# Question 34915: Exact Distance Between Rational Points [Return to catalogue page 175](https://ocme.evemisslab.com/en/questions/page/175/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260923-real-number-rational-points/#q-20260923-real-rational-point-distance-0015) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0068-34915 · q-20260923-real-rational-point-distance-0015 ## Original question and givens Two rational real points are 4/5 and -2/1. Find the exact distance |4/5−-2/1| as a lowest-terms fraction. - G001 — Two rational real points are 4/5 and -2/1. Find the exact distance |4/5−-2/1| 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. - [mko-real-number](https://ocme.evemisslab.com/en/mko/mko-real-number/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001 S002 · calculationBefore Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator := div(4, 5) After Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator = 4/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore source roles = [Real ORIGINAL left point, [4, 5]] After derived = [[4, 5], 4/5] 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 dependenciesRule: Real_BASE_fraction_sign_gcd_normalization_NO_invented_SAFEcap · Uses: G001, S002 S004 · calculationBefore Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator := div(-2, 1) After Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator = -2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [Real ORIGINAL right point, [-2, 1]] After derived = [[-2, 1], -2/1] 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 dependenciesRule: Real_BASE_fraction_sign_gcd_normalization_NO_invented_SAFEcap · Uses: G001, S004 S006 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Real ORIGINAL normalized right numerator times left positive denominator := mul(-2, 5) After Real ORIGINAL normalized right numerator times left positive denominator = -10 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Real distance ORIGINAL signed crossproduct difference := sub(4, -10) After Real distance ORIGINAL signed crossproduct difference = 14 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Real distance ORIGINAL positive denominator product := mul(5, 1) After Real distance ORIGINAL positive denominator product = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [4/5, -2/1, 14, 5] After derived = [14, 5] 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 dependenciesRule: Real_DISTANCE_absolute_difference_NOT_signed_or_squared · Uses: G001, S009 S011 · calculationBefore Real exact distance derived fraction ORIGINAL exact numerator over nonzero denominator := div(14, 5) After Real exact distance derived fraction ORIGINAL exact numerator over nonzero denominator = 14/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore source roles = [Real exact distance derived fraction, [14, 5]] After derived = [[14, 5], 14/5] 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 dependenciesRule: 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 14/5 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: 511257dd5eb422d6a48452ef4fb62a6b7eb397af1ecf8c755e94b2dd1da73cf7 - candidate record SHA-256: 93c1ddf11f1e77c78733c1642cf57a26c1a62f73013477d776c90fd9f173e536 - semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed - question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260923-real-rational-point-distance-0015.json)