dimensional analysis

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In
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
and
science Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...
, dimensional analysis is the analysis of the relationships between different
physical quantities A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For exam ...
by identifying their base quantities (such as length,
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
,
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
, and electric current) and
units of measure A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multi ...
(such as miles vs. kilometres, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The
conversion of units Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors which change the measured quantity value without changing its effects. Overview The process ...
from one dimensional unit to another is often easier within the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
or the SI than in others, due to the regular 10-base in all units. ''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 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 expressed in, e.g. metres and kilograms, seconds and kilograms, metres and seconds. For example, asking whether a kilogram is larger than an hour is meaningless. Any physically meaningful equation, or
inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
, ''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 Derive may refer to: * Derive (computer algebra system), a commercial system made by Texas Instruments * ''Dérive'' (magazine), an Austrian science magazine on urbanism *Dérive, a psychogeographical concept See also * *Derivation (disambiguatio ...
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 in 1822.

# Formulation

The
Buckingham π theorem In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically me ...
describes how every physically meaningful equation involving ''n'' variables can be equivalently rewritten as an equation of dimensionless parameters, where ''m'' is the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of the dimensional
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. 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 Nondimensionalization is the partial or full removal of dimensional analysis, physical dimensions from an mathematical equation, equation involving physical quantity, physical quantities by a suitable substitution of variables. This technique can ...
, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
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 Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
)
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
. The ''dimension'' of a physical quantity is more fundamental than some ''scale'' or
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
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 In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
, 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 symbols:
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
(T), length (L),
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
(M), electric current (I),
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic w ...
(Θ), amount of substance (N) and
luminous intensity In photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human e ...
(J). The symbols are by convention usually written in
roman Roman or Romans most often refers to: *Rome, the capital city of Italy *Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people, the people of ancient Rome *'' Epistle to the Romans'', shortened to ''Romans'', a lette ...
sans serif In typography and lettering, a sans-serif, sans serif, gothic, or simply sans letterform is one that does not have extending features called "serifs" at the end of strokes. Sans-serif typefaces tend to have less stroke width variation than ser ...
typeface. Mathematically, the dimension of the quantity ''Q'' is given by :$\operatornameQ = \mathsf^a\mathsf^b\mathsf^c\mathsf^d\mathsf^e\mathsf^f\mathsf^g$ 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 In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
– for instance, one could replace the dimension (I) of electric current of the SI basis with a dimension (Q) of
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
, since . As examples, the dimension of the physical quantity
speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...
''v'' is :$\operatornamev = \frac = \frac = \mathsf^\mathsf$ and the dimension of the physical quantity force ''F'' is :$\operatornameF = \text \times \text = \text \times \frac = \frac = \mathsf^\mathsf\mathsf$ 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, ; 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.

## Rayleigh's method

In dimensional analysis, Rayleigh's method is a conceptual tool used in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, chemistry, and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
. It expresses a
functional relationship In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
of some variables in the form of an exponential equation. It was named after
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Am ...
. The method involves the following steps: # Gather all the independent variables that are likely to influence the dependent variable. # If ''R'' is a variable that depends upon independent variables ''R''1, ''R''2, ''R''3, ..., ''R''''n'', then the
functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
can be written as . # Write the above equation in the form , where ''C'' is a dimensionless constant and ''a'', ''b'', ''c'', ..., ''m'' are arbitrary exponents. # Express each of the quantities in the equation in some base units in which the solution is required. # By using
dimensional homogeneity In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric current) and units of measur ...
, obtain a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of
simultaneous equations In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single e ...
involving the exponents ''a'', ''b'', ''c'', ..., ''m''. #
Solve Solve may refer to: * Sölve, viking king of Sweden * SOLVE, an American environmental organization * Solve (advertising agency) * "Solve" (song), by Japanese pop band Dream * HSwMS ''Sölve'' See also * Equation solving * Problem solving * ...
these equations to obtain the value of exponents ''a'', ''b'', ''c'', ..., ''m''. # Substitute the values of exponents in the main equation, and form the
non-dimensional A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s by
grouping Grouping may refer to: * Muenchian grouping * Principles of grouping * Railways Act 1921, also known as Grouping Act, a reorganisation of the British railway system * Grouping (firearms), the pattern of multiple shots from a sidearm See also ...
the variables with like exponents. 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 Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
, e.g. 60 km/h. Other relations can involve multiplication (often shown with a centered dot or
juxtaposition Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc. Speech Juxtaposition in literary terms is the showin ...
), powers (like m2 for square metres), or combinations thereof. A set of base units for a
system of measurement A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce. Systems of measurement i ...
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 Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
, 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 .

