# Question 34941: Exact Fraction for a Terminating Decimal [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-terminating-decimal-fraction-0016) 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. R0068-34941 · q-20260923-real-terminating-decimal-fraction-0016 ## Original question and givens Convert the explicit decimal string 0.09 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation. - G001 — Convert the explicit decimal string 0.09 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation. ## 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-terminating-decimal-fraction, [decimal, operation, scale]] After derived = [decimal_to_fraction, 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 Decimal ORIGINAL power10 place1 := mul(1, 10) After Decimal ORIGINAL power10 place1 = 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, S001 S003 · calculationBefore Decimal ORIGINAL power10 place2 := mul(10, 10) After Decimal ORIGINAL power10 place2 = 100 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, S002 S004 · calculationBefore Decimal ORIGINAL whole part times power10 := mul(0, 100) After Decimal ORIGINAL whole part times power10 = 0 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 · calculationBefore Decimal ORIGINAL add ALL fractional digits := sum(0, 9) After Decimal ORIGINAL add ALL fractional digits = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Decimal ORIGINAL apply explicit sign := mul(1, 9) After Decimal ORIGINAL apply explicit sign = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [0.09, 2, 0, 09] After derived = [9, 100] Conditions: none Preserve the ORIGINAL string and place count.Compute whole*10^scale plus ALL fractional digits,then apply the sign,WITHOUT floating parsing.The leading fractional zeros in0.001 still count toward scale. Step identifiers and dependenciesRule: Real_DECIMAL_lexical_places_EXACT_integer_over_power10 · Uses: G001, S006 S008 · calculationBefore Decimal exact derived fraction ORIGINAL exact numerator over nonzero denominator := div(9, 100) After Decimal exact derived fraction ORIGINAL exact numerator over nonzero denominator = 9/100 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 · impliesBefore source roles = [Decimal exact derived fraction, [9, 100]] After derived = [[9, 100], 9/100] 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, S008 ## Reverse checks - S009 · 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. - S008 · premise_and_result_audit — Recompute Decimal exact derived fraction ORIGINAL exact numerator over nonzero denominator with exact rational arithmetic, not rounded decimals. - S007 · 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. - S006 · premise_and_result_audit — Recompute Decimal ORIGINAL apply explicit sign with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Decimal ORIGINAL add ALL fractional digits with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Decimal ORIGINAL whole part times power10 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Decimal ORIGINAL power10 place2 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Decimal ORIGINAL power10 place1 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 9/100 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: 9633be8effeb5e3b0d2f13f8dd05f78a2d7fe953b123b8ef91cd0d09e060ab6a - candidate record SHA-256: 7aea5f0a6c10059bd05f0009f87187d2e9b89ed4cce3914ea74359b6e465d7ee - 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-terminating-decimal-fraction-0016.json)