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R0008-04662 · q-20261008-advance-20261011-r02-arithmetic-operations-0037
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-1/2, b=-10/3, c=12/2. 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=-1/2, b=-10/3, c=12/2. 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(-1/2, -10/3)
After
a+b inside parentheses = -23/6
Conditions: none
For the first expression,evaluate the parenthesized sum first.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
(a+b)c := mul(-23/6, 6)
After
(a+b)c = -23
Conditions: none
Multiply the entire parenthesized value by c.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
bc before addition := mul(-10/3, 6)
After
bc before addition = -20
Conditions: none
For the second expression,compute bc first.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
a+bc := sum(-1/2, -20)
After
a+bc = -41/2
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)=(-23,-41/2)
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.(-23/1, -41/2)
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: 32f08e4567ab7ebce123daee3f8d1e3ae1c55f80d54dc5a4904ef1a2463ab4b7candidate record SHA-256: 993076b837cc048be29e8a281759d73b753151aa55940015036625023b255fd5semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67Lean: NOT_RUN · human review: NOT_PERFORMED