Question 1672: Relative Error with an Explicit Nonzero Reference

Original question and givens

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=45/2, reference=4/2 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.

• 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=45/2, reference=4/2 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.

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-numerical-approximation-error

Step-by-step derivation

reference=2

all values are exact rationals

approximation−reference := sub(45/2, 2)

approximation−reference = 41/2

absolute error := mul(1, 41/2)

absolute error = 41/2

signed error=41/2

absolute error=41/2≥0

reference=2≠0

relative denominator=|reference|=2>0

relative error := div(41/2, 2)

relative error = 41/4

relative error=41/4

ratio may exceed1; it is not a probability

Reverse checks

• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute relative error with exact rational arithmetic, not rounded decimals.
• 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.

Result

(45/2, 2, 41/2, 41/4)

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.