Dimensionless numbers of chemistry

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OR:

A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a
quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a uni ...
to which no physical dimension is assigned, with a corresponding SI
unit of measurement 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 ...
of one (or 1), ISBN 978-92-822-2272-0. which is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics,
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 relat ...
,
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties, ...
,
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
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
. Dimensionless quantities are distinct from quantities that have associated dimensions, such as time (measured in
second The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ea ...
s). Dimensionless units are dimensionless values that serve as
units of measurement 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 ...
plane angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ar ...
s and
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The po ...
s, respectively. For example, optical extent is defined as having units of metres multiplied by steradians.International Commission on Illumination (CIE) e-ILV, CIE S 017:2020 ILV: International Lighting Vocabulary, 2nd edition.
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# History

Quantities having dimension one, ''dimensionless quantities'', regularly occur in sciences, and are formally treated within the field of
dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...
. In the nineteenth century, French mathematician
Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harm ...
and Scottish physicist
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 lig ...
led significant developments in the modern concepts of
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
and
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'' (al ...
. Later work by British physicists
Osborne Reynolds Osborne Reynolds (23 August 1842 – 21 February 1912) was an Irish-born innovator in the understanding of fluid dynamics. Separately, his studies of heat transfer between solids and fluids brought improvements in boiler and condenser design. ...
and
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. Amo ...
contributed to the understanding of dimensionless numbers in physics. Building on Rayleigh's method of dimensional analysis, Edgar Buckingham proved the theorem (independently of French mathematician
Joseph Bertrand Joseph Louis François Bertrand (; 11 March 1822 – 5 April 1900) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics. Biography Joseph Bertrand was the ...
's previous work) to formalize the nature of these quantities. Numerous dimensionless numbers, mostly ratios, were coined in the early 1900s, particularly in the areas of
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 biomed ...
and
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
. Measuring ''ratios'' in the (derived) unit ''dB'' (
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
) finds widespread use nowadays. There have been periodic proposals to "patch" the SI system to reduce confusion regarding physical dimensions. For example, a 2017
op-ed An op-ed, short for "opposite the editorial page", is a written prose piece, typically published by a North-American newspaper or magazine, which expresses the opinion of an author usually not affiliated with the publication's editorial board. ...
in
Nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
(1 page) argued for formalizing the
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
as a physical unit. The idea was rebutted on the grounds that such a change would raise inconsistencies for both established dimensionless groups, like the Strouhal number, and for mathematically distinct entities that happen to have the same units, like torque (a vector product) versus energy (a
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
). In another instance in the early 2000s, the
International Committee for Weights and Measures The General Conference on Weights and Measures (GCWM; french: Conférence générale des poids et mesures, CGPM) is the supreme authority of the International Bureau of Weights and Measures (BIPM), the intergovernmental organization established i ...
discussed naming the unit of 1 as the " uno", but the idea of just introducing a new SI name for 1 was dropped.

# Integers

Integer numbers may be used to represent discrete dimensionless quantities. More specifically, counting numbers can be used to express countable quantities, such as the
number of particles The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter ''N'', is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which ...
and population size. In mathematics, the "number of elements" in a set is termed '' cardinality''. '' Countable nouns'' is a related linguistics concept. Counting numbers, such as number of bits, can be compounded with units of frequency ( inverse second) to derive units of count rate, such as
bits per second In telecommunications and computing, bit rate (bitrate or as a variable ''R'') is the number of bits that are conveyed or processed per unit of time. The bit rate is expressed in the unit bit per second (symbol: bit/s), often in conjunction w ...
.
Count data Count (feminine: countess) is a historical title of nobility in certain European countries, varying in relative status, generally of middling rank in the hierarchy of nobility. Pine, L. G. ''Titles: How the King Became His Majesty''. New York: ...
is a related concept in statistics.

