Dimensionless quantity

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities to which dimensions are regularly assigned are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.


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 (independent 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 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.[4][5][6]

Pure numbers

All pure numbers are dimensionless quantities, for example 1, i, π, e, and φ.[7] Units of number such as the dozen, gross, googol, and Avogadro's number may also be considered dimensionless.

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.[8] 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 radians, degrees (°= π/180) and grads(= 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.

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:[9]

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
  • 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.[10]


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. ^ "The International System of Units (SI)" (PDF). Bureau International des Poids et Mesures. Retrieved 2017-11-03.
  3. ^ Mohr, Peter J.; Phillips, William D. (2015-06-01). "Dimensionless units in the SI". Metrologia. 52.
  4. ^ "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.
  5. ^ "BIPM Consultative Committee for Units (CCU), 16th Meeting" (PDF). Archived from the original (PDF) on 2006-11-30. Retrieved 2010-01-22.
  6. ^ Dybkaer, René (2004). "An ontology on property for physical, chemical, and biological systems". APMIS Suppl. (117): 1–210. PMID 15588029.
  7. ^ Khan Academy (21 April 2011). "Pure Numbers and Significant Digits" – via YouTube.
  8. ^ http://web.mit.edu/6.055/old/S2008/notes/apr02a.pdf
  9. ^ Baez, John (April 22, 2011). "How Many Fundamental Constants Are There?". Retrieved October 7, 2015.
  10. ^ Huba, J. D. (2007). "NRL Plasma Formulary: Dimensionless Numbers of Fluid Mechanics". Naval Research Laboratory. Retrieved October 7, 2015. p. 23–25

External links

API gravity

The American Petroleum Institute gravity, or API gravity, is a measure of how heavy or light a petroleum liquid is compared to water: if its API gravity is greater than 10, it is lighter and floats on water; if less than 10, it is heavier and sinks.

API gravity is thus an inverse measure of a petroleum liquid's density relative to that of water (also known as specific gravity). It is used to compare densities of petroleum liquids. For example, if one petroleum liquid is less dense than another, it has a greater API gravity. Although API gravity is mathematically a dimensionless quantity (see the formula below), it is referred to as being in 'degrees'. API gravity is graduated in degrees on a hydrometer instrument. API gravity values of most petroleum liquids fall between 10 and 70 degrees.

In 1916, the U.S. National Bureau of Standards accepted the Baumé scale, which had been developed in France in 1768, as the U.S. standard for measuring the specific gravity of liquids less dense than water. Investigation by the U.S. National Academy of Sciences found major errors in salinity and temperature controls that had caused serious variations in published values. Hydrometers in the U.S. had been manufactured and distributed widely with a modulus of 141.5 instead of the Baumé scale modulus of 140. The scale was so firmly established that, by 1921, the remedy implemented by the American Petroleum Institute was to create the API gravity scale, recognizing the scale that was actually being used.

Arbitrary unit

In science and technology, an arbitrary unit (abbreviated arb. unit, see below)

or procedure defined unit (p.d.u.)

is a relative unit of measurement to show the ratio of amount of substance, intensity, or other quantities, to a predetermined reference measurement. The reference measurement is typically defined by the local laboratories or dependent on individual measurement apparatus. It is therefore impossible to compare "1 arb. unit" by one measurer and "1000 arb. unit" by another measurer without detailed prior knowledge on how the respective "arbitrary units" were defined; thus, the unit is sometimes called an unknown unit. The unit only serves to compare multiple measurements performed in similar environment, since the ratio between the measurement and the reference is a consistent and dimensionless quantity independent of what actual units are used.

Units of such kind are commonly used in fields such as physiology to indicate substance concentration, and spectroscopy to express spectral intensity.

When the reference measurement is precisely defined and internationally agreed upon, arbitrary units can also be a unit capable of public comparison. One example of a publicly defined arbitrary unit is the WHO International Unit.

