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R0008-04655 · q-20261008-advance-20261011-r02-arithmetic-operations-0030
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-8/2, b=-3/3, c=5/2. Give ((a+b)c, a+bc), preserving parentheses. 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=-8/2, b=-3/3, c=5/2. Give ((a+b)c, a+bc), preserving parentheses. 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 inside parentheses := sum(-4, -1)
After
a+b inside parentheses = -5
Conditions: none
For the first expression,evaluate the parenthesized sum first.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
(a+b)c := mul(-5, 5/2)
After
(a+b)c = -25/2
Conditions: none
Multiply the entire parenthesized value by c.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
bc before addition := mul(-1, 5/2)
After
bc before addition = -5/2
Conditions: none
For the second expression,compute bc first.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
a+bc := sum(-4, -5/2)
After
a+bc = -13/2
Conditions: none
Then add a without borrowing the first expression's grouping.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
((a+b)c,a+bc)=(-25/2,-13/2)
After
coinciding special values do not change the expression trees
Conditions: none
Special zeros or ones may make the values coincide,but do not permit arbitrary removal of parentheses.
Rule: parentheses_preserve_evaluation_tree · 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+bc with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute bc before addition with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute (a+b)c with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute a+b inside parentheses 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.(-25/2, -13/2)
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: 819ba5d0fa40e2d8de8e38680cae27c83d4533974dcb092e6d05cdc597bbafd1candidate record SHA-256: d5da3b7630a6022b0c070361f303cfede080e2b4d383a5e8d9f4d9bea3be7a83semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67Lean: NOT_RUN · human review: NOT_PERFORMED