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R0057-29094 · q-20261013-advance-20261022-r03-complex-number-0044
Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[6, 1], right=[1, 27]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).
G001 — Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[6, 1], right=[1, 27]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).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
source roles = [qf-str22-complex-subtraction, complex-subtraction]
After
derived = [[left, right, result]]
Conditions: none
Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning.
Rule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001
Before
Complex ORIGINAL sub coordinate0 := sub(6, 1)
After
Complex ORIGINAL sub coordinate0 = 5
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Complex ORIGINAL sub coordinate1 := sub(1, 27)
After
Complex ORIGINAL sub coordinate1 = -26
Conditions: none
Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [complex-subtraction, [6, 1], [1, 27]]
After
derived = [[5, -26]]
Conditions: none
From(a+bi)±(c+di)=(a±c)+(b±d)i,combine matching components.Real stays first;subtraction is left-right,not an absolute difference.
Rule: Complex_ordered_real_imaginary_components_and_operand_direction · Uses: G001, S003
S004 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.S003 · premise_and_result_audit — Recompute Complex ORIGINAL sub coordinate1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Complex ORIGINAL sub coordinate0 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.([6, 1], [1, 27], [5, -26])
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: 51524fe9fd752bbcf212150b47b58c0e9050f51c30ee0d663b416d4563d1509ccandidate record SHA-256: 7f5b0bf921633984653b2dacc8440b9be1555e4372dd834dcb188ea8e5b332d3semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5Lean: NOT_RUN · human review: NOT_PERFORMED