Biot number

The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations. It is named after the eighteenth century French physicist Jean-Baptiste Biot (1774–1862), and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body. This ratio determines whether or not the temperatures inside a body will vary significantly in space, while the body heats or cools over time, from a thermal gradient applied to its surface.

In general, problems involving small Biot numbers (much smaller than 1) are thermally simple, due to uniform temperature fields inside the body. Biot numbers much larger than 1 signal more difficult problems due to non-uniformity of temperature fields within the object. It should not be confused with Nusselt number, which employs the thermal conductivity of the fluid and hence is a comparative measure of conduction and convection, both in the fluid.

The Biot number has a variety of applications, including transient heat transfer and use in extended surface heat transfer calculations.

Capillary number

In fluid dynamics, the capillary number (Ca) represents the relative effect of viscous drag forces versus surface tension forces acting across an interface between a liquid and a gas, or between two immiscible liquids. For example, an air bubble in a liquid flow tends to be deformed by the friction of the liquid flow due to viscosity effects, but the surface tension forces tend to minimize the surface. The capillary number is defined as:

where µ is the dynamic viscosity of the liquid, V is a characteristic velocity and is the surface tension or interfacial tension between the two fluid phases.

The capillary number is a dimensionless quantity, hence its value does not depend on the system of units. In the petroleum industry, capillary number is denoted instead of .

For low capillary numbers (a rule of thumb says less than 10−5), flow in porous media is dominated by capillary forces whereas for high capillary number the capillary forces are negligible compared to the viscous forces. Flow through the pores in an oil field reservoir have capillary number on the order of 10−6, whereas flow of oil through an oil well drill pipe has a capillary number on the order of 1.

The capillary number plays a role in the dynamics of capillary flow, in particular it governs the dynamic contact angle of a flowing droplet at an interface.

Chandrasekhar number

The Chandrasekhar number is a dimensionless quantity used in magnetic convection to represent ratio of the Lorentz force to the viscosity. It is named after the Indian astrophysicist Subrahmanyan Chandrasekhar.

The number's main function is as a measure of the magnetic field, being proportional to the square of a characteristic magnetic field in a system.

Dukhin number

The Dukhin number (Du) is a dimensionless quantity that characterizes the contribution of the surface conductivity to various electrokinetic and electroacoustic effects, as well as to electrical conductivity and permittivity of fluid heterogeneous systems.


The humidex (short for humidity index) is an index number used by Canadian meteorologists to describe how hot the weather feels to the average person, by combining the effect of heat and humidity. The term humidex is a Canadian innovation coined in 1965. The humidex is a dimensionless quantity based on the dew point.

Range of humidex: Scale of comfort:

20 to 29: Little to no discomfort

30 to 39: Some discomfort

40 to 45: Great discomfort; avoid exertion

Above 45: Dangerous; heat stroke quite possible

Keulegan–Carpenter number

In fluid dynamics, the Keulegan–Carpenter number, also called the period number, is a dimensionless quantity describing the relative importance of the drag forces over inertia forces for bluff objects in an oscillatory fluid flow. Or similarly, for objects that oscillate in a fluid at rest. For small Keulegan–Carpenter number inertia dominates, while for large numbers the (turbulence) drag forces are important.

The Keulegan–Carpenter number KC is defined as:


The Keulegan–Carpenter number is named after Garbis H. Keulegan (1890–1989) and Lloyd H. Carpenter.

A closely related parameter, also often used for sediment transport under water waves, is the displacement parameter δ:

with A the excursion amplitude of fluid particles in oscillatory flow and L a characteristic diameter of the sediment material. For sinusoidal motion of the fluid, A is related to V and T as A = VT/(2π), and:

The Keulegan–Carpenter number can be directly related to the Navier–Stokes equations, by looking at characteristic scales for the acceleration terms:

Dividing these two acceleration scales gives the Keulegan–Carpenter number.

A somewhat similar parameter is the Strouhal number, in form equal to the reciprocal of the Keulegan–Carpenter number. The Strouhal number gives the vortex shedding frequency resulting from placing an object in a steady flow, so it describes the flow unsteadiness as a result of an instability of the flow downstream of the object. Conversely, the Keulegan–Carpenter number is related to the oscillation frequency of an unsteady flow into which the object is placed.

