In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1.[1] Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and in everyday life (such as in counting). Numerous well-known quantities, such as π, e, and φ, are dimensionless. By contrast, non-dimensionless quantities are measured in units of length, area, time, etc.

Dimensionless quantities are often defined as products or ratios of quantities that are not dimensionless, but whose dimensions cancel out when their powers are multiplied. This is the case, for instance, with the engineering strain, a measure of deformation. It is defined as change in length over initial length but, since these quantities both have dimensions L (length), the result is a dimensionless quantity.


  • Even though a dimensionless quantity has no physical dimension associated with it, it can still have dimensionless units. To show the quantity being measured (for example mass fraction or mole fraction), it is sometimes helpful to use the same units in both the numerator and denominator (kg/kg or mol/mol). The quantity may also be given as a ratio of two different units that have the same dimension (for instance, light years over meters). This may be the case when calculating slopes in graphs, or when making unit conversions. Such notation does not indicate the presence of physical dimensions, and is purely a notational convention. Other common dimensionless units are  % (= 0.01),  ‰ (= 0.001), ppm (= 10−6), ppb (= 10−9), ppt (= 10−12) and angle units (degrees, radians, grad). Units of number such as the dozen and the gross are also dimensionless.
  • The ratio of two quantities with the same dimensions is dimensionless, and has the same value regardless of the units used to calculate them. For instance, if body A exerts a force of magnitude F on body B, and B exerts a force of magnitude f on A, then the ratio F/f is always equal to 1, regardless of the actual units used to measure F and f. This is a fundamental property of dimensionless proportions and follows from the assumption that the laws of physics are independent of the system of units used in their expression. In this case, if the ratio F/f was not always equal to 1, but changed if we switched from SI to CGS, that would mean that Newton's Third Law's truth or falsity would depend on the system of units used, which would contradict this fundamental hypothesis. This assumption that the laws of physics are not contingent upon a specific unit system is the basis for the Buckingham π theorem. 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.

Buckingham π theorem

Another consequence of the Buckingham π theorem of dimensional analysis 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.


The power consumption of a stirrer with a given shape 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:

  • 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 (a dimensionless number describing the fluid flow regime)
  • Power number (describing the stirrer and also involves the density of the fluid)

Standards efforts

The International Committee for Weights and Measures contemplated defining the unit of 1 as the 'uno', but the idea was dropped.[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.
  • Plane angles – An angle is measured as the ratio of the length of a circle's arc subtended by an angle whose vertex is the centre of the circle to some other length. The ratio—i.e., length divided by length—is dimensionless. When using radians as the unit, the length that is compared is the length of the radius of the circle. When using degree as the units, the arc's length is compared to 1/360 of the circumference of the circle.
  • In the case of the dimensionless quantity π, being the ratio of a circle's circumference to its diameter, the number would be constant regardless of what unit is used to measure a circle's circumference and diameter (e.g., centimetres, miles, light-years, etc.), as long as the same unit is used for both.

Dimensionless physical constants

Certain fundamental physical constants, such as the speed of light in a vacuum, the universal gravitational constant, Planck'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. 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 and must be determined experimentally:

List of dimensionless quantities

All numbers are dimensionless quantities. Certain dimensionless quantities of some importance are given below:

Name Standard symbol Definition Field of application
Abbe number V V = \frac{ n_d - 1 }{ n_F - n_C } optics (dispersion in optical materials)
Activity coefficient \gamma \gamma= \frac Template:A chemistry (Proportion of "active" molecules or atoms)
Albedo \alpha {\alpha}= (1-D) \bar \alpha(\theta_i) + D \bar{ \bar \alpha} climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number Ar \mathrm{Ar} = \frac{g L^3 \rho_\ell (\rho - \rho_\ell)}{\mu^2} fluid mechanics (motion of fluids due to density differences)
Arrhenius number \alpha \alpha = \frac{E_a}{RT} chemistry (ratio of activation energy to thermal energy)[5]
Atomic weight M chemistry (mass of atom over one atomic mass unit, u, where carbon-12 is exactly 12 u)
Atwood number A \mathrm{A} = \frac{\rho_1 - \rho_2} {\rho_1 + \rho_2} fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold number Ba \mathrm{Ba} = \frac{\rho d^2 \lambda^{1/2} \gamma}{\mu} fluid mechanics, geology (ratio of grain collision stresses to viscous fluid stresses in flow of a granular material such as grain and sand)[6]
Bejan number
(fluid mechanics)
Be \mathrm{Be} = \frac{\Delta P L^2} {\mu \alpha} fluid mechanics (dimensionless pressure drop along a channel)[7]
Bejan number
Be \mathrm{Be} = \frac{\dot S'_{\mathrm{gen},\, \Delta T}}{\dot S'_{\mathrm{gen},\, \Delta T}+ \dot S'_{\mathrm{gen},\, \Delta p}} thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)[8]
Bingham number Bm \mathrm{Bm} = \frac{ \tau_y L }{ \mu V } fluid mechanics, rheology (ratio of yield stress to viscous stress)[5]
Biot number Bi \mathrm{Bi} = \frac{h L_C}{k_b} heat transfer (surface vs. volume conductivity of solids)
Blake number Bl or B \mathrm{B} = \frac{u \rho}{\mu (1 - \epsilon) D} geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bodenstein number Bo or Bd \mathrm{Bo} = vL/\mathcal{D} = \mathrm{Re}\, \mathrm{Sc} chemistry (residence-time distribution; similar to the axial mass transfer Peclet number)[9]
Bond number Bo \mathrm{Bo} = \frac{\rho a L^2}{\gamma} geology, fluid mechanics, porous media (buoyant versus capilary forces, similar to the Eötvös number) [10]
Brinkman number Br \mathrm{Br} = \frac {\mu U^2}{\kappa (T_w - T_0)} heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Brownell–Katz number NBK \mathrm{N}_\mathrm{BK} = \frac{u \mu}{k_\mathrm{rw}\sigma} fluid mechanics (combination of capillary number and Bond number) [11]
Capillary number Ca \mathrm{Ca} = \frac{\mu V}{\gamma} porous media (viscous forces versus surface tension)
Chandrasekhar number Q \mathrm{Q} = \frac mechanics (the level of damping in a system)
Darcy friction factor Cf or fD fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
Darcy number Da \mathrm{Da} = \frac{K}{d^2} porous media (ratio of permeability to cross-sectional area)
Dean number D \mathrm{D} = \frac{\rho V d}{\mu} \left( \frac{d}{2 R} \right)^{1/2} turbulent flow (vortices in curved ducts)
Deborah number De \mathrm{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}} rheology (viscoelastic fluids)
Decibel dB acoustics, electronics, control theory (ratio of two intensities or powers of a wave)
Drag coefficient cd c_\mathrm{d} = \dfrac{2 F_\mathrm{d}}{\rho v^2 A}\, , aeronautics, fluid dynamics (resistance to fluid motion)
Dukhin number Du \mathrm{Du} = \frac{\kappa^{\sigma}}{L \lambda} optics (slit diffraction)[17]
Froude number Fr \mathrm{Fr} = \frac{v}{\sqrt{g\ell}} fluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
Gain electronics (signal output to signal input)
Gain ratio bicycling (system of representing gearing; length traveled over length pedaled)[18]
Galilei number Ga \mathrm{Ga} = \frac{g\, L^3}{\nu^2} fluid mechanics (gravitational over viscous forces)
Golden ratio \varphi \varphi = \frac{1+\sqrt{5}}{2} \approx 1.61803 mathematics, aesthetics (long side length of self-similar rectangle)
Görtler number G \mathrm{G} = \frac{U_e \theta}{\nu} \left( \frac{\theta}{R} \right)^{1/2} fluid dynamics (boundary layer flow along a concave wall)
Graetz number Gz \mathrm{Gz} = {D_H \over L} \mathrm{Re}\, \mathrm{Pr} heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof number Gr \mathrm{Gr}_L = \frac{g \beta (T_s - T_\infty ) L^3}{\nu ^2} heat transfer, natural convection (ratio of the buoyancy to viscous force)
Gravitational coupling constant \alpha_G \alpha_G=\frac{Gm_e^2}{\hbar c} gravitation (attraction between two massy elementary particles; analogous to the Fine structure constant)