## 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 . Taking a derivative with respect to a quantity divides the dimension by the dimension of the variable that is differentiated with respect to. Thus: * position (''x'') has the dimension L (length); * derivative of position with respect to time (''dx''/''dt'',
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
) has dimension T−1L—length from position, time due to the gradient; * the second derivative (''d'x''/''dt'' = ''d''(''dx''/''dt'') / ''dt'',
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
) has dimension T−2L. Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator. * force has the dimension (mass multiplied by acceleration); * the integral of force with respect to the distance (''s'') the object has travelled ($\textstyle\int F\ ds$,
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...
) has dimension . In economics, one distinguishes between
stocks and flows Stocks are feet restraining devices that were used as a form of corporal punishment and public humiliation. The use of stocks is seen as early as Ancient Greece, where they are described as being in use in Solon's law code. The law describing ...
: 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 theis 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

The most basic rule of dimensional analysis is that of dimensional homogeneity. However, the dimensions form an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
under multiplication, so: 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 is meaningful, but the heterogeneous expression is meaningless. However, ''m''man/''L''2man is fine. Thus, dimensional analysis may be used as a
sanity check A sanity check or sanity test is a basic test to quickly evaluate whether a claim or the result of a calculation can possibly be true. It is a simple check to see if the produced material is rational (that the material's creator was thinking ration ...
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 In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
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 Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between a unit that measures 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 . 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 the unit. For example, because , and bar/bar cancels out, so .

# 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 $\left(n-1\right)$-dimensional, scales as $x^.$ Thus the volume of the ''n''-ball in terms of the radius is $C_nr^n,$ 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 referred to in terms of the distinction between stocks and flows. More generally, dimensional analysis is used in interpreting various
financial ratios A financial ratio or accounting ratio is a relative magnitude of two selected numerical values taken from an enterprise's financial statements. Often used in accounting, there are many standard ratios used to try to evaluate the overall financial ...
, economics ratios, and accounting ratios. * For example, the P/E ratio has dimensions of time (unit: year), and can be interpreted as "years of earnings to earn the price paid". * In economics, debt-to-GDP ratio also has the unit year (debt has a unit of currency, GDP has a unit of currency/year). *
Velocity of money image:M3 Velocity in the US.png, 300px, Similar chart showing the logged velocity (green) of a broader measure of money M3 that covers M2 plus large institutional deposits. The US no longer publishes official M3 measures, so the chart only runs thr ...
has a unit of 1/years (GDP/money supply has a unit of currency/year over currency): how often a unit of currency circulates per year. * Annual continuously compounded interest rates and simple interest rates are often expressed as a percentage (adimensional quantity) while time is expressed as an adimensional quantity consisting of the number of years. However, if the time includes year as the unit of measure, the dimension of the rate is 1/year. Of course, there is nothing special (apart from the usual convention) about using year as a unit of time: any other time unit can be used. Furthermore, if rate and time include their units of measure, the use of different units for each is not problematic. In contrast, rate and time need to refer to a common period if they are adimensional. (Note that effective interest rates can only be defined as adimensional quantities.) * In financial analysis, bond duration can be defined as (''dV''/''dr'')/''V'', where ''V'' is the value of a bond (or portfolio), ''r'' is the continuously compounded interest rate and ''dV''/''dr'' is a derivative. From the previous point, the dimension of ''r'' is 1/time. Therefore, the dimension of duration is time (usually expressed in years) because ''dr'' is in the "denominator" of the derivative.

## Fluid mechanics

In
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
, dimensional analysis is performed 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: $\mathrm = \frac.$ * Froude number (Fr), modeling flow with a free surface: $\mathrm = \frac.$ *
Euler number In mathematics, the Euler numbers are a sequence ''En'' of integers defined by the Taylor series expansion :\frac = \frac = \sum_^\infty \frac \cdot t^n, where \cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a ...
(Eu), used in problems in which pressure is of interest: $\mathrm = \frac.$ * Mach number (Ma), important in high speed flows where the velocity approaches or exceeds the local speed of sound: $\mathrm = \frac,$ where is the local speed of sound.

