dimensionless quantity

{{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. 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 {{pi}} theorem (independently of French mathematician Joseph Bertrand's previous work) to formalize the nature of these quantities.{{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}}

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 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{{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 |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) argued for formalizing the radian as a physical unit. The idea was rebutted{{cite journal |author-last=Wendl |author-first=Michael Christopher |author-link=Michael Christopher Wendl |title=Don't tamper with SI-unit consistency |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 }} 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 another instance in the early 2000s, the International Committee for Weights and Measures discussed naming the unit of 1 as the "uno", but the idea of just introducing a new SI name for 1 was dropped.{{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}}{{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}}{{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}}

{{main|Buckingham π theorem|l1=Buckingham {{pi}} theorem}}

{{Unreferenced section|date=April 2022}}

The Buckingham {{pi}} theorem {{ 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 }}

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 – 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 dependence between a certain number (say, n) of variables can be reduced by the number (say, k) of independent dimensions occurring in those variables to give a set of p = n − k independent, dimensionless quantities. For the purposes of the experimenter, different systems that share the same description by dimensionless quantity are equivalent.

{{Infobox physical quantity

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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.{{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}}{{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}}

The concept is formalized as quantity number of entities (symbol N) in 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}}

Examples include number of particles 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.

Count data is a related concept in statistics.

The concept may be generalized by allowing non-integer numbers to account for fractions of a full item, e.g., number of turns equal to one half.

Dimensionless quantities can be obtained as ratios of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation.{{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}} Examples of quotients of dimension one include calculating slopes or some unit conversion factors. Another set of examples is mass fractions or mole fractions, often written using parts-per notation 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, which characterizes the concentration of ethanol in an alcoholic beverage, could be written as {{nowrap|mL / 100 mL}}.

Other common proportions are percentages % (= 0.01),  ‰ (= 0.001). Some angle units such as 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 mean and is used to measure the dispersion in the 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}}}}.{{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}}

For example, moisture content may be defined as a ratio of volumes (volumetric moisture, m³⋅m⁻³, dimension L{{sup|3}}⋅L{{sup|−3}}) or as a ratio of masses (gravimetric moisture, units kg⋅kg⁻¹, dimension M⋅M{{sup|−1}}); both would be unitless quantities, but of different dimension.

{{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, 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 constants 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:{{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}}

• 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. {{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 }}
• 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_s ≈ 1.
• The tensor-to-scalar ratio $r$, a ratio between the contributions of tensor and scalar modes to the primordial power spectrum observed in the CMB.
• The Immirzi-Barbero parameter $\backslash gamma$, which characterizes the area gap in loop quantum gravity. {{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}}
• emissivity, which is the ratio of actual emitted radiation from a surface to that of an idealized surface at the same temperature

{{Main|List of dimensionless quantities}}

• Lorentz factor{{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}} – 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 $Q^*$, 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 $Q^*\; \backslash approx\; 1$. Flat spread fires such as forest fires have . Fires originating from pressured vessels or pipes, with additional momentum caused by pressure, have $Q^*\backslash gg\; 1$.{{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}}

{{Main|Dimensionless numbers in fluid mechanics}}

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

• Cost of transport is the efficiency 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
• Basic reproduction number is a dimensionless ratio used in epidemiology to quantify the transmissibility of an infection.

• List of dimensionless quantities
• Arbitrary unit
• Dimensional analysis
• Normalization (statistics) and standardized moment, the analogous concepts in statistics
• Orders of magnitude (numbers)
• Similitude (model)

{{Reflist}}

• {{cite journal |title=Redressing grievances with the treatment of dimensionless quantities in SI |author-first=David |author-last=Flater

|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)

• {{Commons category-inline|Dimensionless numbers}}