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Question 34879: Exact Ordering of Rational Points

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0068-34879 · q-20260923-real-rational-order-0004

Original question and givens

The rational points are a=-9/4 and b=-3/4. Use cross multiplication to determine whether a<b, a=b, or a>b.

  • G001 — The rational points are a=-9/4 and b=-3/4. 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.

Step-by-step derivation

S001 · implies

Before

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

After

derived = [cross_multiplication_order, no-invented-domain]

Conditions: none

Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact.

Step identifiers and dependencies

Rule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001

S002 · calculation

Before

Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator := div(-9, 4)

After

Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator = -9/4

Conditions: none

Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

source roles = [Real ORIGINAL left point, [-9, 4]]

After

derived = [[-9, 4], -9/4]

Conditions: none

BASE normalizeFraction requires INTEGER components and nonzero denominator,without invented SAFE/million/100million caps.Normalize denominator sign and gcd;zero is0/1.This derived representation does not alter the ORIGINAL raw pair.

Step identifiers and dependencies

Rule: Real_BASE_fraction_sign_gcd_normalization_NO_invented_SAFEcap · Uses: G001, S002

S004 · calculation

Before

Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator := div(-3, 4)

After

Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator = -3/4

Conditions: none

Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [Real ORIGINAL right point, [-3, 4]]

After

derived = [[-3, 4], -3/4]

Conditions: none

BASE normalizeFraction requires INTEGER components and nonzero denominator,without invented SAFE/million/100million caps.Normalize denominator sign and gcd;zero is0/1.This derived representation does not alter the ORIGINAL raw pair.

Step identifiers and dependencies

Rule: Real_BASE_fraction_sign_gcd_normalization_NO_invented_SAFEcap · Uses: G001, S004

S006 · calculation

Before

Real ORIGINAL normalized left numerator times right positive denominator := mul(-9, 4)

After

Real ORIGINAL normalized left numerator times right positive denominator = -36

Conditions: none

Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Real ORIGINAL normalized right numerator times left positive denominator := mul(-3, 4)

After

Real ORIGINAL normalized right numerator times left positive denominator = -12

Conditions: none

Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [-9/4, -3/4]

After

derived = [-36, -12, <]

Conditions: none

Normalize BOTH denominator signs BEFORE cross-product order comparison to preserve inequality direction.Compare the ORIGINAL rational points exactly,not floating approximations.

Step identifiers and dependencies

Rule: Real_ORDER_normalize_sign_BEFORE_crossproduct_inequality · Uses: G001, S007

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: d41d8caa7fc05f68962d9205ba5d98341187847b0acda017e8a62c07cfc84370
  • candidate record SHA-256: 3b439b2d15242f141897eb3e90099a8865b69b7c9852d6f809cc25eb15c5bc82
  • semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed
  • question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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