Question 6880: Ordered Rational Offsets and Absolute Distances

Original question and givens

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-8/2, ordered points=[-20/3, -9/2, -4/1, -10/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

• G001 — Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-8/2, ordered points=[-20/3, -9/2, -4/1, -10/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

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-real-number

Step-by-step derivation

explicit rational points embedded in the real line

exact finite order and distance data,not real completeness or irrational classification

point0 signed offset := sub(-20/3, -4)

point0 signed offset = -8/3

point0 opposite negative offset := mul(-1, -8/3)

point0 opposite negative offset = 8/3

point=-20/3;origin=-4;offset=-8/3

absolute distance=8/3

point1 signed offset := sub(-9/2, -4)

point1 signed offset = -1/2

point1 opposite negative offset := mul(-1, -1/2)

point1 opposite negative offset = 1/2

point=-9/2;origin=-4;offset=-1/2

absolute distance=1/2

point2 signed offset := sub(-4, -4)

point2 signed offset = 0

point=-4;origin=-4;offset=0

absolute distance=0/1

point3 signed offset := sub(-10/3, -4)

point3 signed offset = 2/3

point=-10/3;origin=-4;offset=2/3

absolute distance=2/3

Reverse checks

• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute point3 signed offset with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute point2 signed offset with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute point1 opposite negative offset with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute point1 signed offset with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute point0 opposite negative offset with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute point0 signed offset 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.

Result

([-8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])

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.