View source for Dimensionless quantity

{{Short description|Quantity with no physical dimension}}
{{Use dmy dates|date=December 2022|cs1-dates=y}}
{{More citations needed|date=March 2017}}

'''Dimensionless quantities''', or quantities of dimension one,<ref>{{cite web |url=http://www.iso.org/sites/JCGM/VIM/JCGM_200e_FILES/MAIN_JCGM_200e/01_e.html#L_1_8 |title='''1.8''' (1.6) '''quantity of dimension one''' dimensionless quantity |work=International vocabulary of metrology — Basic and general concepts and associated terms (VIM) |publisher=[[International Organization for Standardization|ISO]] |date=2008 |access-date=2011-03-22}}</ref> are quantities [[implicitly defined]] in a manner that prevents their aggregation into [[unit of measurement|units of measurement]].<ref name="SI Brochure">{{cite web |url=https://www.bipm.org/en/publications/si-brochure/ |title=SI Brochure: The International System of Units, 9th Edition |publisher=[[International Bureau of Weights and Measures|BIPM]]}} ISBN 978-92-822-2272-0.</ref><ref>{{cite journal |author-last1=Mohr |author-first1=Peter J. |author-last2=Phillips |author-first2=William Daniel |author-link2=William Daniel Phillips |date=2015-06-01 |title=Dimensionless units in the SI |url=https://www.nist.gov/publications/dimensionless-units-si |journal=[[Metrologia]] |language=en |volume=52}}</ref> Typically expressed as [[Ratio|ratios]] that align with another system, these quantities do not necessitate explicitly defined [[Unit of measurement|units]]. For instance, [[alcohol by volume]] (ABV) represents a [[volumetric ratio]]; its value remains independent of the specific [[Unit of volume|units of volume]] used, such as in [[milliliter]]s per milliliter (mL/mL).

The [[1|number one]] is recognized as a dimensionless [[Base unit of measurement|base quantity]].<ref>{{Cite journal |last=Mills |first=I. M. |date=May 1995 |title=Unity as a Unit |url=https://dx.doi.org/10.1088/0026-1394/31/6/013 |journal=Metrologia |language=en |volume=31 |issue=6 |pages=537–541 |doi=10.1088/0026-1394/31/6/013 |bibcode=1995Metro..31..537M |issn=0026-1394}}</ref> [[Radian|Radians]] serve as dimensionless units for [[Angle|angular measurements]], derived from the universal ratio of 2π times the [[radius]] of a circle being equal to its circumference.<ref>{{Cite book |last=Zebrowski |first=Ernest |url=https://books.google.com/books?id=2twRfiUwkxYC&dq=universal+ratio+of+2%CF%80+times+the+radius+of+a+circle+being+equal+to+its+circumference&pg=PR9 |title=A History of the Circle: Mathematical Reasoning and the Physical Universe |date=1999 |publisher=Rutgers University Press |isbn=978-0-8135-2898-4 |language=en}}</ref>

Dimensionless quantities play a crucial role serving as [[Parameter|parameters]] in [[Differential equation|differential equations]] in various technical disciplines. In [[calculus]], concepts like the unitless ratios in [[Limits of integration|limits]] or [[derivative]]s often involve dimensionless quantities. In [[differential geometry]], the use of dimensionless parameters is evident in geometric relationships and transformations. Physics relies on dimensionless numbers like the [[Reynolds number]] in [[fluid dynamics]],<ref>{{Cite book |last1=Cengel |first1=Yunus |url=https://books.google.com/books?id=LIZvEAAAQBAJ&dq=calculus,+concepts+like+the+unitless+ratios+in+limits+or+derivatives+often+involve+dimensionless+quantities&pg=PP1 |title=EBOOK: Fluid Mechanics Fundamentals and Applications (SI units) |last2=Cimbala |first2=John |date=2013-10-16 |publisher=McGraw Hill |isbn=978-0-07-717359-3 |language=en}}</ref> the [[fine-structure constant]] in [[quantum mechanics]],<ref>{{Cite journal |last1=Webb |first1=J. K. |last2=King |first2=J. A. |last3=Murphy |first3=M. T. |last4=Flambaum |first4=V. V. |last5=Carswell |first5=R. F. |last6=Bainbridge |first6=M. B. |date=2011-10-31 |title=Indications of a Spatial Variation of the Fine Structure Constant |url=https://link.aps.org/doi/10.1103/PhysRevLett.107.191101 |journal=Physical Review Letters |volume=107 |issue=19 |pages=191101 |doi=10.1103/PhysRevLett.107.191101|pmid=22181590 |arxiv=1008.3907 |bibcode=2011PhRvL.107s1101W }}</ref> and the [[Lorentz factor]] in [[Special relativity|relativity]].<ref>{{Cite journal |last=Einstein |first=A. |date=2005-02-23 |title=Zur Elektrodynamik bewegter Körper [AdP 17, 891 (1905)] |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.200590006 |journal=Annalen der Physik |language=en |volume=14 |issue=S1 |pages=194–224 |doi=10.1002/andp.200590006}}</ref> In [[chemistry]], [[Equation of state|state properties]] and ratios such as [[Mole fraction|mole fractions]] [[Concentration ratio|concentration ratios]] are dimensionless.<ref>{{Cite journal |last1=Ghosh |first1=Soumyadeep |last2=Johns |first2=Russell T. |date=2016-09-06 |title=Dimensionless Equation of State to Predict Microemulsion Phase Behavior |url=https://pubs.acs.org/doi/10.1021/acs.langmuir.6b02666 |journal=Langmuir |language=en |volume=32 |issue=35 |pages=8969–8979 |doi=10.1021/acs.langmuir.6b02666 |pmid=27504666 |issn=0743-7463}}</ref>

