Question 34886: Exact Ordering of Rational Points

Original question and givens

The rational points are a=-1/2 and b=4/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

• G001 — The rational points are a=-1/2 and b=4/1. Use cross multiplication to determine whether a<b, a=b, or a>b.

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

source roles = [qf-real-rational-order, [left, operation, right]]

derived = [cross_multiplication_order, no-invented-domain]

Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator := div(-1, 2)

Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator = -1/2

source roles = [Real ORIGINAL left point, [-1, 2]]

derived = [[-1, 2], -1/2]

Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator := div(4, 1)

Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator = 4

source roles = [Real ORIGINAL right point, [4, 1]]

derived = [[4, 1], 4/1]

Real ORIGINAL normalized left numerator times right positive denominator := mul(-1, 1)

Real ORIGINAL normalized left numerator times right positive denominator = -1

Real ORIGINAL normalized right numerator times left positive denominator := mul(4, 2)

Real ORIGINAL normalized right numerator times left positive denominator = 8

source roles = [-1/2, 4/1]

derived = [-1, 8, <]

Reverse checks

• S008 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
• S007 · premise_and_result_audit — Recompute Real ORIGINAL normalized right numerator times left positive denominator with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Real ORIGINAL normalized left numerator times right positive denominator with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
• S004 · premise_and_result_audit — Recompute Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
• S002 · premise_and_result_audit — Recompute Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.

Result

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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.