Mach number

In fluid dynamics, the Mach number (M or Ma) (/mɑːk/; German: [max]) is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound.


M is the Mach number,
u is the local flow velocity with respect to the boundaries (either internal, such as an object immersed in the flow, or external, like a channel), and
c is the speed of sound in the medium.

By definition, at Mach 1, the local flow velocity u is equal to the speed of sound. At Mach 0.65, u is 65% of the speed of sound (subsonic), and, at Mach 1.35, u is 35% faster than the speed of sound (supersonic).

The local speed of sound, and thereby the Mach number, depends on the condition of the surrounding medium, in particular the temperature. The Mach number is primarily used to determine the approximation with which a flow can be treated as an incompressible flow. The medium can be a gas or a liquid. The boundary can be traveling in the medium, or it can be stationary while the medium flows along it, or they can both be moving, with different velocities: what matters is their relative velocity with respect to each other. The boundary can be the boundary of an object immersed in the medium, or of a channel such as a nozzle, diffusers or wind tunnels channeling the medium. As the Mach number is defined as the ratio of two speeds, it is a dimensionless number. If M < 0.2–0.3 and the flow is quasi-steady and isothermal, compressibility effects will be small and simplified incompressible flow equations can be used.

The Mach number is named after Austrian physicist and philosopher Ernst Mach, and is a designation proposed by aeronautical engineer Jakob Ackeret. As the Mach number is a dimensionless quantity rather than a unit of measure, with Mach, the number comes after the unit; the second Mach number is Mach 2 instead of 2 Mach (or Machs). This is somewhat reminiscent of the early modern ocean sounding unit mark (a synonym for fathom), which was also unit-first, and may have influenced the use of the term Mach. In the decade preceding faster-than-sound human flight, aeronautical engineers referred to the speed of sound as Mach's number, never Mach 1.


A Machmeter is an aircraft pitot-static system flight instrument that

shows the ratio of the true airspeed to the speed of sound,

a dimensionless quantity called Mach number. This is shown on a Machmeter as a decimal fraction.

An aircraft flying at the speed of sound is flying

at a Mach number of one, expressed as Mach 1.

Magnetic Prandtl number

The Magnetic Prandtl number (Prm) is a dimensionless quantity occurring in magnetohydrodynamics which approximates the ratio of momentum diffusivity (viscosity) and magnetic diffusivity. It is defined as:


At the base of the Sun's convection zone the Magnetic Prandtl number is approximately 10−2, and in the interiors of planets and in liquid-metal laboratory dynamos is approximately 10−5.

Particle number

The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter N, is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is conjugate to the chemical potential. Unlike most physical quantities, particle number is a dimensionless quantity. It is an extensive parameter, as it is directly proportional to the size of the system under consideration, and thus meaningful only for closed systems.

A constituent particle is one that cannot be broken into smaller pieces at the scale of energy k·T involved in the process (where k is the Boltzmann constant and T is the temperature). For example, for a thermodynamic system consisting of a piston containing water vapour, the particle number is the number of water molecules in the system. The meaning of constituent particle, and thereby of particle number, is thus temperature-dependent.

Rate (mathematics)

In mathematics, a rate is the ratio between two related quantities. If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the numerator of the ratio expresses the corresponding rate of change in the other (dependent) variable.

The most common type of rate is "per unit of time", such as speed, heart rate and flux. Ratios that have a non-time denominator include exchange rates, literacy rates and electric field (in volts/meter).

In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate (for example a heart rate is expressed "beats per minute"). A rate defined using two numbers of the same units (such as tax rates) or counts (such as literacy rate) will result in a dimensionless quantity, which can be expressed as a percentage (for example, the global literacy rate in 1998 was 80%) or fraction or as a multiple.

Often rate is a synonym of rhythm or frequency, a count per second (i.e., Hertz); e.g., radio frequencies or heart rate or sample rate.

