Dimensionless quantity

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is applicable. It is also known as a bare number or a quantity of dimension one.^[1] Dimensionless quantities are widely used in many fields, such as mathematics, physics, engineering, and economics. By contrast, examples of quantities with dimensions are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second.

History

Quantities having dimension 1, habitually called 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 (independent of French mathematician Joseph Bertrand's previous work) to formalize the nature of these quantities. Numerous other 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. 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.^[2]^[3]^[4]

Pure numbers

All pure numbers are dimensionless quantities, for example 1, $i$ , $π$ , $e$ , and $φ$ .^[5] Units of number such as the dozen, gross, googol, and Avogadro's number may also be considered dimensionless, although Avogadro's constant is definitely not.

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.^[6] 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 more 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 radians, degrees and grads. 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.

Buckingham $π$ theorem

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 = n − k 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 fundamental dimensions, the 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 = n − k = 5 − 3 = 2 independent dimensionless numbers. These quantities are $\mathrm {Re} ={\frac {\rho nD^{2}}{\mu }}$ , commonly named the Reynolds number which describes the fluid flow regime, and $N_{\mathrm {p} }={\frac {P}{\rho n^{3}D^{5}}}$ , the Power number, which is the dimensionless description of the stirrer.

Dimensionless physical constants

Certain fundamental physical constants, such as the speed of light in a vacuum, the universal gravitational constant, Planck's constant and Boltzmann's 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. 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 and must be determined experimentally:^[7]

α ≈ 1/137, the fine structure constant, which characterizes the magnitude of the electromagnetic interaction between electrons.
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;
α_s, a constant characterizing the strong force;
α_G ≈ 1.75 × 10⁻⁴⁵, the gravitational coupling constant, which characterizes the magnitude of the gravitational interaction between electrons.

Other quantities produced by nondimensionalization

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
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.^[8]
Mach number – ratio of the speed of an object or flow relative to the speed of sound in the fluid.

Beta (plasma physics) - ratio of plasma pressure to magnetic pressure, used in magnetospheric physics as well as fusion plasma physics.

Chemistry

Relative density – density relative to water
Relative atomic mass, Standard atomic weight

Other fields

Cost of transport is the efficiency in moving from one place to another

References

↑ "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.
↑ "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.
↑ "BIPM Consultative Committee for Units (CCU), 16th Meeting" (PDF). Archived from the original (PDF) on 2006-11-30. Retrieved 2010-01-22.
↑ Dybkaer, René (2004). "An ontology on property for physical, chemical, and biological systems". APMIS Suppl. (117): 1–210. PMID 15588029.
↑ https://www.youtube.com/watch?v=Ym5u5IlYWcg
↑ http://web.mit.edu/6.055/old/S2008/notes/apr02a.pdf
↑ Baez, John (April 22, 2011). "How Many Fundamental Constants Are There?". Retrieved October 7, 2015.
↑ Huba, J. D. (2007). "NRL Plasma Formulary: Dimensionless Numbers of Fluid Mechanics". Naval Research Laboratory. Retrieved October 7, 2015. p. 23–25

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