# Ratios, proportions, and angles

Dimensionless quantities are often obtained as
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s of
quantities Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
that are not dimensionless, but whose dimensions cancel out in the mathematical operation. Examples include calculating slopes or unit conversion factors. A more complex example of such a ratio is engineering strain, a measure of physical deformation defined as a change in length divided by the initial length. Since both quantities have the dimension ''length'', their ratio is dimensionless. Another set of examples is mass fractions or
mole fraction In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This e ...
s often written using
parts-per notation In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, the ...
such as ppm (= 10−6), ppb (= 10−9), and ppt (= 10−12), or perhaps confusingly as ratios of two identical units ( kg/kg or mol/mol). For example,
alcohol by volume Alcohol by volume (abbreviated as ABV, abv, or alc/vol) is a standard measure of how much alcohol (ethanol) is contained in a given volume of an alcoholic beverage (expressed as a volume percent). It is defined as the number of millilitres (mL) o ...
, which characterizes the concentration of
ethanol Ethanol (abbr. EtOH; also called ethyl alcohol, grain alcohol, drinking alcohol, or simply alcohol) is an organic compound. It is an alcohol with the chemical formula . Its formula can be also written as or (an ethyl group linked to a hyd ...
in an
alcoholic beverage An alcoholic beverage (also called an alcoholic drink, adult beverage, or a drink) is a drink that contains ethanol, a type of alcohol that acts as a drug and is produced by fermentation of grains, fruits, or other sources of sugar. The con ...
, could be written as . Other common proportions are percentages % (= 0.01),    (= 0.001) and angle units such as turn,
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
, degree (° = ) and grad (= ). In statistics the
coefficient of variation In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as ...
is the ratio of the standard deviation to the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''ari ...
and is used to measure the dispersion in the data. It has been argued that quantities defined as ratios having equal dimensions in numerator and denominator are actually only ''unitless quantities'' and still have physical dimension defined as . For example, moisture content may be defined as a ratio of volumes (volumetric moisture, m3⋅m−3, dimension L⋅L) or as a ratio of masses (gravimetric moisture, units kg⋅kg−1, dimension M⋅M); both would be unitless quantities, but of different dimension.

# Buckingham theorem

The Buckingham theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked by
Boyle's Law Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as: The ...
– they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold. Another consequence of the theorem is that the
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional s ...
dependence between a certain number (say, ''n'') of variables can be reduced by the number (say, ''k'') of
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
s occurring in those variables to give a set of ''p'' = ''n'' − ''k'' independent, dimensionless
quantities Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
. For the purposes of the experimenter, different systems that share the same description by dimensionless
quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a uni ...
are equivalent.

## Example

To demonstrate the application of the theorem, consider the power consumption of a stirrer with a given shape. The power, ''P'', in dimensions · L2/T3 is a function of the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
, ''ρ'' /L3 and the
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the int ...
of the fluid to be stirred, ''μ'' /(L · T) as well as the size of the stirrer given by its
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
, ''D'' and the
angular speed Angular may refer to: Anatomy * Angular artery, the terminal part of the facial artery * Angular bone, a large bone in the lower jaw of amphibians and reptiles * Angular incisure, a small anatomical notch on the stomach * Angular gyrus, a regio ...
of the stirrer, ''n'' /T Therefore, we have a total of ''n'' = 5 variables representing our example. Those ''n'' = 5 variables are built up from ''k'' = 3 fundamental dimensions, the length: L ( SI units: m), time: T ( s), and mass: M ( kg). According to the -theorem, the ''n'' = 5 variables can be reduced by the ''k'' = 3 dimensions to form ''p'' = ''n'' − ''k'' = 5 − 3 = 2 independent dimensionless numbers. Usually, these quantities are chosen as $\mathrm =$, commonly named the
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
which describes the fluid flow regime, and $N_\mathrm = \frac$, the
power number : For Newton number, see also Kissing number in the sphere packing problem. The power number ''N''p (also known as Newton number) is a commonly used dimensionless number relating the resistance force to the inertia force. The power-number ha ...
, which is the dimensionless description of the stirrer. Note that the two dimensionless quantities are not unique and depend on which of the ''n'' = 5 variables are chosen as the ''k'' = 3 independent basis variables, which appear in both dimensionless quantities. The Reynolds number and power number fall from the above analysis if $\rho$, ''n'', and ''D'' are chosen to be the basis variables. If instead, $\mu$, ''n'', and ''D'' are selected, the Reynolds number is recovered while the second dimensionless quantity becomes $N_\mathrm = \frac$. We note that $N_\mathrm$ is the product of the Reynolds number and the power number.

# Dimensionless physical constants

Certain universal dimensioned physical constants, such as the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit for ...
in a vacuum, the universal gravitational constant, the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, the
Coulomb constant The Coulomb constant, the electric force constant, or the electrostatic constant (denoted , or ) is a proportionality constant in electrostatics equations. In SI base units it is equal to .Derived from ''k''e = 1/(4''πε''0) – It was nam ...
, and the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
can be normalized to 1 if appropriate units for time,
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inter ...
,
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 elementa ...
, charge, and temperature are chosen. The resulting system of units is known as the
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 ...
, specifically regarding these five constants, Planck units. However, not all
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant ...
s can be normalized in this fashion. For example, the values of the following constants are independent of the system of units, cannot be defined, and can only be determined experimentally: * ''α'' ≈ 1/137, the fine-structure constant, which characterizes the magnitude of the
electromagnetic interaction In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
between electrons. * ''β'' (or ''μ'') ≈ 1836, the proton-to-electron mass ratio. This ratio is the
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
of the
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mas ...
divided by that of the electron. An analogous ratio can be defined for any
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ant ...
; * ''α''s ≈ 1, a constant characterizing the
strong nuclear force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
coupling strength; * The ratio of the mass of any given elementary particle to the Planck mass, $\sqrt$.