# History

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 Turin ( , Piedmontese: ; it, Torino ) is a city and an important business and cultural centre in Northern Italy. It is the capital city of Piedmont and of the Metropolitan City of Turin, and was the first Italian capital from 1861 to 1865. The ...
Academy of Science. Daviet had the master Lagrange as teacher. 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 In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically me ...
.
Simeon Poisson Simeon () is a given name, from the Hebrew (Biblical Hebrew, Biblical ''Šimʿon'', Tiberian vocalization, Tiberian ''Šimʿôn''), usually transliterated as Shimon. In Greek it is written Συμεών, hence the Latinized spelling Symeon. Meani ...
also treated the same problem of the
parallelogram law In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...
by Daviet, in his treatise of 1811 and 1833 (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.
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
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 Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
in which the gravitational constant ''G'' is taken as unity, thereby defining . By assuming a form of
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
in which the Coulomb constant ''k''e 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 John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Am ...
, 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 T−2L, instead of just the exponents.

# Examples

## A simple example: period of a harmonic oscillator

What is the period of
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
of a mass attached to an ideal linear spring with spring constant suspended in gravity of strength ? That period is the solution for of some dimensionless equation in the variables , , , and . The four quantities have the following dimensions: /T2 and /T2 From these we can form only one dimensionless product of powers of our chosen variables, $G_1$ = $T^2 k/m$ , and putting $G_1 = C$ for some dimensionless constant gives the dimensionless equation sought. The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables; here the term "group" means "collection" rather than mathematical
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. They are often called
dimensionless number A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
s as well. Note that the variable does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines with , , and , because is the only quantity that involves the dimension L. This implies that in this problem the is irrelevant. Dimensional analysis can sometimes yield strong statements about the ''irrelevance'' of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of : it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: $T = \kappa \sqrt\tfrac$, for some dimensionless constant (equal to $\sqrt$ from the original dimensionless equation). When faced with a case where dimensional analysis rejects a variable (, here) that one intuitively expects to belong in a physical description of the situation, another possibility is that the rejected variable is in fact relevant, but that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here. When dimensional analysis yields only one dimensionless group, as here, there are no unknown functions, and the solution is said to be "complete" – although it still may involve unknown dimensionless constants, such as .

## A more complex example: energy of a vibrating wire

Consider the case of a vibrating wire of length ''ℓ'' (L) vibrating with an
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
''A'' (L). The wire has a
linear density Linear density is the measure of a quantity of any characteristic value per unit of length. Linear mass density (titer in textile engineering, the amount of mass per unit length) and linear charge density (the amount of electric charge per unit ...
''ρ'' (M/L) and is under
tension Tension may refer to: Science * Psychological stress * Tension (physics), a force related to the stretching of an object (the opposite of compression) * Tension (geology), a stress which stretches rocks in two opposite directions * Voltage or el ...
''s'' (LM/T2), and we want to know the energy ''E'' (L2M/T2) in the wire. Let ''π''1 and ''π''2 be two dimensionless products of powers of the variables chosen, given by :$\begin \pi_1 &= \frac \\ \pi_2 &= \frac. \end$ The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation :$F\left\left(\frac, \frac\right\right) = 0 ,$ where ''F'' is some unknown function, or, equivalently as :$E = As f\left\left(\frac\right\right) ,$ where ''f'' is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function ''f''. But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to ''ℓ'', and so infer that . The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident. The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a
dimensionless number A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
such as the Reynolds number, which may be interpreted by dimensional analysis.

