Dimensionless quantity

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.[8]

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[9] argued for formalizing the radian as a physical unit. The idea was rebutted[10] 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.[11][12][13]

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.

Integer numbers may be used to represent discrete dimensionless quantities. More specifically, counting numbers can be used to express countable quantities.[14][15] The concept is formalized as quantity number of entities (symbol N) in ISO 80000-1.[4] 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 are often obtained as ratios, the quotient resulting from the division of quantities of the same kind ― that are not dimensionless, but whose dimensions cancel out in the mathematical operation.[4][16] Examples of quotients of dimension one include calculating slopes or some 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. 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.[17] 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.

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:[18]

• 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 ≈ 1, a constant characterizing the strong nuclear force coupling strength;
• The ratio of the mass of any given elementary particle to the Planck mass, ${\textstyle {\sqrt {\hbar c/G}}}$.

Examples of dimensionless material constants include Poisson's ratio and relative atomic mass, refractive index.

Characteristic numbers are abundant in fluid mechanics, thermodynamics, and other areas.

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.

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

• 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. London, UK: Elsevier Ltd. 109: 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)

• Media related to Dimensionless numbers at Wikimedia Commons