Dimensionless quantity

Dimensionless quantities, or quantities of dimension one,^[1] are quantities implicitly defined in a manner that prevents their aggregation into units of measurement.^[2][3] Typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined units. For instance, alcohol by volume (ABV) represents a volumetric ratio; its value remains independent of the specific units of volume used, such as in milliliters per milliliter (mL/mL).

The number one is recognized as a dimensionless base quantity.^[4] Radians serve as dimensionless units for angular measurements, derived from the universal ratio of 2π times the radius of a circle being equal to its circumference.^[5]

Dimensionless quantities play a crucial role serving as parameters in differential equations in various technical disciplines. In calculus, concepts like the unitless ratios in limits or derivatives 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,^[6] the fine-structure constant in quantum mechanics,^[7] and the Lorentz factor in relativity.^[8] In chemistry, state properties and ratios such as mole fractions concentration ratios are dimensionless.^[9]

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. 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 π theorem (independently of French mathematician Joseph Bertrand's previous work) to formalize the nature of these quantities.^[10]

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^[11] argued for formalizing the radian as a physical unit. The idea was rebutted^[12] 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.^[13][14][15]

Buckingham π theorem

Main article: Buckingham π theorem

The Buckingham π theorem ^[16] 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.

Integers

Number of entities
Common symbols	N
SI unit	Unitless
Dimension	1

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.^[17][18] The concept is formalized as quantity number of entities (symbol N) in ISO 80000-1.^[19] 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.

Ratios, proportions, and angles

Dimensionless quantities can be obtained as ratios of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation.^[19][20] 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 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 Q = A/B having equal dimensions in numerator and denominator are actually only unitless quantities and still have physical dimension defined as dim Q = dim A × dim B⁻¹.^[21] For example, moisture content may be defined as a ratio of volumes (volumetric moisture, m³⋅m⁻³, dimension L³⋅L⁻³) or as a ratio of masses (gravimetric moisture, units kg⋅kg⁻¹, dimension M⋅M⁻¹); both would be unitless quantities, but of different dimension.

Dimensionless physical constants

Main article: 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:^[22]

• 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. ^[23]
• 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 ${\displaystyle r}$, a ratio between the contributions of tensor and scalar modes to the primordial power spectrum observed in the CMB. ^[23]
• The Immirzi-Barbero parameter ${\displaystyle \gamma }$, which characterizes the area gap in loop quantum gravity. ^[24]
• emissivity, which is the ratio of actual emitted radiation from a surface to that of an idealized surface at the same temperature

List

Main article: List of dimensionless quantities

Physics and engineering

• Lorentz factor^[25] – 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 ${\displaystyle Q^{*}}$, 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 ${\displaystyle Q^{*}\approx 1}$. Flat spread fires such as forest fires have ${\displaystyle Q^{*}<1}$. Fires originating from pressured vessels or pipes, with additional momentum caused by pressure, have ${\displaystyle Q^{*}\gg 1}$.^[26]

Fluid mechanics

Main article: 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 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