== History ==
{{See also|Dimensional analysis#History}}
Quantities having dimension one, ''dimensionless quantities'', regularly occur in sciences, and are formally treated within the field of [[dimensional analysis]]. In the 19th century, French mathematician [[Joseph Fourier]] and Scottish physicist [[James Clerk Maxwell]] led significant developments in the modern concepts of [[dimension]] and [[Unit (measurement)|unit]]. Later work by British physicists [[Osborne Reynolds]] and [[Lord Rayleigh]] contributed to the understanding of dimensionless numbers in physics. Building on Rayleigh's method of dimensional analysis, [[Edgar Buckingham]] proved the [[Buckingham π theorem|{{pi}} theorem]] (independently of French mathematician [[Joseph Bertrand]]'s previous work) to formalize the nature of these quantities.<ref>{{cite journal |author-last=Buckingham |author-first=Edgar |author-link=Edgar Buckingham |date=1914 |title=On physically similar systems; illustrations of the use of dimensional equations |journal=[[Physical Review]] |volume=4 |issue=4 |pages=345–376 |doi=10.1103/PhysRev.4.345 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.31210014450082&view=1up&seq=905 |bibcode=1914PhRv....4..345B |hdl=10338.dmlcz/101743 |hdl-access=free}}</ref>

Numerous dimensionless numbers, mostly ratios, were coined in the early 1900s, particularly in the areas of [[fluid mechanics]] and [[heat transfer]]. Measuring logarithm of ratios as [[level quantity|''levels'']] in the (derived) unit [[decibel]] (dB) 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]] in [[Nature (journal)|Nature]]<ref name="nature_2017">{{cite journal |title=Lost dimension: A flaw in the SI system leaves physicists grappling with ambiguous units - SI units need reform to avoid confusion |journal=[[Nature (journal)|Nature]] |date=2017-08-10 |volume=548 |issue=7666 |page=135 |doi=10.1038/548135b |pmid=28796224 |bibcode=2017Natur.548R.135. |s2cid=4444368 |language=en |department=This Week: Editorials |issn=1476-4687 |url=https://www.nature.com/articles/548135b.pdf?error=cookies_not_supported&code=87d78113-7ea0-47c0-a0ac-cd3da87c16ba |access-date=2022-12-21 |url-status=live |archive-url=https://web.archive.org/web/20221221120517/https://www.nature.com/articles/548135b.pdf?error=cookies_not_supported&code=87d78113-7ea0-47c0-a0ac-cd3da87c16ba |archive-date=2022-12-21}} (1 page)</ref> argued for formalizing the [[radian]] as a physical unit. The idea was rebutted<ref name="wendl_2017">{{cite journal |author-last=Wendl |author-first=Michael Christopher |author-link=Michael Christopher Wendl |title=Don't tamper with SI-unit consistency |journal=[[Nature (journal)|Nature]] |date=September 2017 |volume=549 |issue=7671 |pages=160 |doi=10.1038/549160d |pmid=28905893 |s2cid=52806576 |language=en |issn=1476-4687|doi-access=free }}</ref> 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 [[Cross product|vector product]]) versus energy (a [[Dot product|scalar product]]). In another instance in the early 2000s, the [[International Committee for Weights and Measures]] discussed naming the unit of 1 as the "[[Uno (unit)|uno]]", but the idea of just introducing a new SI name for 1 was dropped.<ref>{{cite web |url=http://www.bipm.fr/utils/common/pdf/CCU15.pdf |title=BIPM Consultative Committee for Units (CCU), 15th Meeting |date=17–18 April 2003 |access-date=2010-01-22 |url-status=dead |archive-url=https://web.archive.org/web/20061130201238/http://www.bipm.fr/utils/common/pdf/CCU15.pdf |archive-date=2006-11-30}}</ref><ref>{{cite web |url=http://www.bipm.fr/utils/common/pdf/CCU16.pdf |title=BIPM Consultative Committee for Units (CCU), 16th Meeting |access-date=2010-01-22 |url-status=dead |archive-url=https://web.archive.org/web/20061130200835/http://www.bipm.fr/utils/common/pdf/CCU16.pdf |archive-date=2006-11-30}}</ref><ref>{{cite journal |author-last=Dybkær |author-first=René |author-link=René Dybkær |title=An ontology on property for physical, chemical, and biological systems |journal=APMIS Suppl. |issue=117 |pages=1–210 |date=2004 |pmid=15588029 |url=http://www.iupac.org/publications/ci/2005/2703/bw1_dybkaer.html}}</ref>


