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In dimensional analysis, a dimensionless quantity is a quantity without a physical unit and is thus a pure number. Such a number is typically defined as a product or ratio of quantities that have units individually, but cancel out in the combination.

Dimensionless quantities are widely used in the fields of mathematics, physics, engineering, and economics, but also in everyday life.



  • A dimensionless quantity has no physical unit associated with it. However, it is sometimes helpful to use the same units in both the numerator and denominator, such as kg/kg, to show the quantity being measured (for example to distinguish a mass ratio from a volume ratio).
  • A dimensionless proportion has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the SI system of units or the imperial system of units. This doesn't hold for all dimensionless quantities; it is guaranteed to hold only for proportions.

Buckingham π theorem

According to the Buckingham π theorem of dimensional analysis, the functional dependence between a certain number (e.g., n) of variables can be reduced by the number (e.g., 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 which share the same description by dimensionless quantity are equivalent.



The power consumption of a stirrer with a particular geometry is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n = 5 variables representing our example.

Those n = 5 variables are built up from k = 3 dimensions which are:

  • Length: L (m)
  • Time: T (s)
  • 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 which are in case of the stirrer

  • Reynolds number (This is a very important dimensionless number; it describes the fluid flow regime)
  • Power number (describes the stirrer and also involves the density of the fluid).

Standards efforts

The CIPM Consultative Committee for Units contemplated defining the unit of 1 as the 'uno', but the idea was dropped.[1][2][3][4]


Consider this example: Sarah says, "Out of every 10 apples I gather, 1 is rotten.". The rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity. Another more typical example in physics and engineering is the measure of plane angles. Angles are typically measured as the ratio of the length of an arc lying on a circle (with its center being the vertex of the angle) swept out by the angle, compared to some other length. The ratio, length divided by length, is dimensionless. When using the unit radians, the length that is compared is the length of the radius of the circle. When using the unit degree, the length that is compared is 1/360 of the circumference of the circle.

In case of dimensionless quantities the unit is a quotient of like dimensioned quantities that can be reduced to a number (kg/kg = 1, μg/g = 10−6). Dimensionless quantities can also carry dimensionless units like % (= 0.01), ppt (= 10−3), ppm (= 10−6), ppb (= 10−9).

List of dimensionless quantities

There are infinitely many dimensionless quantities and they are often called numbers. Some of those that are used most often have been given names, as in the following list of examples (alphabetical order):

