, dimensional analysis is the analysis of the relationships between different physical quantities
by identifying their base quantities
(such as length
, and electric charge
) and units of measure
(such as miles vs. kilometres, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The conversion of units
from one dimensional unit to another is often easier within the metric
system than in others, due to the regular 10-base in all units. 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
''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 originally expressed in differing units of measure, e.g. yards and metres, pounds(mass) and kilograms, 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 originally expressed in, e.g. meters and kilograms, seconds and kilograms, meters and seconds. For example, asking whether a kilogram 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, and of dimensional analysis, was introduced by Joseph Fourier
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/1 h. Other relations can involve multiplication
(often shown with a centered dot
), powers (like m2
for square metres), or combinations thereof.
A set of base unit
s 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
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 has units of mass (kg) times units of acceleration (m⋅s−2
). The newton is defined as .
Percentages and derivatives
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 .
Taking a derivative with respect to a quantity adds the dimension of the variable one is differentiating with respect to, in the denominator. Thus:
* position (''x'') has the dimension L (length);
* derivative of position with respect to time (''dx''/''dt'', velocity
) has dimension LT−1
—length from position, time due to the gradient;
* the second derivative (''d'x''/''dt'' = ''d''(''dx''/''dt'') / ''dt'', acceleration
) has dimension LT−2
In economics, one distinguishes between stocks and flows
: a stock has units of "units" (say, widgets or dollars), while a flow is a derivative of a stock, and has units of "units/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 ratio
s 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 units of years, 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.
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 . The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to . Since any quantity can be multiplied by 1 without changing it, the expression "" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including units. For example, because , and bar/bar cancels out, so .
The most basic rule of dimensional analysis is that of dimensional homogeneity.
:: 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:
:: 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 perfect 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
denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression is meaningful, but the heterogeneous expression is meaningless. However, ''m''man
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.
This has the implication that most mathematical functions, particularly the transcendental function
s, must have a dimensionless quantity, a pure number, as the argument
and must return a dimensionless number as a result. This is clear because many transcendental functions can be expressed as an infinite power series
with dimensionless coefficients.
All powers of ''x'' must have the same dimension for the terms to be commensurable. But if ''x'' is not dimensionless, then the different powers of ''x'' will have different, incommensurable dimensions. However, power functions
including root functions
may have a dimensional argument and will return a result having dimension that is the same power applied to the argument dimension. This is because power functions and root functions are, loosely, just an expression of multiplication of quantities.
Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although torque
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 units. For example, to compare 32 metre
s with 35 yard
s, 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 conversion factors must take between units that measure the same dimension: 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.
The factor-label method for converting units
The factor-label method is the sequential application of conversion factors expressed as fractions and arranged so that any dimensional unit appearing in both the numerator and denominator of any of the fractions can be cancelled out until only the desired set of dimensional units is obtained. For example, 10 miles per hour
can be converted to meters per second
by using a sequence of conversion factors as shown below:
Each conversion factor is chosen based on the relationship between one of the original units and one of the desired units (or some intermediary unit), before being re-arranged to create a factor that cancels out the original unit. For example, as "mile" is the numerator in the original fraction and
, "mile" will need to be the denominator in the conversion factor. Dividing both sides of the equation by 1 mile yields
, which when simplified results in the dimensionless
. Multiplying any quantity (physical quantity or not) by the dimensionless 1 does not change that quantity. Once this and the conversion factor for seconds per hour have been multiplied by the original fraction to cancel out the units ''mile'' and ''hour'', 10 miles per hour converts to 4.4704 meters per second.
As a more complex example, the concentration
of nitrogen oxides
) in the flue gas
from an industrial furnace
can be converted to a mass flow rate
expressed in grams per hour (i.e., g/h) of
by using the following information as shown below:
concentration := 10 parts per million
by volume = 10 ppmv = 10 volumes/106
molar mass := 46 kg/kmol = 46 g/mol
; Flow rate of flue gas := 20 cubic meters per minute = 20 m3
: The flue gas exits the furnace at 0 °C temperature and 101.325 kPa absolute pressure.
: The molar volume
of a gas at 0 °C temperature and 101.325 kPa is 22.414 m3
After canceling out any dimensional units that appear both in the numerators and denominators of the fractions in the above equation, the NOx
concentration of 10 ppmv
converts to mass flow rate of 24.63 grams per hour.
Checking equations that involve dimensions
The factor-label method can also be used on any mathematical equation to check whether or not the dimensional units on the left hand side of the equation are the same as the dimensional units on the right hand side of the equation. Having the same units on both sides of an equation does not ensure that the equation is correct, but having different units on the two sides (when expressed in terms of base units) of an equation implies that the equation is wrong.