1. "1.8 (1.6) quantity of dimension one dimensionless quantity". International vocabulary of metrology — Basic and general concepts and associated terms (VIM). ISO. 2008. Retrieved 2011-03-22.
2. "SI Brochure: The International System of Units, 9th Edition". BIPM. ISBN 978-92-822-2272-0.
3. Mohr, Peter J.; Phillips, William Daniel (2015-06-01). "Dimensionless units in the SI". Metrologia. 52.
4. Mills, I. M. (May 1995). "Unity as a Unit". Metrologia. 31 (6): 537–541. Bibcode:1995Metro..31..537M. doi:10.1088/0026-1394/31/6/013. ISSN 0026-1394.
5. Zebrowski, Ernest (1999). A History of the Circle: Mathematical Reasoning and the Physical Universe. Rutgers University Press. ISBN 978-0-8135-2898-4.
6. Cengel, Yunus; Cimbala, John (2013-10-16). EBOOK: Fluid Mechanics Fundamentals and Applications (SI units). McGraw Hill. ISBN 978-0-07-717359-3.
7. Webb, J. K.; King, J. A.; Murphy, M. T.; Flambaum, V. V.; Carswell, R. F.; Bainbridge, M. B. (2011-10-31). "Indications of a Spatial Variation of the Fine Structure Constant". Physical Review Letters. 107 (19): 191101. arXiv:1008.3907. Bibcode:2011PhRvL.107s1101W. doi:10.1103/PhysRevLett.107.191101. PMID 22181590.
8. Einstein, A. (2005-02-23). "Zur Elektrodynamik bewegter Körper [AdP 17, 891 (1905)]". Annalen der Physik. 14 (S1): 194–224. doi:10.1002/andp.200590006.
9. Ghosh, Soumyadeep; Johns, Russell T. (2016-09-06). "Dimensionless Equation of State to Predict Microemulsion Phase Behavior". Langmuir. 32 (35): 8969–8979. doi:10.1021/acs.langmuir.6b02666. ISSN 0743-7463. PMID 27504666.
10. Buckingham, Edgar (1914). "On physically similar systems; illustrations of the use of dimensional equations". Physical Review. 4 (4): 345–376. Bibcode:1914PhRv....4..345B. doi:10.1103/PhysRev.4.345. hdl:10338.dmlcz/101743.
11. "Lost dimension: A flaw in the SI system leaves physicists grappling with ambiguous units - SI units need reform to avoid confusion" (PDF). This Week: Editorials. Nature. 548 (7666): 135. 2017-08-10. Bibcode:2017Natur.548R.135.. doi:10.1038/548135b. ISSN 1476-4687. PMID 28796224. S2CID 4444368. Archived (PDF) from the original on 2022-12-21. Retrieved 2022-12-21. (1 page)
12. Wendl, Michael Christopher (September 2017). "Don't tamper with SI-unit consistency". Nature. 549 (7671): 160. doi:10.1038/549160d. ISSN 1476-4687. PMID 28905893. S2CID 52806576.
13. "BIPM Consultative Committee for Units (CCU), 15th Meeting" (PDF). 17–18 April 2003. Archived from the original (PDF) on 2006-11-30. Retrieved 2010-01-22.
14. "BIPM Consultative Committee for Units (CCU), 16th Meeting" (PDF). Archived from the original (PDF) on 2006-11-30. Retrieved 2010-01-22.
15. Dybkær, René (2004). "An ontology on property for physical, chemical, and biological systems". APMIS Suppl. (117): 1–210. PMID 15588029.
16. Buckingham, E. (1914). "On Physically Similar Systems; Illustrations of the Use of Dimensional Equations". Physical Review. 4 (4): 345–376. Bibcode:1914PhRv....4..345B. doi:10.1103/physrev.4.345.
17. Rothstein, Susan (2017). Semantics for Counting and Measuring. Key Topics in Semantics and Pragmatics. Cambridge University Press. p. 206. ISBN 978-1-107-00127-5. Retrieved 2021-11-30.
18. Berch, Daniel B.; Geary, David Cyril; Koepke, Kathleen Mann (2015). Development of Mathematical Cognition: Neural Substrates and Genetic Influences. Elsevier Science. p. 13. ISBN 978-0-12-801909-2. Retrieved 2021-11-30.
19. "ISO 80000-1:2022(en) Quantities and units — Part 1: General". iso.org. Retrieved 2023-07-23.
20. "7.3 Dimensionless groups" (PDF). Massachusetts Institute of Technology. Retrieved 2023-11-03.
21. Johansson, Ingvar (2010). "Metrological thinking needs the notions of parametric quantities, units and dimensions". Metrologia. 47 (3): 219–230. Bibcode:2010Metro..47..219J. doi:10.1088/0026-1394/47/3/012. ISSN 0026-1394. S2CID 122242959.
22. Baez, John Carlos (2011-04-22). "How Many Fundamental Constants Are There?". Retrieved 2015-10-07.
23. Navas, S.; Amsler, C.; Gutsche, T.; Hanhart, C.; Hernández-Rey, J. J.; Lourenço, C.; Masoni, A.; Mikhasenko, M.; Mitchell, R. E.; Patrignani, C.; Schwanda, C.; Spanier, S.; Venanzoni, G.; Yuan, C. Z.; Agashe, K. (2024-08-01). "Review of Particle Physics". Physical Review D. 110 (3). doi:10.1103/PhysRevD.110.030001. hdl:20.500.11850/695340. ISSN 2470-0010.
24. Rovelli, Carlo (2004). Quantum Gravity. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511755804. ISBN 978-0-521-71596-6.
25. Einstein, A. (2005-02-23). "Zur Elektrodynamik bewegter Körper [AdP 17, 891 (1905)]". Annalen der Physik. 14 (S1): 194–224. doi:10.1002/andp.200590006.
26. Zukoski, Edward E. (1986). "Fluid Dynamic Aspects of Room Fires" (PDF). Fire Safety Science. Retrieved 2022-06-13.

Further reading

• Flater, David (October 2017) [2017-05-20, 2017-03-23, 2016-11-22]. Written at National Institute of Standards and Technology, Gaithersburg, Maryland, USA. "Redressing grievances with the treatment of dimensionless quantities in SI". Measurement. 109. London, UK: Elsevier Ltd.: 105–110. Bibcode:2017Meas..109..105F. doi:10.1016/j.measurement.2017.05.043. eISSN 1873-412X. ISSN 0263-2241. PMC 7727271. PMID 33311828. NIHMS1633436. (15 pages)

External links

• Media related to Dimensionless numbers at Wikimedia Commons