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R0002-01691 · q-20261013-advance-20261020-r03-numerical-approximation-error-0066
Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=33/1, reference=7/4, tolerance=4/3≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).
G001 — Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=33/1, reference=7/4, tolerance=4/3≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).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.
Before
reference=7/4
After
all values are exact rationals
Conditions: none
This is exact error arithmetic against a specified reference,without floating-point rounding that could alter an equality boundary.
Rule: exact_error_model · Uses: G001
Before
approximation−reference := sub(33, 7/4)
After
approximation−reference = 125/4
Conditions: none
Compute the signed error as approximation minus reference.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
absolute error := mul(1, 125/4)
After
absolute error = 125/4
Conditions: none
Use the sign to take the absolute value;the result is nonnegative.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
signed error=125/4
After
absolute error=125/4≥0
Conditions: none
Absolute error has the original quantity's units;a negative reference does not make it negative.
Rule: absolute_error_definition · Uses: G001, S003
Before
absolute error=125/4; tolerance=4/3
After
absolute error≤tolerance: No
Conditions: none
The contract uses≤,not<;an error exactly equal to tolerance passes.
Rule: inclusive_tolerance · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute absolute error with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute approximation−reference with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(33, 7/4, 125/4, 4/3, No)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
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