# Question 1653: Relative Error with an Explicit Nonzero Reference

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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.

R0002-01653 · q-20261013-advance-20261020-r03-numerical-approximation-error-0028



## 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=7/3, reference=3/3 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=7/3, reference=3/3 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](https://ocme.evemisslab.com/en/mko/mko-numerical-approximation-error/)





## Step-by-step derivation

S001 · impliesBefore

reference=1

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.

Step identifiers and dependenciesRule: exact_error_model · Uses: G001





S002 · calculationBefore

approximation−reference := sub(7/3, 1)

After

approximation−reference = 4/3

Conditions: none

Compute the signed error as approximation minus reference.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

absolute error := mul(1, 4/3)

After

absolute error = 4/3

Conditions: none

Use the sign to take the absolute value;the result is nonnegative.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

signed error=4/3

After

absolute error=4/3≥0

Conditions: none

Absolute error has the original quantity's units;a negative reference does not make it negative.

Step identifiers and dependenciesRule: absolute_error_definition · Uses: G001, S003





S005 · impliesBefore

reference=1≠0

After

relative denominator=|reference|=1>0

Conditions: none

Divide by the absolute reference;this definition does not apply at reference0.

Step identifiers and dependenciesRule: relative_denominator_guard · Uses: G001, S004





S006 · calculationBefore

relative error := div(4/3, 1)

After

relative error = 4/3

Conditions: none

Divide by the positive denominator for a dimensionless ratio,without multiplying by100.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

relative error=4/3

After

ratio may exceed1; it is not a probability

Conditions: none

Relative error may exceed1;only a separate percent conversion multiplies by100,so ratio2 is not2%.

Step identifiers and dependenciesRule: relative_error_scope · Uses: G001, S006









## 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

(7/3, 1, 4/3, 4/3)

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: ae78c74872e56cfe6755c77c6ef762cde259779b4cfff152a51df36af52d291a
- candidate record SHA-256: 90a63d7f155682b0bc8dda60fc057575db43b64c3e5ad93bc26864d1d865e5d6
- semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861
- question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8
- Lean: NOT_RUN · human review: NOT_PERFORMED

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