For example, check the Universal Gas Law
equation of , when:
* the pressure ''P'' is in pascals (Pa)
* the volume ''V'' is in cubic meters (m3
* the amount of substance ''n'' is in moles (mol)
* the universal gas law constant ''R''
is 8.3145 Pa⋅m3
* the temperature ''T'' is in kelvins (K)
As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's right hand side are cancelled out, both sides of the equation have the same dimensional units. Dimensional analysis can be used as a tool to construct equations that relate non-associated physico-chemical properties. The equations may reveal hitherto unknown or overlooked properties of matter, in the form of left-over dimensions — dimensional adjusters — that can then be assigned physical significance. It is important to point out that such 'mathematical manipulation' is neither without prior precedent, nor without considerable scientific significance. Indeed, the Planck's constant, a fundamental constant of the universe, was 'discovered' as a purely mathematical abstraction or representation that built on the Rayleigh-Jeans Equation for preventing the ultraviolet catastrophe. It was assigned and ascended to its quantum physical significance either in tandem or post mathematical dimensional adjustment – not earlier.
The factor-label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0. (Ratio scale
in Stevens's typology) Most units fit this paradigm. An example for which it cannot be used is the conversion between degrees Celsius
s (or degrees Fahrenheit
). Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between degrees Celsius and degrees Fahrenheit there is neither a constant difference nor a constant ratio. There is, however, an affine transform
, rather than a linear transform
) between them.
For example, the freezing point of water is 0 °C and 32 °F, and a 5 °C change is the same as a 9 °F change. Thus, to convert from units of Fahrenheit to units of Celsius, one subtracts 32 °F (the offset from the point of reference), divides by 9 °F and multiplies by 5 °C (scales by the ratio of units), and adds 0 °C (the offset from the point of reference). Reversing this yields the formula for obtaining a quantity in units of Celsius from units of Fahrenheit; one could have started with the equivalence between 100 °C and 212 °F, though this would yield the same formula at the end.
Hence, to convert the numerical quantity value of a temperature ''T''
in degrees Fahrenheit to a numerical quantity value ''T''
in degrees Celsius, this formula may be used:
− 32) × 5/9.
To convert ''T''
in degrees Celsius to ''T''
in degrees Fahrenheit, this formula may be used:
× 9/5) + 32.
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.
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
while the surface area, being
-dimensional, scales as
Thus the volume of the ''n''-ball in terms of the radius is
for some constant
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 referred to in terms of the distinction between stocks and flows
. More generally, dimensional analysis is used in interpreting various financial ratios
, economics ratios, and accounting ratios.
* For example, the P/E ratio
has dimensions of time (units of years), and can be interpreted as "years of earnings to earn the price paid".
* In economics, debt-to-GDP ratio
also has units of years (debt has units of currency, GDP has units of currency/year).
* In financial analysis, some bond duration
types also have dimension of time (unit of years) and can be interpreted as "years to balance point between interest payments and nominal repayment".
* Velocity of money
has units of 1/years (GDP/money supply has units of currency/year over currency): how often a unit of currency circulates per year.
* Interest rates are often expressed as a percentage, but more properly percent per annum, which has dimensions of 1/years.
In fluid mechanics
, dimensional analysis is performed in order to obtain dimensionless pi terms
or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable pi terms or groups, it is possible to develop a similar set of pi terms for a model that has the same dimensional relationships. In other words, pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include:
* Reynolds number
(Re), generally important in all types of fluid problems:
* Froude number
(Fr), modeling flow with a free surface:
* Euler number
(Eu), used in problems in which pressure is of interest:
* Mach number
(Ma), important in high speed flows where the velocity approaches or exceeds the local speed of sound:
where: is the local speed of sound.
The origins of dimensional analysis have been disputed by historians.
The first written application of dimensional analysis has been credited to an article of François Daviet
at the Turin
Academy of Science. Daviet had the master Lagrange
His fundamental works are contained in acta of the Academy dated 1799.
This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually later formalized in the Buckingham π theorem
also treated the same problem of the parallelogram law
by Daviet, in his treatise of 1811
(vol I, p.39). In the second edition of 1833, Poisson explicitly introduces the term ''dimension'' instead of the Daviet ''homogeneity''.
In 1822, the important Napoleonic scientist Joseph Fourier
made the first credited important contributions based on the idea that physical laws like
should be independent of the units employed to measure the physical variables.
played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived.
Although Maxwell defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of Newton's law of universal gravitation
in which the gravitational constant
''G'' is taken as unity, thereby defining .
By assuming a form of Coulomb's law
in which Coulomb's constant
is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were ,
which, after substituting his equation for mass, results in charge having the same dimensions as mass, viz. .
Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time in this way in 1872 by Lord Rayleigh
, who was trying to understand why the sky is blue. Rayleigh first published the technique in his 1877 book ''The Theory of Sound''.
The original meaning of the word ''dimension'', in Fourier's ''Theorie de la Chaleur'', was the numerical value of the exponents of the base units. For example, acceleration was considered to have the dimension 1 with respect to the unit of length, and the dimension −2 with respect to the unit of time. This was slightly changed by Maxwell, who said the dimensions of acceleration are LT−2
, instead of just the exponents.
The Buckingham π theorem
describes how every physically meaningful equation involving ''n'' variables can be equivalently rewritten as an equation of 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 natural units
of nature. This gives 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 basic physical dimensions such as length
, each raised to a rational power
. The ''dimension'' of a physical quantity is more fundamental than some ''scale'' unit
used to express the amount of that physical quantity. For example, ''mass'' is a dimension, while the kilogram
is a particular scale unit chosen to express a quantity of mass. Except for natural units
, the choice of scale is cultural and arbitrary.
There are many possible choices of basic physical dimensions. The SI standard
recommends the usage of the following dimensions and corresponding symbols: length
(T), 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
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 linearly independent basis
. For instance, one could replace the dimension of electric current
(I) of the SI basis with a dimension of electric charge
(Q), since Q = IT.
As examples, the dimension of the physical quantity speed
and the dimension of the physical quantity force
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 that have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity,
although this does not invalidate the usefulness of dimensional analysis.
The dimensions that can be formed from a given collection of basic physical dimensions, such as M, L, and T, form an abelian group
: The identity is written as 1; , and the inverse to L is 1/L or L−1
. L raised to any rational power ''p'' is a member of the group, having an inverse of L−''p''
. The operation of the group is multiplication, having the usual rules for handling exponents ().
This group can be described as a vector space
over the rational numbers, with for example dimensional symbol M''i''
corresponding to the vector . When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the vector space. When measurable quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication
in the vector space.
A basis for such a vector space of dimensional symbols is called a set of base quantities
, and all other vectors are called derived units. As in any vector space, one may choose different bases
, which yields different systems of units (e.g., choosing
whether the unit for charge is derived from the unit for current, or vice versa).
The group identity 1, the dimension of dimensionless quantities, corresponds to the origin in this vector space.
The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The nullity
describes some number (e.g., ''m'') of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, . (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and exponentiating
) together the measured quantities to produce something with the same units as some derived quantity ''X'' can be expressed in the general form
Consequently, every possible commensurate
equation for the physics of the system can be rewritten in the form
Knowing this restriction can be a powerful tool for obtaining new insight into the system.
The dimension of physical quantities of interest in mechanics
can be expressed in terms of base dimensions M, L, and T – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a change of basis
. The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not entirely arbitrary, because they must form a basis
: they must span
the space, and be linearly independent
For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to M, L, T: the former can be expressed as = ML/T2
L, M, while the latter can be expressed as M, L, = (ML/F)1/2
On the other hand, length, velocity and time do not form a set of base dimensions for mechanics, for two reasons:
* There is no way to obtain mass – or anything derived from it, such as force – without introducing another base dimension (thus, they do not ''span the space'').
* Velocity, being expressible in terms of length and time (V = L/T), is redundant (the set is not ''linearly independent'').
Other fields of physics and chemistry
Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of M, L, T, and Q, where Q represents the dimension of electric charge
. In thermodynamics
, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the amount of substance
(the number of molecules divided by the Avogadro constant
, ≈ ) is defined as a base dimension, N, as well.
In the interaction of relativistic plasma
with strong laser pulses, a dimensionless relativistic similarity parameter
, connected with the symmetry properties of the collisionless Vlasov equation
, is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features.
Polynomials and transcendental functions
arguments to transcendental function
s such as exponential
ic functions, or to inhomogeneous polynomials
, must be dimensionless quantities
. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.)
While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity log(a/b) = log a − log b, where the logarithm is taken in any base, holds for dimensionless numbers a and b, but it does ''not'' hold if a and b are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.
Similarly, while one can evaluate monomials
) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for ''x''2
, the expression (3 m)2
= 9 m2
makes sense (as an area), while for ''x''2
+ ''x'', the expression (3 m)2
+ 3 m = 9 m2
+ 3 m does not make sense.
However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example,
This is the height to which an object rises in time ''t'' if the acceleration of gravity is 9.8 meter per second per second and the initial upward speed is 500 meter per second. It is not necessary for ''t'' to be in ''seconds''. For example, suppose ''t'' = 0.01 minutes. Then the first term would be
The value of a dimensional physical quantity ''Z'' is written as the product of a unit 'Z''
within the dimension and a dimensionless numerical factor, ''n''.
[For a review of the different conventions in use see: ]