Unit commensurability
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 charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms vs. grams) and tracking these dimensions as calculations or comparisons are performed. Converting from one dimensional unit to another is often somewhat complex. Dimensional analysis, or more specifically the factor-label method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra.
The concept of physical dimension was introduced by Joseph Fourier in 1822. Physical quantities that are of the same kind (also called commensurable) have the same dimension (length, time, mass) and can be directly compared to each other, even if they are originally expressed in differing units of measure (such as inches and meters, or pounds and newtons). If physical quantities have different dimensions (such as length vs. mass), they cannot be expressed in terms of similar units and cannot be compared in quantity (also called incommensurable). For example, asking whether a kilogram is greater than, equal to, or less than an hour is meaningless.
Any physically meaningful equation (and likewise any inequality and inequation) will 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 derived equations and computations. 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.
Many parameters and measurements in the physical sciences and engineering are expressed as a concrete number – a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.g. 60 miles per hour or 1.4 kilometers per second. Compound relations with "per" are expressed with division, e.g. 60 mi/1 h. Other relations can involve multiplication (often shown with ⋅ or ), powers (like m2 for square meters), or combinations thereof.
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