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Dimensionless quantity

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Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as 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 per milliliter (mL/mL).

The number one is recognized as a dimensionless base quantity. serve as dimensionless units for Angle, derived from the universal ratio of 2π times the radius of a circle being equal to its circumference.

Ernest Zebrowski (1999). 9780813528984, Rutgers University Press. . ISBN 9780813528984

Dimensionless quantities play a crucial role serving as in differential equations in various technical disciplines. In calculus, concepts like the unitless ratios in limits or 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,

(2013). 9780077173593, McGraw Hill. . ISBN 9780077173593

the fine-structure constant in quantum mechanics, and the Lorentz factor in relativity. In chemistry, state properties and ratios such as concentration ratios are dimensionless.

History

Quantities having dimension , 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 theorem (independently of French mathematician Joseph Bertrand's previous work) to formalize the nature of these quantities.

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 level quantity 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 (1 page) argued for formalizing the radian as a physical unit. The idea was rebutted 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 Cross product) versus energy (a Dot 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.

Buckingham theorem

The Buckingham 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 occurring in those variables to give a set of p = n − k independent, dimensionless quantity. For the purposes of the experimenter, different systems that share the same description by dimensionless quantity are equivalent.

Integers

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.

(2025). 9781107001275, Cambridge University Press. . ISBN 9781107001275

(2025). 9780128019092, Elsevier Science. . ISBN 9780128019092

The concept is formalized as quantity number of entities (symbol N) in ISO 80000-1. Examples include number of particles and population size. In mathematics, the "number of elements" in a set is termed cardinality. is a related linguistics concept. Counting numbers, such as number of , 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 of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation. Examples of quotients of dimension one include calculating or some unit conversion factors. Another set of examples is mass fractions or , 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 (kilogram/kg or mol/mol). For example, alcohol by volume, which characterizes the concentration of ethanol in an alcoholic beverage, could be written as .

Other common proportions are percentages % (= 0.01), per mil (= 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 average and is used to measure the dispersion in the statistical data.

It has been argued that quantities defined as ratios having equal dimensions in numerator and denominator are actually only unitless quantities and still have physical dimension defined as . For example, moisture content may be defined as a ratio of volumes (volumetric moisture, m³⋅m⁻³, dimension L⋅L) or as a ratio of masses (gravimetric moisture, units kg⋅kg⁻¹, dimension M⋅M); both would be unitless quantities, but of different dimension.

Dimensionless physical constants

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, electric 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:

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.
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 $r$ , a ratio between the contributions of tensor and scalar modes to the primordial power spectrum observed in the CMB.
The Immirzi-Barbero parameter $\gamma$ , which characterizes the area gap in loop quantum gravity.
Carlo Rovelli (2025). 9780521715966, Cambridge University Press. . ISBN 9780521715966
emissivity, which is the ratio of actual emitted radiation from a surface to that of an Black body at the same temperature

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Physics and engineering

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

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.

Dimensionless quantity

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