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Question 34923: Exact Distance Between 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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R0068-34923 · q-20260923-real-rational-point-distance-0023

Original question and givens

Two rational real points are 4/1 and -5/1. Find the exact distance |4/1−-5/1| as a lowest-terms fraction.

  • G001 — Two rational real points are 4/1 and -5/1. Find the exact distance |4/1−-5/1| as a lowest-terms fraction.

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-point-distance, [left, operation, right]]

After

derived = [exact_rational_distance, 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(4, 1)

After

Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator = 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, [4, 1]]

After

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

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(-5, 1)

After

Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator = -5

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, [-5, 1]]

After

derived = [[-5, 1], -5/1]

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(4, 1)

After

Real ORIGINAL normalized left numerator times right positive denominator = 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, S005

S007 · calculation

Before

Real ORIGINAL normalized right numerator times left positive denominator := mul(-5, 1)

After

Real ORIGINAL normalized right numerator times left positive denominator = -5

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 · calculation

Before

Real distance ORIGINAL signed crossproduct difference := sub(4, -5)

After

Real distance ORIGINAL signed crossproduct difference = 9

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, S007

S009 · calculation

Before

Real distance ORIGINAL positive denominator product := mul(1, 1)

After

Real distance ORIGINAL positive denominator product = 1

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, S008

S010 · implies

Before

source roles = [4/1, -5/1, 9, 1]

After

derived = [9, 1]

Conditions: none

Real-line distance is|left−right|.Retain the signed difference BEFORE taking abs.Equal points have distance0;do not return signed or squared distance or guess a sign by reordering points.

Step identifiers and dependencies

Rule: Real_DISTANCE_absolute_difference_NOT_signed_or_squared · Uses: G001, S009

S011 · calculation

Before

Real exact distance derived fraction ORIGINAL exact numerator over nonzero denominator := div(9, 1)

After

Real exact distance derived fraction ORIGINAL exact numerator over nonzero denominator = 9

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, S010

S012 · implies

Before

source roles = [Real exact distance derived fraction, [9, 1]]

After

derived = [[9, 1], 9/1]

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, S011

Reverse checks

  • S012 · 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.
  • S011 · premise_and_result_audit — Recompute Real exact distance derived fraction ORIGINAL exact numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
  • S010 · 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.
  • S009 · premise_and_result_audit — Recompute Real distance ORIGINAL positive denominator product with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Real distance ORIGINAL signed crossproduct difference with exact rational arithmetic, not rounded decimals.
  • 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

9/1

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: f2c3f88148611b04af113f7632b8b9233c01044c4d6183113af0fcb17c32999b
  • candidate record SHA-256: ca4c1223146140e8ca0d7322b83cb2671281aca27b6aea6906c7a7f99421de4b
  • semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed
  • question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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