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In
dimensional analysis 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 measure ...
, a dimensionless quantity is a
quantity Quantity 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 in terms of a unit of measu ...
to which no physical dimension is assigned, also known as a bare, pure, or scalar quantity or a quantity of dimension one, with a corresponding unit of measurement in the SI of the unit 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
,
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
,
chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other . ...

chemistry
,
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 ...

engineering
, and
economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a bran ...

economics
. Dimensionless quantities are distinct from quantities that have associated dimensions, such as
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

time
(measured in
second The second (symbol: s, also abbreviated: sec) is the base unit of time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, th ...
s). However, the symbols rad and sr are written explicitly where appropriate, in order to emphasize that, for radians or steradians, the quantity being considered is, or involves the plane angle or solid angle respectively. For example,
etendueEtendue or étendue (; ) is a property of light in an optics, optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics. From the source point of v ...

etendue
is defined as having units of meters times steradians.International Commission on Illumination (CIE) e-ILV, CIE S 017:2020 ILV: International Lighting Vocabulary, 2nd edition.
/ref>


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 quantity, base quantities (such as length, mass, time, and electric current) and units of measure ...
. In the nineteenth century, French mathematician
Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Republic (french: link=no, Ré ...

Joseph Fourier
and Scottish physicist
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In classica ...

James Clerk Maxwell
led significant developments in the modern concepts of
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
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 (album), ...
. Later work by British physicists
Osborne Reynolds Osborne Reynolds FRS (23 August 1842 – 21 February 1912) was an innovator in the understanding of fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids ...

Osborne Reynolds
and
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was a British scientist who made extensive contributions to both theoretical A theory is a rational type of abstract thinking about a phenomenon A phenome ...

Lord Rayleigh
contributed to the understanding of dimensionless numbers in physics. Building on Rayleigh's method of dimensional analysis,
Edgar Buckingham Edgar Buckingham (July 8, 1867 in Philadelphia Philadelphia, colloquially Philly, is a city in the state of Pennsylvania in the United States. It is the sixth-most populous city in the United States and the most populous city in the state o ...
proved the theorem (independently of French mathematician
Joseph Bertrand Joseph Louis François Bertrand (11 March 1822 – 5 April 1900) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...

Joseph Bertrand
'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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in o ...
and
heat transfer Heat transfer is a discipline of thermal engineering Thermal engineering is a specialized sub-discipline of mechanical engineering Mechanical engineering is an engineering Engineering is the use of scientific method, scientific pri ...

heat transfer
. Measuring ''ratios'' in the (derived) unit ''dB'' (
decibel The decibel (symbol: dB) is a relative equal to one tenth of a bel (B). It expresses the ratio of two values of a on a . Two signals whose differ by one decibel have a power ratio of 101/10 (approximately ) or root-power ratio of 10 (approxim ...

decibel
) finds widespread use nowadays. 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 in ...
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 A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surrounding ...
and
population size In population genetics and population ecology, population size (usually denoted ''N'') is the number of individual organisms in a population. Population size is directly associated with amount of genetic drift, and is the underlying cause of effects ...
. In mathematics, the "number of elements" in a set is termed ''
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
''. ''
Countable noun In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include ...
s'' is a related linguistics concept.


Ratios, proportions, and angles

Dimensionless quantities are often obtained as
ratio In mathematics, a ratio indicates 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 ...

ratio
s of
quantities Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact of existence. The term has a history of use reaching ba ...
that are not dimensionless, but whose dimensions cancel out in the mathematical operation. Examples include calculating
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

slope
s or unit conversion factors. A more complex example of such a ratio is
engineering strain In engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encom ...
, 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 Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they underg ...
s often written using
parts-per notation In science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable explanations and predictio ...
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 (also called ethyl alcohol, grain alcohol, drinking alcohol, or simply alcohol) is an organic Organic may refer to: * Organic, of or relating to an organism, a living entity * Organic, of or relating to an anatomical organ (anatomy), ...

ethanol
in an
alcoholic beverage An alcoholic drink is a drink A drink (or beverage) is a liquid A liquid is a nearly incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric process, isochoric flow) refers to a fluid f ...
, could be written as . Other common proportions are percentages % (= 0.01),    (= 0.001) and angle units such as
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

radian
,
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
(° = ) and grad (= ). In
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

statistics
the
coefficient of variation In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expr ...
is the ratio of the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

standard deviation
to the
mean There are several kinds of mean in mathematics, especially in statistics. For a data set, the ''arithmetic mean'', also known as arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by ...

mean
and is used to measure the
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variation ...
in the
data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to correspond to experience. Standard reference works are often used ...
. 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 Image:soil-phase-diagram.svg, 300px, Soil morphology, Soil composition by Volume and Mass, by phase: air, water, void (pores filled with water or air), soil, and total. Water content or moisture content is the quantity of water contained in a materi ...
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 Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
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 – 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) In architecture File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted).jpg, upright=1.45, alt=Pl ...
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 * Independent ...
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
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 Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact of existence. The term has a history of use reaching ba ...
. For the purposes of the experimenter, different systems that share the same description by dimensionless
quantity Quantity 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 in terms of a unit of measu ...
are equivalent.