Saturation vapor density

Saturation vapor density (SVD) is a concept closely tied with saturation vapor pressure (SVP). It can be used to calculate exact quantity of water vapor in the air from a relative humidity (RH = % local air humidity measured / local total air humidity possible ) Given an RH percentage, the density of water in the air is given by RH × SVD = Actual Vapor Density. Alternatively, RH can be found by RH = Actual Vapor Density ∕ SVD. As relative humidity is a dimensionless quantity (often expressed in terms of a percentage), vapor density can be stated in units of grams or kilograms per cubic meter.

For low temperatures (below approximately 400 K), SVD can be approximated from the SVP by the ideal gas law: P V = n R T  where P is the SVP, V is the volume, n is the number of moles, R is the gas constant and T is the temperature in kelvins. The number of moles is related to density by n = m ∕ M, where m is the mass of water present and M is the molar mass of water (18.01528 grams/mole). Thus, setting V to 1 cubic meter, we get P M/R T = m/V = density.

The values shown at hyperphysics-sources indicate that the saturated vapor density is 4.85 g/m3 at 273 K, at which the saturated vapor pressure is 4.58 mm of Hg or 610.616447 Pa (760 mm of Hg ≈ 1 atm = 1.01325 * 105 Pa).

Therefore, for particular mole number and volume the saturated vapor pressure will not change if the temperature remains constant.


In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of artificial fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity measured at different times are different but the corresponding dimensionless quantity at given value of remain invariant. It happens if the quantity exhibits dynamic scaling. The idea is just an extension of the idea of similarity of two triangles. Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide.


Sinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve. This dimensionless quantity can also be rephrased as the "actual path length" divided by the "shortest path length" of a curve.

The value ranges from 1 (case of straight line) to infinity (case of a closed loop, where the shortest path length is zero) or for an infinitely-long actual path.

Specific ventilation

In respiratory physiology, specific ventilation is defined as the ratio of the volume of gas entering a region of the lung (ΔV) following an inspiration, divided by the end-expiratory volume (V0) of that same lung region:

SV = ​ΔV⁄V0

It is a dimensionless quantity. For the whole human lung, given an indicative tidal volume of 0.6 L and a functional residual capacity of 2.5 L, average SV is of the order of 0.24.

The distribution of specific ventilation within the lung can be inferred using Multiple Breath Washout (MBW) experiments or imaging techniques such as Positron Emission Tomography (PET) using 13N, Magnetic Resonance Imaging (MRI) using either hyperpolarized gas (3He, 129Xe) or proton MRI (oxygen enhanced imaging).

Standard score

In statistics, the standard score is the signed fractional number of standard deviations by which the value of an observation or data point is above the mean value of what is being observed or measured. Observed values above the mean have positive standard scores, while values below the mean have negative standard scores.

It is calculated by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. It is a dimensionless quantity. This conversion process is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see normalization for more).

Standard scores are also called z-values, z-scores, normal scores, and standardized variables. They are most frequently used to compare an observation to a theoretical deviate, such as a standard normal deviate.

Computing a z-score requires knowing the mean and standard deviation of the complete population to which a data point belongs; if one only has a sample of observations from the population, then the analogous computation with sample mean and sample standard deviation yields the t-statistic.

Volume fraction

In chemistry, the volume fraction φi is defined as the volume of a constituent Vi divided by the volume of all constituents of the mixture V prior to mixing:

Being dimensionless, its unit is 1; it is expressed as a number, e.g., 0.18. It is the same concept as volume percent (vol%) except that the latter is expressed with a denominator of 100, e.g., 18%.

The volume fraction coincides with the volume concentration in ideal solutions where the volumes of the constituents are additive (the volume of the solution is equal to the sum of the volumes of its ingredients).

The sum of all volume fractions of a mixture is equal to 1:

The volume fraction (percentage by volume, vol%) is one way of expressing the composition of a mixture with a dimensionless quantity; mass fraction (percentage by weight, wt%) and mole fraction (percentage by moles, mol%) are others.

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