Name Standard Symbol Definition Field of application
Abbe number V optics (dispersion in optical materials)
Activity coefficient γ chemistry (Proportion of "active" molecules or atoms)
Albedo α climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number Ar motion of fluids due to density differences
Arrhenius Number α Ratio of activation energy to thermal energy[5]
Atomic weight M chemistry
Bagnold number Ba flow of bulk solids such as grain and sand. [6]
Bingham Number Bm Bm = \frac{ \tau_yL }{ \mu V } Ratio of yield stress to viscous stress[5]
Biot number Bi surface vs. volume conductivity of solids
Bodenstein number residence-time distribution
Bond number Bo capillary action driven by buoyancy [7]
Brinkman number Br heat transfer by conduction from the wall to a viscous fluid
Brownell Katz number combination of capillary number and Bond number
Capillary number Ca fluid flow influenced by surface tension
Coefficient of static friction μs friction of solid bodies at rest
Coefficient of kinetic friction μk friction of solid bodies in translational motion
Colburn j factor dimensionless heat transfer coefficient
Courant-Friedrich-Levy number ν numerical solutions of hyperbolic PDEs [8]
Damkohler number Da reaction time scales vs. transport phenomena
Darcy friction factor Cf or f fluid flow
Dean number D vortices in curved ducts
Deborah number De rheology of viscoelastic fluids
Decibel dB ratio of two intensities of sound
Drag coefficient Cd flow resistance
Euler's number e mathematics
Eckert number Ec convective heat transfer
Ekman number Ek geophysics (frictional (viscous) forces)
Elasticity (economics) E widely used to measure how demand or supply responds to price changes
Eötvös number Eo determination of bubble/drop shape
Ericksen number Er liquid crystal flow behavior
Euler number Eu hydrodynamics (pressure forces vs. inertia forces)
Fanning friction factor f fluid flow in pipes [9]
Feigenbaum constants α,δ chaos theory (period doubling) [10]
Fine structure constant α quantum electrodynamics (QED)
f-number f optics, photography
Foppl–von Karman number thin-shell buckling
Fourier number Fo heat transfer
Fresnel number F slit diffraction [11]
Froude number Fr wave and surface behaviour
Gain electronics (signal output to signal input)
Galilei number Ga gravity-driven viscous flow
Golden ratio \varphi mathematics and aesthetics
Graetz number Gz heat flow
Grashof number Gr free convection
Hatta number Ha adsorption enhancement due to chemical reaction
Hagen number Hg forced convection
Hydraulic gradient i groundwater flow
Karlovitz number turbulent combustion turbulent combustion
Keulegan–Carpenter number KC ratio of drag force to inertia for a bluff object in oscillatory fluid flow
Knudsen number Kn continuum approximation in fluids
Kt/V medicine
Kutateladze number K counter-current two-phase flow
Laplace number La free convection within immiscible fluids
Lewis number Le ratio of mass diffusivity and thermal diffusivity
Lift coefficient CL lift available from an airfoil at a given angle of attack
Lockhart-Martinelli parameter χ flow of wet gases [12]
Lundquist number S ratio of a resistive time to an Alfvén wave crossing time in a plasma
Mach number M gas dynamics
Magnetic Reynolds number Rm magnetohydrodynamics
Manning roughness coefficient n open channel flow (flow driven by gravity) [13]
Marangoni number Mg Marangoni flow due to thermal surface tension deviations
Morton number Mo determination of bubble/drop shape
Nusselt number Nu heat transfer with forced convection
Ohnesorge number Oh atomization of liquids, Marangoni flow
Péclet number Pe advectiondiffusion problems
Peel number adhesion of microstructures with substrate [14]
Pi π mathematics (ratio of a circle's circumference to its diameter)
Poisson's ratio ν elasticity (load in transverse and longitudinal direction)
Power factor electronics (real power to apparent power)
Power number Np power consumption by agitators
Prandtl number Pr convection heat transfer (thickness of thermal and momentum boundary layers)
Pressure coefficient CP pressure experienced at a point on an airfoil
Radian rad measurement of angles
Rayleigh number Ra buoyancy and viscous forces in free convection
Refractive index n electromagnetism, optics
Reynolds number Re Re = \frac{vL\rho}{\mu} Ratio of fluid inertial and viscous forces[5]
Relative density RD hydrometers, material comparisons
Richardson number Ri effect of buoyancy on flow stability [15]
Rockwell scale mechanical hardness
Rolling resistance coefficient Crr C_{rr} = \frac{N_f}{F} Vehicle dynamics
Rossby number Ro inertial forces in geophysics
Rouse number Z or P Sediment transport
Schmidt number Sc fluid dynamics (mass transfer and diffusion) [16]
Shape factor H ratio of displacement thickness to momentum thickness in boundary layer flow
Sherwood number Sh mass transfer with forced convection
Sommerfeld number boundary lubrication [17]
Stanton number St heat transfer in forced convection
Stefan number Ste heat transfer during phase change
Stokes number Stk particle dynamics
Strain ε materials science, elasticity
Strouhal number Sr continuous and pulsating flow [18]
Taylor number Ta rotating fluid flows
Ursell number U nonlinearity of surface gravity waves on a shallow fluid layer
van 't Hoff factor i quantitative analysis (Kf and Kb)
Wallis parameter J* nondimensional superficial velocity in multiphase flows
Weaver flame speed number laminar burning velocity relative to hydrogen gas [19]
Weber number We multiphase flow with strongly curved surfaces
Weissenberg number Wi viscoelastic flows [20]
Womersley number α continuous and pulsating flows [21]

Dimensionless physical constants

Certain fundamental physical constants, such as the speed of light in a vacuum, the universal gravitational constant, and the constants of Planck and Boltzmann, are normalized to 1 if the units for time, length, mass, charge, and temperature are chosen appropriately. The resulting system of units is known as natural or Planck units. However, a handful of dimensionless physical constants cannot be eliminated in any system of units; their values must be determined experimentally. The resulting constants include:

See also


External links


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