Dimensionless quantities, also known as quantities of dimension one^[1] are implicitly defined in a manner that prevents their aggregation into units of measurement.^[2][3] Typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined units. For instance, alcohol by volume (ABV) represents a volumetric ratio. Its derivation remains independent of the specific units of volume used; any common unit may be applied. Notably, ABV is never expressed as milliliters per milliliter, underscoring its dimensionless nature.

The number one is recognized as a dimensionless base quantity.^[4] Radians serve as dimensionless units for angular measurements, derived from the universal ratio of 2π times the radius of a circle being equal to its circumference.^[5]

Dimensionless quantities play a crucial role serving as parameters in differential equations in various technical disciplines. In calculus, concepts like the unitless ratios in limits or derivatives 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,^[6] the fine-structure constant in quantum mechanics,^[7] and the Lorentz factor in relativity.^[8] In chemistry, state properties and ratios such as mole fractions concentration ratios are dimensionless.^[9]

History

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.^[10]

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^[11] argued for formalizing the radian as a physical unit. The idea was rebutted^[12] 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.^[13][14][15]

Buckingham π theorem

The Buckingham π theorem ^[16] 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.

Integers

Integer numbers may represent dimensionless quantities. Complex numbers can represent discrete quantities, which can also be dimensionless. More specifically, counting numbers can be used to express countable quantities.^[17][18] The concept is formalized as quantity number of entities (symbol N) in ISO 80000-1.^[19] 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.

Ratios, proportions, and angles

Dimensionless quantities can be obtained as ratios of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation.^[19][20] Examples of quotients of dimension one include calculating slopes or some unit conversion factors. 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.^[21] 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.

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, 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:^[22]

• 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;
• Planck mass ratio of the mass of any given elementary particle, ${\textstyle {\sqrt {\hbar c/G))}$.

List

Physics and engineering

• Lorentz Factor^[23] – 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
• 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.
• 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.^[24]
• Zukoski number, usually noted ${\displaystyle Q^{*))$, 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 ${\displaystyle Q^{*}\approx 1}$. Flat spread fires such as forest fires have ${\displaystyle Q^{*}<1}$. Fires originating from pressured vessels or pipes, with additional momentum caused by pressure, have ${\displaystyle Q^{*}\gg 1}$.^[25]
• Eckert number
• Biot number
• Grashof number

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.