== Buckingham {{pi}} theorem ==
{{main|Buckingham π theorem|l1=Buckingham {{pi}} theorem}}
{{Unreferenced section|date=April 2022}}
The Buckingham {{pi}} theorem <ref>{{ cite journal  | title=On Physically Similar Systems; Illustrations of the Use of Dimensional Equations | year=1914 | pages=345–376 | journal=Physical Review | doi=10.1103/physrev.4.345 | volume=4 | issue=4 | url=http://dx.doi.org/10.1103/PhysRev.4.345 | last1=Buckingham | first1= E. | bibcode=1914PhRv....4..345B }}
</ref> 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 (mathematics)|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]] – 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 [[Function (mathematics)|functional]] dependence between a certain number (say, ''n'') of [[variable (mathematics)|variables]] can be reduced by the number (say, ''k'') of [[independent variable|independent]] [[dimension]]s occurring in those variables to give a set of ''p'' = ''n'' − ''k'' independent, dimensionless [[quantity|quantities]]. For the purposes of the experimenter, different systems that share the same description by dimensionless [[quantity]] are equivalent.

== Integers ==
{{Infobox physical quantity
| name = Number of entities
| othernames = 
| width =
| background =
| image = 
| caption = 
| unit = [[Unitless]]
| otherunits =
| symbols = ''N''
| baseunits =
| dimension = [[Dimension one|1]]
| extensive =
| intensive =
| conserved =
| transformsas =
| derivations =
}}

[[Integer numbers]] may represent dimensionless quantities. They can represent discrete quantities, which can also be dimensionless.  
More specifically, [[counting numbers]] can be used to express '''countable quantities'''.<ref name="Rothstein2017">{{cite book |author-last=Rothstein |author-first=Susan |author-link=Susan Rothstein |title=Semantics for Counting and Measuring |publisher=[[Cambridge University Press]] |series=Key Topics in Semantics and Pragmatics |date=2017 |isbn=978-1-107-00127-5 |url=https://books.google.com/books?id=yV5UDgAAQBAJ&pg=PA206 |access-date=2021-11-30 |page=206}}</ref><ref>{{cite book |author-last1=Berch |author-first1=Daniel B. |author-last2=Geary |author-first2=David Cyril |author-link2=David Cyril Geary |author-last3=Koepke |author-first3=Kathleen Mann |title=Development of Mathematical Cognition: Neural Substrates and Genetic Influences |publisher=[[Elsevier Science]] |date=2015 |isbn=978-0-12-801909-2 |url=https://books.google.com/books?id=XS9OBQAAQBAJ&pg=PR13 |access-date=2021-11-30 |page=13}}</ref>
The concept is formalized as quantity '''number of entities''' (symbol ''N'') in [[ISO 80000-1]].<ref name="ISO 80000-1">{{cite web |title=ISO 80000-1:2022(en) Quantities and units — Part 1: General |website=iso.org |url=https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en |ref={{sfnref |iso.org}} |access-date=2023-07-23}}</ref>
Examples include [[number of particles]] and [[population size]]. In mathematics, the "number of elements" in a set is termed ''[[cardinality]]''. ''[[Countable noun]]s'' is a related linguistics concept.
Counting numbers, such as number of [[bit]]s, can be compounded with units of frequency ([[inverse second]]) to derive units of count rate, such as [[bits per second]].
[[Count data]] is a related concept in statistics.
The concept may be generalized by allowing [[non-integer number]]s to account for fractions of a full item, e.g., [[number of turns]] equal to one half.