# Other quantities produced by nondimensionalization

Physics often uses dimensionless
quantities Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
to simplify the characterization of systems with multiple interacting physical phenomena. These may be found by applying the Buckingham theorem or otherwise may emerge from making
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
unitless by the process of
nondimensionalization Nondimensionalization is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are ...
. Engineering, economics, and other fields often extend these ideas in
design A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process or the result of that plan or specification in the form of a prototype, product, or process. The verb ''to design'' ...
and analysis of the relevant systems.

## Physics and engineering

*
Fresnel number The Fresnel number (''F''), named after the physicist Augustin-Jean Fresnel, is a dimensionless number occurring in optics, in particular in scalar diffraction theory. Definition For an electromagnetic wave passing through an aperture and hitti ...
– wavenumber over distance *
Mach number Mach number (M or Ma) (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Moravian physicist and philosopher Ernst Mach. : \mathrm = \ ...
– ratio of the speed of an object or flow relative to the speed of sound in the fluid. *
Beta (plasma physics) The beta of a plasma, symbolized by ''β'', is the ratio of the plasma pressure (''p'' = ''n'' ''k''B ''T'') to the magnetic pressure (''p''mag = ''B''²/2 ''μ''0). The term is commonly used in studies of the Sun and Earth's magnetic field, ...
– ratio of plasma pressure to magnetic pressure, used in magnetospheric physics as well as fusion plasma physics. * Damköhler numbers (Da) – used in chemical engineering to relate the chemical reaction timescale (reaction rate) to the transport phenomena rate occurring in a system. * Thiele modulus – describes the relationship between diffusion and reaction rate in porous catalyst pellets with no mass transfer limitations. * Numerical aperture – characterizes the range of angles over which the system can accept or emit light. *
Sherwood number The Sherwood number (Sh) (also called the mass transfer Nusselt number) is a dimensionless number used in mass-transfer operation. It represents the ratio of the convective mass transfer to the rate of diffusive mass transport, and is named in ho ...
– (also called the mass transfer
Nusselt number In thermal fluid dynamics, the Nusselt number (, after Wilhelm Nusselt) is the ratio of convective to conductive heat transfer at a boundary in a fluid. Convection includes both advection (fluid motion) and diffusion (conduction). The conduc ...
) is a dimensionless number used in mass-transfer operation. It represents the ratio of the convective mass transfer to the rate of diffusive mass transport. * Schmidt number – defined as the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes. *
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
is commonly used in fluid mechanics to characterize flow, incorporating both properties of the fluid and the flow. It is interpreted as the ratio of inertial forces to viscous forces and can indicate flow regime as well as correlate to frictional heating in application to flow in pipes. * Zukoski number, usually noted Q*, is the ratio of the heat release rate of a fire to the enthalpy of the gas flow rate circulating through the fire. Accidental and natural fires usually have a Q* of ~1. Flat spread fires such as forest fires have Q*<1. Fires originating from pressured vessels or pipes, with additional momentum caused by pressure, have Q*>>>1.

## Chemistry

* Relative density – density relative to water *
Relative atomic mass Relative atomic mass (symbol: ''A''; sometimes abbreviated RAM or r.a.m.), also known by the deprecated synonym atomic weight, is a dimensionless physical quantity defined as the ratio of the average mass of atoms of a chemical element in a gi ...
, Standard atomic weight *
Equilibrium constant The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
(which is sometimes dimensionless)

## Other fields

* Cost of transport is the
efficiency Efficiency is the often measurable ability to avoid wasting materials, energy, efforts, money, and time in doing something or in producing a desired result. In a more general sense, it is the ability to do things well, successfully, and without ...
in moving from one place to another * Elasticity is the measurement of the proportional change of an economic variable in response to a change in another

*
Arbitrary unit In science and technology, an arbitrary unit (abbreviated arb. unit, '' see below'') or procedure defined unit (p.d.u.) is a relative unit of measurement to show the ratio of amount of substance, intensity, or other quantities, to a predetermined ...
*
Dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...
*
Normalization (statistics) In statistics and applications of statistics, normalization can have a range of meanings. In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging ...
and
standardized moment In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invaria ...
, the analogous concepts in statistics *
Orders of magnitude (numbers) This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantities and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as ...
*
Similitude (model) Similitude is a concept applicable to the testing of engineering models. A model is said to have similitude with the real application if the two share geometric similarity, kinematic similarity and dynamic similarity. ''Similarity'' and ''simili ...
* List of dimensionless quantities