## A third example: demand versus capacity for a rotating disc

Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness ''t'' (L) and radius ''R'' (L). The disc has a density ''ρ'' (M/L3), rotates at an angular velocity ''ω'' (T−1) and this leads to a stress ''S'' (T−2L−1M) in the material. There is a theoretical linear elastic solution, given by Lame, to this problem when the disc is thin relative to its radius, the faces of the disc are free to move axially, and the plane stress constitutive relations can be assumed to be valid. As the disc becomes thicker relative to the radius then the plane stress solution breaks down. If the disc is restrained axially on its free faces then a state of plane strain will occur. However, if this is not the case then the state of stress may only be determined though consideration of three-dimensional elasticity and there is no known theoretical solution for this case. An engineer might, therefore, be interested in establishing a relationship between the five variables. Dimensional analysis for this case leads to the following (5 − 3 = 2) non-dimensional groups: : demand/capacity = ''ρR'ω''/''S'' : thickness/radius or aspect ratio = ''t''/''R'' Through the use of numerical experiments using, for example, the
finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
, the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure. As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs

# Properties

## Mathematical properties

The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
: The
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...
is written as 1; , and the inverse of L is 1/L or L−1. L raised to any integer power ''p'' is a member of the group, having an inverse of L−''p'' or 1/L''p''. The operation of the group is multiplication, having the usual rules for handling exponents (). Physically, 1/L can be interpreted as reciprocal length, and 1/T as reciprocal time (see
reciprocal second The inverse second or reciprocal second (s−1) is a unit of frequency, defined as the multiplicative inverse of the second (a unit of time). It is dimensionally equivalent to: * the unit hertz – the SI unit for frequency * the unit radian per s ...
). An abelian group is equivalent to a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
over the integers, with the dimensional symbol corresponding to the tuple . 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 module. When measurable quantities are raised to an integer power, the same is done to the dimensional symbols attached to those quantities; this corresponds to
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
in the module. A basis for such a module of dimensional symbols is called a set of base quantities, and all other vectors are called derived units. As in any module, 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, the dimension of dimensionless quantities, corresponds to the origin in this module, $\left(0, 0, 0\right)$. In certain cases, one can define fractional dimensions, specifically by formally defining fractional powers of one-dimensional vector spaces, like $V^$. However, it is not possible to take arbitrary fractional powers of units, due to representation-theoretic obstructions. One can work with vector spaces with given dimensions without needing to use units (corresponding to coordinate systems of the vector spaces). For example, given dimensions M and L, one has the vector spaces $V^M$ and $V^L$, and can define $V^ := V^M \otimes V^L$ as the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
. Similarly, the dual space can be interpreted as having "negative" dimensions. This corresponds to the fact that under the natural pairing between a vector space and its dual, the dimensions cancel, leaving a
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
scalar. 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 unit as some derived quantity ''X'' can be expressed in the general form :$X = \prod_^m \left(\pi_i\right)^\,.$ Consequently, every possible commensurate equation for the physics of the system can be rewritten in the form :$f\left(\pi_1,\pi_2, ..., \pi_m\right)=0\,.$ Knowing this restriction can be a powerful tool for obtaining new insight into the system.

## Mechanics

The dimension of physical quantities of interest in
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to object ...
can be expressed in terms of base dimensions T, L, and M – 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 In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consider ...
. 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 (linear algebra), basis: they must Linear span, span the space, and be
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to T, L, M: the former can be expressed as [F = LM/T2], L, M, while the latter can be expressed as [T = (LM/F)1/2], L, M. 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 T, L, M and Q, where Q represents the dimension of
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
. 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 also defined as a base dimension, N. 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

Scalar (physics), Scalar arguments to transcendental functions such as Exponential function, exponential, Trigonometric function, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless number, 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 (''x''''n'') 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, :$\tfrac \cdot \left(\mathrm\right) \cdot t^2 + \left(\mathrm\right) \cdot t.$ This is the height to which an object rises in time ''t'' if the acceleration of gravity is 9.8 and the initial upward speed is 500 . It is not necessary for ''t'' to be in ''seconds''. For example, suppose ''t'' = 0.01 minutes. Then the first term would be :$\begin &\tfrac \cdot \left(\mathrm\right) \cdot \left(\mathrm\right)^2 \\\left[10pt\right] = &\tfrac \cdot -9.8 \cdot \left\left(0.01^2\right\right) \left(\mathrm\right)^2 \cdot \mathrm \\\left[10pt\right] = &\tfrac \cdot -9.8 \cdot \left\left(0.01^2\right\right) \cdot 60^2 \cdot \mathrm. \end$

## Incorporating units

The value of a dimensional physical quantity ''Z'' is written as the product of a Unit of measurement, unit [''Z''] within the dimension and a dimensionless numerical factor, ''n''.For a review of the different conventions in use see: :$Z = n \times \left[Z\right] = n \left[Z\right]$ When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in the same unit so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 metre added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. A Conversion of units, conversion factor, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed: :$1 \ \mbox = 0.3048 \ \text \$ is identical to $1 = \frac.\$ The factor $0.3048 \ \frac$ is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to the same unit so that their numerical values can be added or subtracted. Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.

