# Dimensional analysis

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides.

mko-dimensional-analysis · concept · v0.10.0



## Mathematical statement

Relative to basic dimensions L,M,T, write [Q]=L^a M^b T^c, represented by (a,b,c). Products add vectors, quotients subtract them and integer powers scale them. Quantities to be equated or added must have matching dimensions.





## Representative formula (exact TeX)

d(Q)





## Assumptions


- Exercises use only the three basic dimensions L,M,T and integer exponents.
- Dimensional consistency is necessary but not sufficient for a physical equation; dimensionless and dimensioned constants must be explicitly specified.





## Symbol scope


- Q — physical quantity
- L — length dimension
- M — mass dimension
- T — time dimension





## Intuition, definition, and misconceptions

Velocity’s dimension comes from length divided by time. Products and quotients combine exponents separately for each basic dimension.

A pure numerical constant cannot fix inconsistent dimensions. Consistent dimensions may still accompany an incorrect coefficient, direction or physical assumption.

Dimensional analysis does not track numerical unit-conversion factors and does not replace experiments, boundary conditions or model validation.

Do not treat dimensional consistency as proof of an equation or identify unit names completely with dimension exponents.





## Adopted derivation guide

[Dimensional analysis: checking ½mv²=mgh with exponent vectors and ruling out v=2gh](https://ocme.evemisslab.com/en/derivations/derivation-mko-dimensional-analysis-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Ratio and proportion](https://ocme.evemisslab.com/en/mko/mko-ratio-proportion/) — Division and dimensionless ratios use ratio semantics.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — Exponent vectors use addition, subtraction and integer scaling.
- [Equation](https://ocme.evemisslab.com/en/mko/mko-equation/) — Compare the dimensions of both equation sides.





## Verification boundary

This slice checks only L,M,T exponent vectors; consistency is not promoted to physical correctness.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add SI unit conversions, Buckingham Pi and nondimensionalization.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/dimensional-analysis.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-dimensional-analysis/index.md)