Editing Dimensional analysis (section)

{{Short description|Analysis of the relationships between different physical quantities}}
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In [[engineering]] and [[science]], '''dimensional analysis''' is the analysis of the relationships between different [[physical quantities]] by identifying their [[base quantities]] (such as [[length]], [[mass]], [[time]], and [[electric current]]) and [[units of measurement]] (such as metres and grams) and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to [[conversion of units]] from one dimensional unit to another, which can be used to evaluate scientific formulae.

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''Commensurable'' physical quantities are of the same [[Kind of quantity|kind]] and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years. ''Incommensurable'' physical [[quantities]] are of different [[Kind of quantity|kinds]] and have different dimensions, and can not be directly compared to each other, no matter what [[Unit of measurement|units]] they are expressed in, e.g. metres and grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless.

Any physically meaningful [[equation]], or [[inequality (mathematics)|inequality]], ''must'' have the same dimensions on its left and right sides, a property known as ''dimensional homogeneity''. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on [[Formal proof|derived]] equations and [[computation]]s. It also serves as a guide and constraint in deriving equations that may describe a physical [[system]] in the absence of a more rigorous derivation.

The concept of '''physical dimension''' or '''quantity dimension''', and of dimensional analysis, was introduced by [[Joseph Fourier]] in 1822.<ref name="Bolster">{{cite journal|last1=Bolster|first1=Diogo|last2=Hershberger|first2=Robert E.|last3=Donnelly|first3=Russell E.|title=Dynamic similarity, the dimensionless science|url=https://pubs.aip.org/physicstoday/article-abstract/64/9/42/413713/Dynamic-similarity-the-dimensionless|journal=Physics Today|doi=10.1063/PT.3.1258|date=September 2011|volume=64|issue=9|pages=42–47|bibcode=2011PhT....64i..42B |url-access=subscription}}</ref>{{rp|42}}