# Question 1073: Distance on the Real Number Line

[Return to catalogue page 6](https://ocme.evemisslab.com/en/questions/page/6/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260928-absolute-value-foundations/#q-20260928-absolute-value-distance-0023)

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.

R0001-01073 · q-20260928-absolute-value-distance-0023



## Original question and givens

On the real number line, a=10 and b=15. Compute the distance |a-b| exactly.


- G001 — On the real number line, a=10 and b=15. Compute the distance |a-b| exactly.





## 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-absolute-value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/)





## Step-by-step derivation

S001 · impliesBefore

x∈R

After

|x|=x if x≥0; |x|=−x if x<0

Conditions: none

The sign branches cover every real number, with0 in the nonnegative branch.

Step identifiers and dependenciesRule: absolute_definition · Uses: G001





S002 · calculationBefore

a−b := sub(10, 15)

After

a−b = -5

Conditions: none

Compute the signed difference first, not the distance itself.

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





S003 · impliesBefore

a−b=-5

After

distance=|a−b|=5

Conditions: none

Use the sign branch for the difference; the resulting distance is nonnegative.

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





S004 · calculationBefore

b−a := sub(15, 10)

After

b−a = 5

Conditions: none

Swapping the points reverses the difference's sign.

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





S005 · impliesBefore

|a−b|=5; |b−a|=5

After

swapping endpoints preserves distance

Conditions: none

Endpoint order changes but the absolute values agree, completing the reverse audit.

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









## Reverse checks


- 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 — Recompute b−a with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Recompute a−b with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Use ≥0 rather than >0 so that0 is not omitted.





## Result

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: e0035cb11f38f9d932621f282d5125a40f6ea11b3e09e5570c519980e8c079cd
- candidate record SHA-256: 91a03837a0c869a5259e1352fa6f30f56bbfa35bd5781b9437eefafbaa8430ef
- semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc
- question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260928-absolute-value-distance-0023.json)