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A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity[citation needed] as well as quantity of dimension one)[1] is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1),[2][3] which is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Dimensionless quantities are distinct from quantities that have associated dimensions, such as time (measured in seconds). Dimensionless units are dimensionless values that serve as units of measurement for expressing other quantities, such as radians (rad) or steradians (sr) for plane angles and solid angles, respectively.[2] For example, optical extent is defined as having units of metres multiplied by steradians.[4]

## 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 nineteenth 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.[5]

Numerous dimensionless numbers, mostly ratios, were coined in the early 1900s, particularly in the areas of fluid mechanics and heat transfer. Measuring ratios in the (derived) unit dB (decibel) 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[6] argued for formalizing the radian as a physical unit. The idea was rebutted[7] 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.[8][9][10]

## Integers

Integer numbers may be used to represent discrete dimensionless quantities. More specifically, counting numbers can be used to express countable quantities,[11][12] such as the 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.

## Ratios, proportions, and angles

Dimensionless quantities are often obtained as ratios of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation.[13] 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 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) and angle units such as turn, radian, degree (° = π/180) and grad (= π/200). 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.[14] 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.

## Buckingham π theorem

 Main article: Buckingham π theorem
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The Buckingham π 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 – 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 = nk independent, dimensionless quantities. For the purposes of the experimenter, different systems that share the same description by dimensionless quantity are equivalent.

### Example

To demonstrate the application of the π theorem, consider the power consumption of a stirrer with a given shape. The power, P, in dimensions [M · L2/T3], is a function of the density, ρ [M/L3], and the viscosity of the fluid to be stirred, μ [M/(L · T)], as well as the size of the stirrer given by its diameter, D [L], and the 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 independent dimensions, e.g., length: L (SI units: m), time: T (s), and mass: M (kg).

According to the π-theorem, the n = 5 variables can be reduced by the k = 3 dimensions to form p = nk = 5 − 3 = 2 independent dimensionless numbers. Usually, these quantities are chosen as ${\textstyle \mathrm {Re} ={\frac {\rho nD^{2)){\mu ))}$, commonly named the Reynolds number which describes the fluid flow regime, and ${\textstyle N_{\mathrm {p} }={\frac {P}{\rho n^{3}D^{5))))$, 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 dimensionally independent basis variables, which, in this example, appear in both dimensionless quantities. The Reynolds number and power number fall from the above analysis if ${\textstyle \rho }$, n, and D are chosen to be the basis variables. If, instead, ${\textstyle \mu }$, n, and D are selected, the Reynolds number is recovered while the second dimensionless quantity becomes ${\textstyle N_{\mathrm {Rep} }={\frac {P}{\mu D^{3}n^{2))))$. We note that ${\textstyle N_{\mathrm {Rep} ))$ is the product of the Reynolds number and the power number.

## 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:[15]

## Other quantities produced by nondimensionalization

 Main article: List of dimensionless quantities

Physics often uses dimensionless quantities to simplify the characterization of systems with multiple interacting physical phenomena. These may be found by applying the Buckingham π theorem or otherwise may emerge from making partial differential equations unitless by the process of nondimensionalization. Engineering, economics, and other fields often extend these ideas in design and analysis of the relevant systems.

### Physics and engineering

• Fresnel number – wavenumber over distance
• Mach number – ratio of the speed of an object or flow relative to the speed of sound in the fluid.
 Further information: Dimensionless numbers in fluid mechanics
• Beta (plasma physics) – 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 – (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 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.[16]
• 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.[17]

### 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

## 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. ^ a b "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. ^ "17-21-048 | CIE". cie.co.at. Retrieved 2023-02-20.
5. ^
6. ^ "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)
7. ^ 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.
8. ^ "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.
9. ^ "BIPM Consultative Committee for Units (CCU), 16th Meeting" (PDF). Archived from the original (PDF) on 2006-11-30. Retrieved 2010-01-22.
10. ^ Dybkær, René (2004). "An ontology on property for physical, chemical, and biological systems". APMIS Suppl. (117): 1–210. PMID 15588029.
11. ^ 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.
12. ^ 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.
13. ^
14. ^ 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.
15. ^ Baez, John Carlos (2011-04-22). "How Many Fundamental Constants Are There?". Retrieved 2015-10-07.
16. ^ Huba, Joseph D. (2007). "NRL Plasma Formulary: Dimensionless Numbers of Fluid Mechanics". Naval Research Laboratory. pp. 23–25. Retrieved 2015-10-07.
17. ^ Zukoski, Edward E. (1986). "Fluid Dynamic Aspects of Room Fires" (PDF). Fire Safety Science. Retrieved 2022-06-13.