Dimensionless quantity

Dimensional analysis Quantity Buckingham π theorem

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned, also known as a bare, pure, or scalar quantity or a quantity of dimension one,[1] with a corresponding unit of measurement in the SI of the unit 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. An example of a quantity that has a dimension is time, measured in seconds.


Quantities having dimension 1, 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.[4]

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.

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.[5][6][7]

Pure numbers

All pure numbers are dimensionless quantities, for example 1, i, π, e, and φ.[8] Units of number such as the dozen, gross, googol, and Avogadro's number may also be considered dimensionless.[9]

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.[10] 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 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.[11] 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

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.


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 = nk = 5 − 3 = 2 independent dimensionless numbers. These quantities are , commonly named the Reynolds number which describes the fluid flow regime, and , the power number, which is the dimensionless description of the stirrer.

Dimensionless physical constants

Certain universal dimensioned physical constants, such as the speed of light in a vacuum, the universal gravitational constant, Planck's constant, Coulomb'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, 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:[12]

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


Other fields

See also


  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 (SI)". BIPM. Retrieved 2019-11-22.
  3. ^ Mohr, Peter J.; Phillips, William D. (2015-06-01). "Dimensionless units in the SI". Metrologia. 52.
  4. ^ 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. hdl:10338.dmlcz/101743.CS1 maint: ref=harv (link)
  5. ^ "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.
  6. ^ "BIPM Consultative Committee for Units (CCU), 16th Meeting" (PDF). Archived from the original (PDF) on 2006-11-30. Retrieved 2010-01-22.
  7. ^ Dybkaer, René (2004). "An ontology on property for physical, chemical, and biological systems". APMIS Suppl. (117): 1–210. PMID 15588029.
  8. ^ Khan Academy (21 April 2011). "Pure Numbers and Significant Digits" – via YouTube.
  9. ^ Křen, Petr (2019). "Why dimensionless units should not be used in physics". arXiv:1911.10030 [physics.gen-ph].
  10. ^ http://web.mit.edu/6.055/old/S2008/notes/apr02a.pdf
  11. ^ 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.
  12. ^ Baez, John (April 22, 2011). "How Many Fundamental Constants Are There?". Retrieved October 7, 2015.
  13. ^ Huba, J. D. (2007). "NRL Plasma Formulary: Dimensionless Numbers of Fluid Mechanics". Naval Research Laboratory. Retrieved October 7, 2015. p. 23–25