Question 29027: The Negative Real Contribution from the Product of Imaginary Parts

Original question and givens

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=[a,b]=[-11, -2], right=[c,d]=[-1, -7]. Give (left,right,[a*c−b*d,a*d+b*c]).

• 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=[a,b]=[-11, -2], right=[c,d]=[-1, -7]. Give (left,right,[a*c−b*d,a*d+b*c]).

Definition and scope

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.

• mko-complex-number

Step-by-step derivation

source roles = [qf-str22-complex-multiplication, complex-multiplication]

derived = [[left, right, result]]

Complex ORIGINAL real-real product := mul(-11, -1)

Complex ORIGINAL real-real product = 11

Complex ORIGINAL imaginary-imaginary product := mul(-2, -7)

Complex ORIGINAL imaginary-imaginary product = 14

Complex ORIGINAL real-imaginary product := mul(-11, -7)

Complex ORIGINAL real-imaginary product = 77

Complex ORIGINAL imaginary-real product := mul(-2, -1)

Complex ORIGINAL imaginary-real product = 2

Complex ORIGINAL real part ac minus bd := sub(11, 14)

Complex ORIGINAL real part ac minus bd = -3

Complex ORIGINAL imaginary coefficient ad plus bc := sum(77, 2)

Complex ORIGINAL imaginary coefficient ad plus bc = 79

source roles = [[-11, -2], [-1, -7]]

derived = [11, 14, 77, 2, [-3, 79]]

Reverse checks

• S008 · 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.
• S007 · premise_and_result_audit — Recompute Complex ORIGINAL imaginary coefficient ad plus bc with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Complex ORIGINAL real part ac minus bd with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Complex ORIGINAL imaginary-real product with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Complex ORIGINAL real-imaginary product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Complex ORIGINAL imaginary-imaginary product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Complex ORIGINAL real-real product 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.

Result

([-11, -2], [-1, -7], [-3, 79])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.