## Position vs displacement

Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. (In mathematics scalars are considered a special case of vectors; vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an origin (mathematics), origin. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change). Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable: * adding two displacements should yield a new displacement (walking ten paces then twenty paces gets you thirty paces forward), * adding a displacement to a position should yield a new position (walking one block down the street from an intersection gets you to the next intersection), * subtracting two positions should yield a displacement, * but one may ''not'' add two positions. This illustrates the subtle distinction between ''affine'' quantities (ones modeled by an affine space, such as position) and ''vector'' quantities (ones modeled by a vector space, such as displacement). * Vector quantities may be added to each other, yielding a new vector quantity, and a vector quantity may be added to a suitable affine quantity (a vector space ''Group action (mathematics), acts on'' an affine space), yielding a new affine quantity. * Affine quantities cannot be added, but may be subtracted, yielding ''relative'' quantities which are vectors, and these ''relative differences'' may then be added to each other or to an affine quantity. Properly then, positions have dimension of ''affine'' length, while displacements have dimension of ''vector'' length. To assign a number to an ''affine'' unit, one must not only choose a unit of measurement, but also a Origin (mathematics), point of reference, while to assign a number to a ''vector'' unit only requires a unit of measurement. Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis. This distinction is particularly important in the case of temperature, for which the numeric value of absolute zero is not the origin 0 in some scales. For absolute zero, : −273.15 °C ≘ 0 K = 0 °R ≘ −459.67 °F, where the symbol ≘ means ''corresponds to'', since although these values on the respective temperature scales correspond, they represent distinct quantities in the same way that the distances from distinct starting points to the same end point are distinct quantities, and cannot in general be equated. For temperature differences, : 1 K = 1 °C ≠ 1 °F = 1 °R. (Here °R refers to the Rankine scale, not the Réaumur scale). Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 K (although the ratio is not a constant value). But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1 K means the absolute temperature equal to −272.15 °C, or the temperature difference equal to 1 °C.

## Orientation and frame of reference

Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a ''direction''. (This issue does not arise in 1 dimension, or rather is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in a multi-dimensional space, one also needs an orientation: they need to be compared to a frame of reference. This leads to the #Extensions, extensions discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.

# Extensions

## Huntley's extensions

Huntley has pointed out that a dimensional analysis can become more powerful by discovering new independent dimensions in the quantities under consideration, thus increasing the rank $m$ of the dimensional matrix. He introduced two approaches: * The magnitudes of the components of a vector are to be considered dimensionally independent. For example, rather than an undifferentiated length dimension L, we may have Lx represent dimension in the x-direction, and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar, vector, or tensor) must be dimensionally consistent. * Mass as a measure of the quantity of matter is to be considered dimensionally independent from mass as a measure of inertia.

### Directed dimensions

As an example of the usefulness of the first approach, suppose we wish to calculate the trajectory#Range and height, distance a cannonball travels when fired with a vertical velocity component $V_\mathrm$ and a horizontal velocity component $V_\mathrm$, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then , the distance travelled, with dimension L, $V_\mathrm$, $V_\mathrm$, both dimensioned as T−1L, and the downward acceleration of gravity, with dimension T−2L. With these four quantities, we may conclude that the equation for the range may be written: :$R \propto V_\text^a\,V_\text^b\,g^c.\,$ Or dimensionally :$\mathsf = \left\left(\frac\right\right)^ \left\left(\frac\right\right)^c\,$ from which we may deduce that $a + b + c = 1$ and $a + b + 2c = 0$, which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions T and L, and four parameters, with one equation. However, if we use directed length dimensions, then $V_\mathrm$ will be dimensioned as T−1L, $V_\mathrm$ as T−1L, as L and as T−2L. The dimensional equation becomes: :$\mathsf_\mathrm = \left\left(\frac\right\right)^a \left\left(\frac\right\right)^b \left\left(\frac\right\right)^c$ and we may solve completely as $a = 1$, $b = 1$ and $c = -1$. The increase in deductive power gained by the use of directed length dimensions is apparent. Huntley's concept of directed length dimensions however has some serious limitations: * It does not deal well with vector equations involving the ''cross product'', * nor does it handle well the use of ''angles'' as physical variables. It also is often quite difficult to assign the L, L, L, L, symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries? Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's directed length dimensions to real problems.

