Dimensional Analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions (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 measurable (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 (inches and meters, pounds and newtons). If physical quantities have different dimensions (length verses mass), they cannot be compared by similar units (are incommensurable) and cannot be compared in quantity. Hence, it is meaningless to ask whether a kilogram is greater than, equal to, or less than an hour.
Any physically meaningful equation (and likewise any inequality and inequation) will have the same dimensions on their left and right sides, a property known as "dimensional homogeneity". Checking for dimensional homogeneity is a common application of dimensional analysis. Dimensional analysis is also routinely used as a check of the plausibility of derived equations and computations. It is generally used to categorize types of physical quantities and units based on their relationship to or dependence on other units.
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 km 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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