== Ratios, proportions, and angles ==
Dimensionless quantities can be obtained as [[ratio]]s of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation.<ref name="ISO 80000-1" /><ref>{{Cite web |title=7.3 Dimensionless groups |url=http://web.mit.edu/6.055/old/S2008/notes/apr02a.pdf |access-date=3 November 2023 |website=[[Massachusetts Institute of Technology]]}}</ref> Examples of quotients of dimension one include calculating [[slope]]s or some [[Conversion of units|unit conversion factors]]. Another set of examples is [[mass fraction (chemistry)|mass fraction]]s or [[mole fraction]]s, often written using [[parts-per notation]] such as ppm (=&nbsp;10<sup>−6</sup>), ppb (=&nbsp;10<sup>−9</sup>), and ppt (=&nbsp;10<sup>−12</sup>), or perhaps confusingly as ratios of two identical units ([[kilogram|kg]]/kg or [[mole (unit)|mol]]/mol). For example, [[alcohol by volume]], which characterizes the concentration of [[ethanol]] in an [[alcoholic beverage]], could be written as {{nowrap|mL / 100 mL}}.

Other common proportions are percentages [[%]]&nbsp;(=&nbsp;0.01), &nbsp;[[per mil|‰]]&nbsp;(=&nbsp;0.001). Some angle units such as [[turn (angle)|turn]], [[radian]], and [[steradian]] are defined as ratios of quantities of the same kind. In [[statistics]] the [[coefficient of variation]] is the ratio of the [[standard deviation]] to the [[average|mean]] and is used to measure the [[Statistical dispersion|dispersion]] in the [[statistical data|data]].

It has been argued that quantities defined as ratios {{nowrap|1=''Q'' = ''A''/''B''}} having equal dimensions in numerator and denominator are actually only ''unitless quantities'' and still have physical dimension defined as {{nowrap|1=dim ''Q'' = dim ''A'' × dim ''B''{{i sup|−1}}}}.<ref name="Johansson2010">{{cite journal |author-last=Johansson |author-first=Ingvar |title=Metrological thinking needs the notions of parametric quantities, units and dimensions |journal=[[Metrologia]] |volume=47 |issue=3 |date=2010 |pages=219–230 |issn=0026-1394 |doi=10.1088/0026-1394/47/3/012 |bibcode=2010Metro..47..219J |s2cid=122242959}}</ref>
For example, [[moisture content]] may be defined as a ratio of volumes (volumetric moisture, m<sup>3</sup>⋅m<sup>−3</sup>, dimension L{{sup|3}}⋅L{{sup|−3}}) or as a ratio of masses (gravimetric moisture, units kg⋅kg<sup>−1</sup>, dimension M⋅M{{sup|−1}}); both would be unitless quantities, but of different dimension.

== Dimensionless physical constants ==
{{main|Dimensionless physical constant}}

Certain universal dimensioned physical constants, such as the [[speed of light]] in vacuum, the [[universal gravitational constant]], the [[Planck constant]], the [[Coulomb constant]], and the [[Boltzmann constant]] can be normalized to 1 if appropriate units for [[time]], [[length]], [[mass]], [[electric charge|charge]], and [[temperature]] are chosen. The resulting [[system of units]] is known as the [[natural units]], specifically regarding these five constants, [[Planck units]]. However, not all [[physical 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:<ref>{{cite web |url=http://math.ucr.edu/home/baez/constants.html |title=How Many Fundamental Constants Are There? |author-last=Baez |author-first=John Carlos |author-link=John Carlos Baez |date=2011-04-22 |access-date=2015-10-07}}</ref>

* [[engineering strain]], a measure of physical deformation defined as a change in length divided by the initial length.

