Editing Dimensional analysis (section)

== Dimensional homogeneity (commensurability) {{anchor|Dimensional homogeneity|Commensurability}} ==
{{See also|Apples and oranges}}
{{further|Kind of quantity}}

The most basic rule of dimensional analysis is that of dimensional homogeneity.<ref>{{cite book |first1=John |last1=Cimbala |first2=Yunus |last2=Çengel |year=2006 |title=Essential of Fluid Mechanics: Fundamentals and Applications |publisher=McGraw-Hill |chapter=§7-2 Dimensional homogeneity |chapter-url=http://highered.mcgraw-hill.com/sites/0073138355/student_view0/chapter7/ |isbn=9780073138350 |page=203–}}</ref> 
{{block indent|Only commensurable quantities (physical quantities having the same dimension) may be ''compared'', ''equated'', ''added'', or ''subtracted''.}}
However, the dimensions form an [[abelian group]] under multiplication, so:
{{block indent|One may take ''ratios'' of ''incommensurable'' quantities (quantities with different dimensions), and ''multiply'' or ''divide'' them.}}

For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometre, as these have different dimensions, nor to add 1 hour to 1 kilometre. However, it makes sense to ask whether 1 mile is more, the same, or less than 1 kilometre, being the same dimension of physical quantity even though the units are different. On the other hand, if an object travels 100&nbsp;km in 2 hours, one may divide these and conclude that the object's average speed was 50&nbsp;km/h.

The rule implies that in a physically meaningful ''expression'' only quantities of the same dimension can be added, subtracted, or compared. For example, if {{math|''m''<sub>man</sub>}}, {{math|''m''<sub>rat</sub>}} and {{math|''L''<sub>man</sub>}} denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression {{math|''m''<sub>man</sub> + ''m''<sub>rat</sub>}} is meaningful, but the heterogeneous expression {{math|''m''<sub>man</sub> + ''L''<sub>man</sub>}} is meaningless. However, {{math|''m''<sub>man</sub>/''L''<sup>2</sup><sub>man</sub>}} is fine. Thus, dimensional analysis may be used as a [[sanity check]] of physical equations: the two sides of any equation must be commensurable or have the same dimensions.

Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although [[torque]] and energy share the dimension {{dimanalysis|length=2|mass=1|time=−2}}, they are fundamentally different physical quantities.

To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is first to convert them all to the same unit. For example, to compare 32 metres with 35 yards, use {{nowrap|1=1&nbsp;yard = 0.9144&nbsp;m}} to convert 35 yards to 32.004&nbsp;m.

A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables.<ref>{{Cite book |last1=de Jong |first1=Frits J. |url=https://archive.org/details/dimensionalanaly0000jong |title=Dimensional analysis for economists |last2=Quade |first2=Wilhelm |publisher=North Holland |year=1967 |page=[https://archive.org/details/dimensionalanaly0000jong/page/28 28] |url-access=registration}}</ref> For example, [[Newton's laws of motion]] must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between two units that measure the same dimension must take multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in metres.