### Quantity of matter

In Huntley's second approach, he holds that it is sometimes useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (inertial mass), and mass as a measure of the quantity of matter. Quantity of matter is defined by Huntley as a quantity (a) proportional to inertial mass, but (b) not implicating inertial properties. No further restrictions are added to its definition. For example, consider the derivation of Poiseuille's Law. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass, we may choose as the relevant variables: There are three fundamental variables, so the above five equations will yield two independent dimensionless variables: :$\pi_1 = \frac$ :$\pi_2 = \frac$ If we distinguish between inertial mass with dimension $M_\text$ and quantity of matter with dimension $M_\text$, then mass flow rate and density will use quantity of matter as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written: :$C = \frac$ where now only is an undetermined constant (found to be equal to $\pi/8$ by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield Poiseuille's law. Huntley's recognition of quantity of matter as an independent quantity dimension is evidently successful in the problems where it is applicable, but his definition of quantity of matter is open to interpretation, as it lacks specificity beyond the two requirements (a) and (b) he postulated for it. For a given substance, the SI dimension amount of substance, with unit Mole (unit), mole, does satisfy Huntley's two requirements as a measure of quantity of matter, and could be used as a quantity of matter in any problem of dimensional analysis where Huntley's concept is applicable.

## Siano's extension: orientational analysis

Angles are, by convention, considered to be dimensionless quantities. As an example, consider again the projectile problem in which a point mass is launched from the origin at a speed and angle above the ''x''-axis, with the force of gravity directed along the negative ''y''-axis. It is desired to find the range , at which point the mass returns to the ''x''-axis. Conventional analysis will yield the dimensionless variable , but offers no insight into the relationship between and . Siano has suggested that the directed dimensions of Huntley be replaced by using ''orientational symbols'' to denote vector directions, and an orientationless symbol 10. Thus, Huntley's L becomes L1 with L specifying the dimension of length, and specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that , the following multiplication table for the orientation symbols results: Note that the orientational symbols form a group (the Klein four-group or "Viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem". Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of . For angles, consider an angle that lies in the z-plane. Form a right triangle in the z-plane with being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation and the side opposite has an orientation . Since (using to indicate orientational equivalence) we conclude that an angle in the xy-plane must have an orientation , which is not unreasonable. Analogous reasoning forces the conclusion that has orientation while has orientation 10. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form , where and are real scalars. Note that an expression such as $\sin\left(\theta+\pi/2\right)=\cos\left(\theta\right)$ is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written: :$\sin\left\left(a\,1_\text + b\,1_\text\right\right) = \sin\left\left(a\,1_\text\right) \cos\left(b\,1_\text\right\right) + \sin\left\left(b\,1_\text\right) \cos\left(a\,1_\text\right\right),$ which for $a = \theta$ and $b = \pi/2$ yields $\sin\left(\theta\,1_\text + \left[\pi/2\right]\,1_\text\right) = 1_\text\cos\left(\theta\,1_\text\right)$. Siano distinguishes between geometric angles, which have an orientation in 3-dimensional space, and phase angles associated with time-based oscillations, which have no spatial orientation, i.e. the orientation of a phase angle is $1_0$. The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive more information about acceptable solutions of physical problems. In this approach, one solves the dimensional equation as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral, putting it into Canonical form, normal form. The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols. The solution is then more complete than the one that dimensional analysis alone gives. Often, the added information is that one of the powers of a certain variable is even or odd. As an example, for the projectile problem, using orientational symbols, , being in the xy-plane will thus have dimension and the range of the projectile will be of the form: :$R = g^a\,v^b\,\theta^c\text\mathsf\,1_\mathrm \sim \left\left(\frac\right\right)^a \left\left(\frac\right\right)^b\,1_\mathsf^c.\,$ Dimensional homogeneity will now correctly yield and , and orientational homogeneity requires that $1_x /\left(1_y^a 1_z^c\right)=1_z^=1$. In other words, that must be an odd integer. In fact, the required function of theta will be which is a series consisting of odd powers of . It is seen that the Taylor series of and are orientationally homogeneous using the above multiplication table, while expressions like and are not, and are (correctly) deemed unphysical. Siano's orientational analysis is compatible with the conventional conception of angular quantities as being dimensionless, and within orientational analysis, the radian may still be considered a dimensionless unit. The orientational analysis of a quantity equation is carried out separately from the ordinary dimensional analysis, yielding information that supplements the dimensional analysis.