* [[fine-structure constant]], ''α'' ≈ 1/137 which characterizes the magnitude of the [[electromagnetic interaction]] between electrons. <ref name=":0">{{Cite journal |last=Navas |first=S. |last2=Amsler |first2=C. |last3=Gutsche |first3=T. |last4=Hanhart |first4=C. |last5=Hernández-Rey |first5=J. J. |last6=Lourenço |first6=C. |last7=Masoni |first7=A. |last8=Mikhasenko |first8=M. |last9=Mitchell |first9=R. E. |last10=Patrignani |first10=C. |last11=Schwanda |first11=C. |last12=Spanier |first12=S. |last13=Venanzoni |first13=G. |last14=Yuan |first14=C. Z. |last15=Agashe |first15=K. |date=2024-08-01 |title=Review of Particle Physics |url=https://link.aps.org/doi/10.1103/PhysRevD.110.030001 |journal=Physical Review D |language=en |volume=110 |issue=3 |doi=10.1103/PhysRevD.110.030001 |issn=2470-0010|hdl=20.500.11850/695340 |hdl-access=free }}</ref>
* ''β'' (or ''μ'') ≈ 1836, the [[proton-to-electron mass ratio]]. This ratio is the [[rest mass]] of the [[proton]] divided by that of the [[electron]]. An analogous ratio can be defined for any [[elementary particle]].
* [[Strong force]] coupling strength ''α''<sub>s</sub> ≈ 1.
* The tensor-to-scalar ratio <math>r</math>, a ratio between the contributions of tensor and scalar modes to the primordial power spectrum observed in the [[Cosmic microwave background|CMB]]. <ref name=":0" />
* The [[Immirzi parameter|Immirzi-Barbero]] parameter <math>\gamma</math>, which characterizes the area gap in [[loop quantum gravity]]. <ref>{{Cite book |last=Rovelli |first=Carlo |url=https://www.cambridge.org/core/books/quantum-gravity/9EEB701AAB938F06DCF151AACE1A445D |title=Quantum Gravity |date=2004 |publisher=Cambridge University Press |isbn=978-0-521-71596-6 |series=Cambridge Monographs on Mathematical Physics |location=Cambridge |doi=10.1017/cbo9780511755804}}</ref>
* [[emissivity]], which is the ratio of actual emitted radiation from a surface to that of an [[Black body|idealized surface]] at the same [[temperature]]

== List ==
{{Main|List of dimensionless quantities}}

=== Physics and engineering ===
* [[Lorentz factor]]<ref>{{Cite journal |last=Einstein |first=A. |date=2005-02-23 |title=Zur Elektrodynamik bewegter Körper [AdP 17, 891 (1905)] |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.200590006 |journal=Annalen der Physik |language=en |volume=14 |issue=S1 |pages=194–224 |doi=10.1002/andp.200590006}}</ref> – parameter used in the context of special relativity for time dilation, length contraction, and relativistic effects between observers moving at different velocities
* [[Fresnel number]] – wavenumber (spatial frequency) over distance

* [[Beta (plasma physics)]] – ratio of plasma pressure to magnetic pressure, used in magnetospheric physics as well as fusion plasma physics.
* [[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.
* Zukoski number, usually noted <math>Q^*</math>, 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 <math>Q^* \approx 1</math>. Flat spread fires such as forest fires have <math>Q^*<1</math>. Fires originating from pressured vessels or pipes, with additional momentum caused by pressure, have <math>Q^*\gg 1</math>.<ref>{{cite web |url=https://authors.library.caltech.edu/21188/1/287_Zukoski_EE_1985.pdf |title=Fluid Dynamic Aspects of Room Fires |author-last=Zukoski |author-first=Edward E. |date=1986 |publisher=Fire Safety Science |access-date=2022-06-13}}</ref>

==== Fluid mechanics ====
{{Main|Dimensionless numbers in fluid mechanics}}

=== 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
* [[Basic reproduction number]] is a dimensionless ratio used in epidemiology to quantify the transmissibility of an infection.

== See also ==
* [[List of dimensionless quantities]]
* [[Arbitrary unit]]
* [[Dimensional analysis]]
* [[Normalization (statistics)]] and [[standardized moment]], the analogous concepts in [[statistics]]
* [[Orders of magnitude (numbers)]]
* [[Similitude (model)]]

==References==
{{Reflist}}

==Further reading==
* {{cite journal |title=Redressing grievances with the treatment of dimensionless quantities in SI |author-first=David |author-last=Flater 
|journal=[[Measurement (journal)|Measurement]] |issn=0263-2241 |eissn=1873-412X |publisher=[[Elsevier Ltd.]] |publication-place=London, UK |location=[[National Institute of Standards and Technology]], Gaithersburg, Maryland, USA |date=October 2017 |orig-date=2017-05-20, 2017-03-23, 2016-11-22 |volume=109 |pages=105–110 |pmc=7727271 |id=NIHMS1633436 |pmid=33311828 |doi=10.1016/j.measurement.2017.05.043 |bibcode=2017Meas..109..105F }} [https://web.archive.org/web/20221220195918/https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7727271/pdf/nihms-1633436.pdf] (15 pages)

==External links==
* {{Commons category-inline|Dimensionless numbers}}

[[Category:Dimensionless quantities| ]]