Example

To demonstrate the application of the theorem, consider the
power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...
consumption of a with a given shape. The power, ''P'', in dimensions · L2/T3 is a function of the
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

density
, ''ρ'' /L3 and the
viscosity The viscosity of a fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, ...

viscosity
of the fluid to be stirred, ''μ'' /(L · T) as well as the size of the stirrer given by its
diameter In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

diameter
, ''D'' and the
angular speed Angular frequency ''ω'' (in radians per second), is larger than frequency ''ν'' (in cycles per second, also called Hertz, Hz), by a factor of . This figure uses the symbol ''ν'', rather than ''f'' to denote frequency. In [ hysics, angular freque ...

angular speed
of the stirrer, ''n'' [1/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: meters, 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 The Reynolds number () helps predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers flows tend to be turbulent In fluid dynam ...
which describes the fluid flow regime, and N_\mathrm = \frac, the power number, 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 A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
in a vacuum, the universal gravitational constant, ,
Coulomb's constant The Coulomb constant, the electric force constant, or the electrostatic constant (denoted , or ) is a proportionality constant in electrostatics equations. In SI units it is equal to .Derived from ''k''e = 1/(4''πε''0) – It was named aft ...
, and
Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
can be normalized to 1 if appropriate units for
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

time
,
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

length
,
mass Mass is the quantity Quantity 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 ...
,
charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * Charge (David Ford album), ''Charge'' (David Ford album) * Charge (Machel Montano album), ''Charge'' (Mac ...
, and
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

temperature
are chosen. The resulting
system of units A system of measurement is a collection of units of measurement A unit of measurement is a definite magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude ...
is known as the
natural units In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Phy ...
, specifically regarding these five constants,
Planck units In particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particl ...
. 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 (mathematics), constant value in time. It is contrasted with a ...
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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
, which characterizes the magnitude of the
electromagnetic interaction Electromagnetism is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ...
between electrons. * ''β'' (or ''μ'') ≈ 1836, the
proton-to-electron mass ratio In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succes ...
. 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 Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** ...
of the
proton A proton is a subatomic particle, symbol or , with a positive electric charge of +1''e'' elementary charge and a mass slightly less than that of a neutron. Protons and neutrons, each with masses of approximately one atomic mass unit, are collecti ...

proton
divided by that of the
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

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 the fundamental fermions (quarks, leptons, antiquarks, and a ...
; * ''α''s ≈ 1, a constant characterizing the
strong nuclear force In nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied. Nuclear physics should not be confused with atomic physics, which ...
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 Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact of existence. The term has a history of use reaching ba ...
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 , a partial differential equation (PDE) is an equation which imposes relations between the various s of a . The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved ...
unitless by the process of
nondimensionalization Nondimensionalization is the partial or full removal of physical dimensions from an equation In mathematics, an equation is a statement that asserts the equality (mathematics), equality of two Expression (mathematics), expressions, which are c ...
. 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 A prototype is an early sample, mode ...

design
and analysis of the relevant systems.


Physics and engineering

*
Fresnel number perfect lens having Fresnel number equal to 100. Adopted wavelength for propagation is 1  µm. Image:half inch perfect lens real amp Fresnel number 001 at focus.png, 210px, Aperture real amplitude as estimated at focus of a half inch perfect ...
– wavenumber over distance *
Mach number #REDIRECT Mach number#REDIRECT Mach number 300px, An F/A-18 Hornet creating a vapor cone at transonic speed">vapor_cone.html" ;"title="F/A-18 Hornet creating a vapor cone">F/A-18 Hornet creating a vapor cone at transonic speed just before reachi ...
– 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 Pressure (symbol: ''p'' or ''P'') is the force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge o ...
– ratio of plasma pressure to magnetic pressure, used in magnetospheric physics as well as fusion plasma physics. *
Damköhler numbers The Damköhler numbers (Da) are dimensionless numbers used in chemical engineering to relate the chemical reaction timescale (reaction rate) to the transport phenomena rate occurring in a system. It is named after German chemist Gerhard Damköhler. ...
(Da) – used in chemical engineering to relate the chemical reaction timescale (reaction rate) to the transport phenomena rate occurring in a system. *
Thiele modulus The Thiele modulus was developed by Ernest Thiele Ernest W. Thiele (1895–1993) was an influential chemical engineering researcher at Standard Oil (then Amoco, now BP) and Professor of Chemical Engineering at the University of Notre Dame. He is kn ...
– describes the relationship between diffusion and reaction rate in porous catalyst pellets with no mass transfer limitations. *
Numerical aperture of light goes through a flat plane of glass, its half-angle changes to . Due to Snell's law, the numerical aperture remains the same:\text = n_1 \sin \theta_1 = n_2 \sin\theta_2. In optics Optics is the branch of physics Physics (from ...

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 hon ...
– (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. * 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 The Reynolds number () helps predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers flows tend to be turbulent In fluid dynam ...
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.


Chemistry

* Relative density – density relative to water * Relative atomic mass, Standard atomic weight * Equilibrium constant (which is sometimes dimensionless)


Other fields

* Cost of transport is the efficiency in moving from one place to another * Elasticity (economics), Elasticity is the measurement of the proportional change of an economic variable in response to a change in another


See also

* Arbitrary unit * Dimensional analysis * Normalization (statistics) and standardized moment, the analogous concepts in
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

statistics
* Orders of magnitude (numbers) * Similitude (model) * List of dimensionless quantities


References


External links

*{{Commons category-inline, Dimensionless numbers Dimensionless numbers, Mathematical concepts Physical constants