# Dimensionless concepts

## Constants

The dimensionless constants that arise in the results obtained, such as the C in the Poiseuille's Law problem and the $\kappa$ in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "back of the envelope" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.

## Formalisms

Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the Ising model can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, $\xi$ ) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be $\sim 1/\xi^$ where $d$ is the dimension of the lattice. It has been argued by some physicists, e.g., Michael Duff (physicist), Michael J. Duff, that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: ''Speed of light, c'', ''Planck constant, ħ'', and ''Gravitational constant, G'', in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other. Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants ''ħ'', ''c'', and ''G'' (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit $c\rightarrow \infty$, $\hbar\rightarrow 0$ and $G\rightarrow 0$. In problems involving a gravitational field the latter limit should be taken such that the field stays finite.

# Dimensional equivalences

Following are tables of commonly occurring expressions in physics, related to the dimensions of energy, momentum, and force.

## Natural units

If , where ''c'' is the speed of light and ''ħ'' is the reduced Planck constant, and a suitable fixed unit of energy is chosen, then all quantities of time ''T'', length ''L'' and mass ''M'' can be expressed (dimensionally) as a power of energy ''E'', because length, mass and time can be expressed using speed ''v'', action ''S'', and energy ''E'': :$t = \frac,\quad L = \frac, \quad M = \frac$ though speed and action are dimensionless ( and ) – so the only remaining quantity with dimension is energy. In terms of powers of dimensions: :$\mathsf^n = \mathsf^p\mathsf^q\mathsf^r = \mathsf^$ This is particularly useful in particle physics and high energy physics, in which case the energy unit is the electron volt (eV). Dimensional checks and estimates become very simple in this system. However, if electric charges and currents are involved, another unit to be fixed is for electric charge, normally the electron charge ''e'' though other choices are possible.

*
Buckingham π theorem In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically me ...
* Dimensionless numbers in fluid mechanics * Fermi problem, Fermi estimate – used to teach dimensional analysis * Numerical-value equation * Rayleigh's method of dimensional analysis * Similitude (model) – an application of dimensional analysis * System of measurement

## Related areas of mathematics

* Covariance and contravariance of vectors * Exterior algebra * Geometric algebra * Quantity calculus

## Programming languages

Dimensional correctness as part of Type system#Type checking, type checking has been studied since 1977. Implementations for Ada and C++ were described in 1985 and 1988. Kennedy's 1996 thesis describes an implementation in Standard ML, and later in F Sharp (programming language), F#. There are implementations for Haskell (programming language), Haskell, OCaml, and Rust (programming language), Rust, Python, and a code checker for Fortran.
Griffioen's 2019 thesis extended Kennedy's Hindley–Milner type system to support Hart's matrices. McBride and Nordvall-Forsberg show how to use dependent types to extend type systems for units of measure.

# References

* * * * * * * * * A
postscript
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*[http://www.boost.org/doc/libs/1_66_0/doc/html/boost_units.html A C++ implementation of compile-time dimensional analysis in the Boost open-source libraries]
Buckingham's pi-theoremQuantity System calculator for units conversion based on dimensional approach

Units, quantities, and fundamental constants project dimensional analysis maps
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## Converting units

*[http://www.chem.tamu.edu/class/fyp/mathrev/mr-da.html Math Skills Review]
U.S. EPA tutorialA Discussion of UnitsShort Guide to Unit ConversionsChapter 11: Behavior of Gases
''Chemistry: Concepts and Applications'', Denton independent school District
www.gnu.org/software/units
free program, very practical {{Authority control Dimensional analysis, Measurement Conversion of units of measurement Chemical engineering Mechanical engineering Environmental engineering