Hatta number Ha \mathrm{Ha} = \frac{N_{\mathrm{A}0}}{N_{\mathrm{A}0}^{\mathrm{phys}}} chemical engineering (adsorption enhancement due to chemical reaction)
Hagen number Hg \mathrm{Hg} = -\frac{1}{\rho}\frac{\mathrm{d} p}{\mathrm{d} x}\frac{L^3}{\nu^2} heat transfer (ratio of the buoyancy to viscous force in forced convection)
Hydraulic gradient i i = \frac{\mathrm{d}h}{\mathrm{d}l} = \frac{h_2 - h_1}{\mathrm{length}} fluid mechanics, groundwater flow (pressure head over distance)
Iribarren number Ir \mathrm{Ir} = \frac{\tan \alpha}{\sqrt{H/L_0}} wave mechanics (breaking surface gravity waves on a slope)
Jakob Number Ja \mathrm{Ja} = \frac{c_p (T_\mathrm{s} - T_\mathrm{sat}) }{\Delta H_{\mathrm{f}} } chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change)[19]
Karlovitz number Ka \mathrm{Ka} = k t_c turbulent combustion (characteristic flow time times flame stretch rate)
Keulegan–Carpenter number KC \mathrm{K_C} = \frac{V\,T}{L} fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number Kn \mathrm{Kn} = \frac {\lambda}{L} gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kt/V Kt/V medicine (hemodialysis and peritoneal dialysis treatment; dimensionless time)
Kutateladze number Ku \mathrm{Ku} = \frac{U_h \rho_g^{1/2}}{\left({\sigma g (\rho_l - \rho_g)}\right)^{1/4}} fluid mechanics (counter-current two-phase flow)[20]
Laplace number La \mathrm{La} = \frac{\sigma \rho L}{\mu^2} fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis number Le \mathrm{Le} = \frac{\alpha}{D} = \frac{\mathrm{Sc}}{\mathrm{Pr}} heat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficient CL C_\mathrm{L} = \frac{L}{q\,S} aerodynamics (lift available from an airfoil at a given angle of attack)
Lockhart–Martinelli parameter \chi \chi = \frac{m_\ell}{m_g} \sqrt{\frac{\rho_g}{\rho_\ell}} two-phase flow (flow of wet gases; liquid fraction)[21]
Love numbers h, k, l geophysics (solidity of earth and other planets)
Lundquist number S S = \frac{\mu_0LV_A}{\eta} plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma)
Mach number M or Ma \mathrm{M} = \frac} gas dynamics (compressible flow; dimensionless velocity)
Magnetic Reynolds number Rm \mathrm{R}_\mathrm{m} = \frac{U L}{\eta} magnetohydrodynamics (ratio of magnetic advection to magnetic diffusion)
Manning roughness coefficient n open channel flow (flow driven by gravity)[22]
Marangoni number Mg \mathrm{Mg} = - {\frac{\mathrm{d}\sigma}{\mathrm{d}T}}\frac{L \Delta T}{\eta \alpha} fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
Morton number Mo \mathrm{Mo} = \frac{g \mu_c^4 \, \Delta \rho}{\rho_c^2 \sigma^3} fluid dynamics (determination of bubble/drop shape)
Nusselt number Nu \mathrm{Nu} =\frac{hd}{k} heat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge number Oh \mathrm{Oh} = \frac{ \mu}{ \sqrt{\rho \sigma L }} = \frac{\sqrt{\mathrm{We}}}{\mathrm{Re}} fluid dynamics (atomization of liquids, Marangoni flow)
Péclet number Pe \mathrm{Pe} = \frac{du\rho c_p}{k} = \mathrm{Re}\, \mathrm{Pr} heat transfer (advectiondiffusion problems; total momentum transfer to molecular heat transfer)
Peel number NP N_\mathrm{P} = \frac{\text{Restoring force}}{\text{Adhesive force}} coating (adhesion of microstructures with substrate)[23]
Perveance K {K} = \fracTemplate:I 0\,\fracTemplate:2{\mathrm{d}\varepsilon_\mathrm{axial}} elasticity (load in transverse and longitudinal direction)
Porosity \phi \phi = \frac{V_\mathrm{V}}{V_\mathrm{T}} geology, porous media (void fraction of the medium)
Power factor P/S electronics (real power to apparent power)
Power number Np N_{p} = {P\over \rho n^{3} d^{5}} electronics (power consumption by agitators; resistance force versus inertia force)
Prandtl number Pr \mathrm{Pr} = \frac{\nu}{\alpha} = \frac{c_p \mu}{k} heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Prater number β \beta = \frac{-\Delta H_r D_{TA}^e C_{AS}}{\lambda^e T_s} reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet)[24]
Pressure coefficient CP C_p = {p - p_\infty \over \frac{1}{2} \rho_\infty V_\infty^2} aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
Q factor Q physics, engineering (damping of oscillator or resonator; energy stored versus energy lost)
Radian measure rad \text{arc length}/\text{radius} mathematics (measurement of planar angles, 1 radian = 180/π degrees)
