Analysis of the relationships between different physical quantities
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.
Commensurable physical quantities are of the same 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 kinds and have different dimensions, and can not be directly compared to each other, no matter what 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, "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 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.
The concept of physical dimension or quantity dimension, and of dimensional analysis, was introduced by Joseph Fourier in 1822.
Formulation.
The Buckingham π theorem describes how every physically meaningful equation involving "n" variables can be equivalently rewritten as an equation of "n" − "m" dimensionless parameters, where "m" is the rank of the dimensional matrix. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.
A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or physical constants of nature. This may give insight into the fundamental properties of the system, as illustrated in the examples below.
The dimension of a physical quantity can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionally rational) power. The "dimension" of a physical quantity is more fundamental than some "scale" or unit used to express the amount of that physical quantity. For example, "mass" is a dimension, while the kilogram is a particular reference quantity chosen to express a quantity of mass. The choice of unit is arbitrary, and its choice is often based on historical precedent. Natural units, being based on only universal constants, may be thought of as being "less arbitrary".
There are many possible choices of base physical dimensions. The SI standard selects the following dimensions and corresponding dimension symbols: 
 time (T), length (L), mass (M), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J).
The symbols are by convention usually written in roman sans serif typeface. Mathematically, the dimension of the quantity "Q" is given by 
 formula_1
where "a", "b", "c", "d", "e", "f", "g" are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a basis – for instance, one could replace the dimension (I) of electric current of the SI basis with a dimension (Q) of electric charge, since Q = TI.
A quantity that has only "b" ≠ 0 (with all other exponents zero) is known as a geometric quantity. A quantity that has only both "a" ≠ 0 and "b" ≠ 0 is known as a kinematic quantity. A quantity that has only all of "a" ≠ 0, "b" ≠ 0, and "c" ≠ 0 is known as a dynamic quantity.
A quantity that has all exponents null is said to have dimension one.
The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have conversion factors that relate them. For example, 1 in = 2.54 cm; in this case 2.54 cm/in is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity.
There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity, although this does not invalidate the usefulness of dimensional analysis.
Simple cases.
As examples, the dimension of the physical quantity speed "v" is
 formula_2
The dimension of the physical quantity acceleration "a" is
 formula_3
The dimension of the physical quantity force "F" is
 formula_4
The dimension of the physical quantity pressure "P" is 
 formula_5
The dimension of the physical quantity energy "E" is
 formula_6
The dimension of the physical quantity power "P" is
 formula_7
The dimension of the physical quantity electric charge "Q" is
 formula_8
The dimension of the physical quantity voltage "V" is
 formula_9
The dimension of the physical quantity capacitance "C" is
 formula_10
Rayleigh's method.
In dimensional analysis, Rayleigh's method is a conceptual tool used in physics, chemistry, and engineering. It expresses a functional relationship of some variables in the form of an exponential equation. It was named after Lord Rayleigh.
The method involves the following steps:
As a drawback, Rayleigh's method does not provide any information regarding number of dimensionless groups to be obtained as a result of dimensional analysis.
Concrete numbers and base 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 kilometres per hour or 1.4 kilometres per second. Compound relations with "per" are expressed with division, e.g. 60 km/h. Other relations can involve multiplication (often shown with a centered dot or juxtaposition), powers (like m2 for square metres), or combinations thereof.
A set of base units for a system of measurement is a conventionally chosen set of units, none of which can be expressed as a combination of the others and in terms of which all the remaining units of the system can be expressed. For example, units for length and time are normally chosen as base units. Units for volume, however, can be factored into the base units of length (m3), thus they are considered derived or compound units.
Sometimes the names of units obscure the fact that they are derived units. For example, a newton (N) is a unit of force, which may be expressed as the product of mass (with unit kg) and acceleration (with unit m⋅s−2). The newton is defined as 1 N = 1 kg⋅m⋅s−2.
Percentages, derivatives and integrals.
Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since 1% = 1/100.
Taking a derivative with respect to a quantity divides the dimension by the dimension of the variable that is differentiated with respect to. Thus:
Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.
In economics, one distinguishes between stocks and flows: a stock has a unit (say, widgets or dollars), while a flow is a derivative of a stock, and has a unit of the form of this unit divided by one of time (say, dollars/year).
In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, debt-to-GDP ratios are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency)—but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance) and thus debt-to-GDP should have the unit year, which indicates that debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.
Dimensional homogeneity (commensurability).
The most basic rule of dimensional analysis is that of dimensional homogeneity. 
 ;templatestyles src="Block indent/styles.css"/ ;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:
 ;templatestyles src="Block indent/styles.css"/ ;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 km in 2 hours, one may divide these and conclude that the object's average speed was 50 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 "m"man, "m"rat and "L"man denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression "m"man + "m"rat is meaningful, but the heterogeneous expression "m"man + "L"man is meaningless. However, "m"man/"L"2man 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 , 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 1 yard = 0.9144 m to convert 35 yards to 32.004 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. 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.
Conversion factor.
In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a "conversion factor". For example, kPa and bar are both units of pressure, and 100 kPa = 1 bar. The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1. Since any quantity can be multiplied by 1 without changing it, the expression "100 kPa / 1 bar" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including the unit. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500, and bar/bar cancels out, so 5 bar = 500 kPa.
Applications.
Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications outside of those fields as well.
Mathematics.
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an "n"-ball (the solid ball in "n" dimensions), or the area of its surface, the "n"-sphere: being an "n"-dimensional figure, the volume scales as "x""n", while the surface area, being ("n" − 1)-dimensional, scales as "x""n"−1. Thus the volume of the "n"-ball in terms of the radius is "C""n""r""n", for some constant "C""n". Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.
Finance, economics, and accounting.
In finance, economics, and accounting, dimensional analysis is most commonly