaf758a06de8937f721483af4c488b37598359b3candidate record SHA-256: c10a4b95ecd731d9bb916e8ba5fa40e51fd713b9b63eb32d2596f358634720dfsemantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67Lean: NOT_RUN · human review: NOT_PERFORMED