Rayleigh number Ra \mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infin) x^3 heat transfer (buoyancy versus viscous forces in free convection)
Refractive index n n=\frac{c}{v} electromagnetism, optics (speed of light in a vacuum over speed of light in a material)
Relative density RD RD = \frac{\rho_\mathrm{substance}}{\rho_\mathrm{reference}} hydrometers, material comparisons (ratio of density of a material to a reference material—usually water)
Relative permeability \mu_r \mu_r = \frac{\mu}{\mu_0} magnetostatics (ratio of the permeability of a specific medium to free space)
Relative permittivity \varepsilon_r \varepsilon_{r} = \frac{C_{x}} {C_{0}} electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum)
Reynolds number Re \mathrm{Re} = \frac{vL\rho}{\mu} fluid mechanics (ratio of fluid inertial and viscous forces)[5]
Richardson number Ri \mathrm{Ri} = \frac{gh}{u^2} = \frac{1}{\mathrm{Fr}^2} fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)[25]
Rockwell scale mechanical hardness (indentation hardness of a material)
Rolling resistance coefficient Crr C_{rr} = \frac{F}{N_f} vehicle dynamics (ratio of force needed for motion of a wheel over the normal force)
Roshko number Ro \mathrm{Ro} = {f L^{2}\over \nu} =\mathrm{St}\,\mathrm{Re} fluid dynamics (oscillating flow, vortex shedding)
Rossby number Ro \mathrm{Ro}=\frac{U}{Lf} geophysics (ratio of inertial to Coriolis force)
Rouse number P or Z \mathrm{P} = \frac{w_s}{\kappa u_*} sediment transport (ratio of the sediment fall velocity and the upwards velocity of grain)
Schmidt number Sc \mathrm{Sc} = \frac{\nu}{D} mass transfer (viscous over molecular diffusion rate)[26]
Shape factor H H = \frac {\delta^*}{\theta} boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number Sh \mathrm{Sh} = \frac{K L}{D} mass transfer (forced convection; ratio of convective to diffusive mass transport)
Shields parameter \tau_* or \theta \tau_{\ast} = \frac{\tau}{(\rho_s - \rho) g D} sediment transport (threshold of sediment movement due to fluid motion; dimensionless shear stress)
Sommerfeld number S \mathrm{S} = \left( \frac{r}{c} \right)^2 \frac {\mu N}{P} hydrodynamic lubrication (boundary lubrication)[27]
Specific gravity SG (same as Relative density)
Stanton number St \mathrm{St} = \frac{h}{c_p \rho V} = \frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}} heat transfer and fluid dynamics (forced convection)
Stefan number Ste \mathrm{Ste} = \frac{c_p \Delta T}{L} phase change, thermodynamics (ratio of sensible heat to latent heat)
Stokes number Stk or Sk \mathrm{Stk} = \frac{\tau U_o}{d_c} particles suspensions (ratio of characteristic time of particle to time of flow)
Strain \epsilon \epsilon = \cfrac{\partial{F}}{\partial{X}} - 1 materials science, elasticity (displacement between particles in the body relative to a reference length)
Strouhal number St or Sr \mathrm{St} = {\omega L\over v} fluid dynamics (continuous and pulsating flow; nondimensional frequency)[28]
Stuart number N \mathrm{N} = \frac {B^2 L_{c} \sigma}{\rho U} = \frac{\mathrm{Ha}^2}{\mathrm{Re}} magnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor number Ta \mathrm{Ta} = \frac{4\Omega^2 R^4}{\nu^2} fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
Ursell number U \mathrm{U} = \frac{H\, \lambda^2}{h^3} wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Vadasz number Va \mathrm{Va} = \frac{\phi\, \mathrm{Pr}}{\mathrm{Da}} porous media (governs the effects of porosity \phi, the Prandtl number and the Darcy number on flow in a porous medium) [29]
van 't Hoff factor i i = 1 + \alpha (n - 1) quantitative analysis (Kf and Kb)
Wallis parameter j* j^* = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2} multiphase flows (nondimensional superficial velocity)[30]
Weaver flame speed number Wea \mathrm{Wea} = \frac{w}{w_\mathrm{H}} 100 combustion (laminar burning velocity relative to hydrogen gas)[31]
Weber number We \mathrm{We} = \frac{\rho v^2 l}{\sigma} multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg number Wi \mathrm{Wi} = \dot{\gamma} \lambda viscoelastic flows (shear rate times the relaxation time)[32]
Womersley number \alpha \alpha = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2} biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)[33]

See also


External links

  • How Many Fundamental Constants Are There?"
  • Huba, J. D., 2007, 25
  • Sheppard, Mike, 2007, "Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants."