Dimensionless quantity

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. 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.

Properties

• 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 = n − k independent, dimensionless quantities. For the purposes of the experimenter, different systems that share the same description by dimensionless quantity are equivalent.

Example

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 = n − k = 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.

Examples

• 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.

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	$\gamma$	$\gamma= \frac {{a}}{{x}}$	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	$\mathit{Ar} = \frac{g L^3 \rho_\ell (\rho - \rho_\ell)}{\mu^2}$	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)
Atomic weight	M		chemistry (mass of atom over one atomic mass unit, u, where carbon-12 is exactly 12 u)
Atwood number	A	$\mathit{A} = \frac{\rho_1 - \rho_2} {\rho_1 + \rho_2}$	fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold number	Ba	$\mathit{Ba} = \frac{\rho d^2 \lambda^{1/2} \gamma}{\mu}$	fluid mechanics, geology (ratio of grain collision stresses to viscous fluid stresses in flow of a granular material such as grain and sand)
Bejan number (thermodynamics)	Be	$\mathit{Be} = \frac{\dot S'_{\mathrm{gen},\, \Delta T}}{\dot S'_{\mathrm{gen},\, \Delta T}+ \dot S'_{\mathrm{gen},\, \Delta p}}$	thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)
Bejan number (fluid mechanics)	Be	$\mathit{Be} = \frac{\Delta P L^2} {\mu \alpha}$	fluid mechanics (dimensionless pressure drop along a channel)
Bingham number	Bm	$\mathit{Bm} = \frac{ \tau_y L }{ \mu V }$	fluid mechanics, rheology (ratio of yield stress to viscous stress)
Biot number	Bi	$\mathit{Bi} = \frac{h L_C}{k_b}$	heat transfer (surface vs. volume conductivity of solids)
Blake number	Bl or B	$\mathit{B} = \frac{u \rho}{\mu (1 - \epsilon) D}$	geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bodenstein number	Bo	$\mathit{Bo} = vL/\mathcal{D} = \mathit{Re}\, \mathit{Sc}$	chemistry (residence-time distribution)
Bond number	Bo	$\mathit{Bo} = \frac{\rho a L^2}{\gamma}$	geology, fluid mechanics, porous media (buoyant versus capilary forces, similar to the Eötvös number)
Brinkman number	Br	$\mathit{Br} = \frac {\mu U^2}{\kappa (T_w - T_0)}$	heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Brownell–Katz number			combination of capillary number and Bond number
Capillary number	Ca	$\mathit{Ca} = \frac{\mu V}{\gamma}$	porous media (viscous forces versus surface tension)
Chandrasekhar number	Q	$\mathit{Q} = \frac{{B_0}^2 d^2}{\mu_0 \rho \nu \lambda}$	magnetohydrodynamics (ratio of the Lorentz force to the viscosity in magnetic convection)
Coefficient of static friction	µs		mechanics (friction of solid bodies at rest)
Coefficient of kinetic friction	µk		mechanics (friction of solid bodies in translational motion)
Colburn J factors	JM, JH, JD		turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Courant–Friedrich–Levy number	C or 𝜈	$C = \frac {u\,\Delta t} {\Delta x}$	mathematics (numerical solutions of hyperbolic PDEs)
Damkohler number	Da	$\mathit{Da} = k \tau$	chemistry (reaction time scales vs. residence time)
Damping ratio	ζ	$\zeta = \frac{c}{2 \sqrt{km}}$	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)
Dean number	D	$\mathit{D} = \frac{\rho V d}{\mu} \left( \frac{d}{2 R} \right)^{1/2}$	turbulent flow (vortices in curved ducts)
Deborah number	De	$\mathit{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}}$	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	$\mathit{Du} = \frac{\kappa^{\sigma}}{{\Kappa_m} a}$	colloid science (ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems)
Eckert number	Ec	$\mathit{Ec} = \frac{V^2}{c_p\Delta T}$	convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Ekman number	Ek	$\mathit{Ek} = \frac{\nu}{2D^2\Omega\sin\varphi}$	geophysics (viscous versus Coriolis forces)
Elasticity (economics)	E		economics (response of demand or supply to price changes)
Entropy (physics)	σ		physics, information theory (log of the number of states)
Eötvös number	Eo	$\mathit{Eo}=\frac{\Delta\rho \,g \,L^2}{\sigma}$	fluid mechanics (shape of bubbles or drops)
Ericksen number	Er	$\mathit{Er}=\frac{\mu v L}{K}$	fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number	Eu	$\mathit{Eu}=\frac{\Delta{}p}{\rho V^2}$	hydrodynamics (stream pressure versus inertia forces)
Euler's number	e	$e = \sum_{k = 0}^n \frac{1}{k!} \approx 2.71828$	mathematics (base of the natural logarithm)
Excess temperature coefficient	Θr	$\Theta_r = \frac{c_p (T-T_e)}{U_e^2/2}$	heat transfer, fluid dynamics (change in internal energy versus kinetic energy)
Fanning friction factor	f		fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)
Feigenbaum constants	$\alpha$, δ	$\alpha \approx 2.50290,$ $\ \delta \approx 4.66920$	chaos theory (period doubling)
Fine structure constant	$\alpha$	$\alpha = \frac{e^2}{2\varepsilon_0 hc}$	quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction)
f-number	f	$f = \frac {{\ell}}{{D}}$	optics, photography (ratio of focal length to diameter of aperture)
Föppl–von Kármán number	$\gamma$	$\gamma = \frac{Y r^2}{\kappa}$	virology, solid mechanics (thin-shell buckling)
Fourier number	Fo	$\mathit{Fo} = \frac{\alpha t}{L^2}$	heat transfer, mass transfer (ratio of diffusive rate versus storage rate)
Fresnel number	F	$\mathit{F} = \frac{a^{2}}{L \lambda}$	optics (slit diffraction)
Froude number	Fr	$\mathit{Fr} = \frac{v}{\sqrt{g\ell}}$	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)
Galilei number	Ga	$\mathit{Ga} = \frac{g\, L^3}{\nu^2}$	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	$\mathit{G} = \frac{U_e \theta}{\nu} \left( \frac{\theta}{R} \right)^{1/2}$	fluid dynamics (boundary layer flow along a concave wall)
Graetz number	Gz	$\mathit{Gz}={D_H \over L} \mathit{Re}\, \mathit{Pr}$	heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof number	Gr	$\mathit{Gr_L} = \frac{g \beta (T_s - T_\infty ) L^3}{\nu ^2}$	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	$\mathit{Ha} = \frac{N_{\mathrm{A}0}}{N_{\mathrm{A}0}^{\mathrm{phys}}}$	chemical engineering (adsorption enhancement due to chemical reaction)
Hagen number	Hg	$\mathit{Hg} = -\frac{1}{\rho}\frac{d p}{d x}\frac{L^3}{\nu^2}$	heat transfer (ratio of the buoyancy to viscous force in forced convection)
Hydraulic gradient	i	$i = \frac{dh}{dl} = \frac{h_2 - h_1}{\mathrm{length}}$	fluid mechanics, groundwater flow (pressure head over distance)
Iribarren number	Ir	$\mathit{Ir} = \frac{\tan \alpha}{\sqrt{H/L_0}}$	wave mechanics (breaking surface gravity waves on a slope)
Jakob Number	Ja	$\mathit{Ja} = \frac{c_p (T_\mathrm{s} - T_\mathrm{sat}) }{\Delta H_{\mathrm{f}} }$	chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change)
Karlovitz number	Ka	$\mathit{Ka} = k t_c$	turbulent combustion (characteristic flow time times flame stretch rate)
Keulegan–Carpenter number	KC	$\mathit{K_C} = \frac{V\,T}{L}$	fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number	Kn	$\mathit{Kn} = \frac {\lambda}{L}$	gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kt/V			medicine
Kutateladze number	K		counter-current two-phase flow
Laplace number	La	$\mathit{La} = \frac{\sigma \rho L}{\mu^2}$	fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis number	Le	$\mathit{Le} = \frac{\alpha}{D} = \frac{\mathit{Sc}}{\mathit{Pr}}$	heat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficient	CL		lift available from an airfoil at a given angle of attack
Lockhart–Martinelli parameter	$\chi$		flow of wet gases
Love numbers	h, k, l		geophysics (solidity of earth and other planets)
Lundquist number	S		ratio of a resistive time to an Alfvén wave crossing time in a plasma
Mach number	M or Ma	$M = \frac {{v}}{{v_{sound}}}$	gas dynamics
Magnetic Reynolds number	RM		magnetohydrodynamics
Manning roughness coefficient	n		open channel flow (flow driven by gravity)
Marangoni number	Mg		Marangoni flow due to thermal surface tension deviations
Morton number	Mo		determination of bubble/drop shape
Mpemba number	KM		thermal conduction and diffusion in freezing of a solution
Nusselt number	Nu	$Nu =\frac{hd}{k}$	heat transfer with forced convection
Ohnesorge number	Oh		atomization of liquids, Marangoni flow
Péclet number	Pe	$Pe = \frac{du\rho c_p}{k} = Re\, Pr$	advection–diffusion problems; relates total momentum transfer to molecular heat transfer.
Peel number			adhesion of microstructures with substrate
Perveance	K		measure of the strength of space charge in a charged particle beam
Pi	π	$\pi = \frac{C}{d} \approx 3.14159$	mathematics (ratio of a circle's circumference to its diameter)
Poisson's ratio	𝜈		elasticity (load in transverse and longitudinal direction)
Porosity	ϕ		geology
Power factor			electronics (real power to apparent power)
Power number	Np		power consumption by agitators
Prandtl number	Pr	$Pr = \frac{\nu}{\alpha} = \frac{c_p \mu}{k}$	ratio of viscous diffusion rate over thermal diffusion rate
Prater number			ratio of heat evolution to heat conduction within a catalyst pellet
Pressure coefficient	CP		pressure experienced at a point on an airfoil
Q factor	Q	$Q$	describes how under-damped an oscillator or resonator is
Radian measure	rad	$\mathrm{arc~length/radius}$	mathematics (measurement of planar angles, 1 radian = 180/π degrees)
Rayleigh number	Ra		buoyancy and viscous forces in free convection
Refractive index	n		electromagnetism, optics
Relative density	RD	$RD = \frac{\rho_\mathrm{substance}}{\rho_\mathrm{reference}}$	hydrometers, material comparisons
Relative permeability	µr		electrodynamics (e.g., magnetostatics)
Relative permittivity	εr		electrodynamics (e.g., electrostatics)
Reynolds number	Re	$Re = \frac{vL\rho}{\mu}$	ratio of fluid inertial and viscous forces
Richardson number	Ri		effect of buoyancy on flow stability
Rockwell scale			mechanical hardness
Rolling resistance coefficient	Crr	$C_{rr} = \frac{N_f}{F}$	vehicle dynamics
Roshko number	Ro	$\mathit{Ro} = {f L^{2}\over \nu} =\mathit{St}\,\mathit{Re}$	fluid dynamics (oscillating flow, vortex shedding)
Rossby number	Ro		inertial forces in geophysics
Rouse number	Z or P		sediment transport
Schmidt number	Sc		fluid dynamics (mass transfer and diffusion)
Shape factor	H		ratio of displacement thickness to momentum thickness in boundary layer flow
Sherwood number	Sh		mass transfer with forced convection
Shields parameter	τ∗ or θ		threshold of sediment movement due to fluid motion
Sommerfeld number			boundary lubrication
Specific gravity		same as Relative density
Stanton number	St	$\mathit{St} = \frac{h}{c_p \rho V} = \frac{\mathit{Nu}}{\mathit{Re}\,\mathit{Pr}}$	heat transfer and fluid dynamics (forced convection)
Stefan number	Ste	$Ste = \frac{C_p\Delta T}{L}$	heat transfer during phase change
Stokes number	Stk or Sk	$Stk = \frac{\tau\,U_o}{d_c}$	particle dynamics in a fluid stream
Strain	ϵ	$\epsilon = \cfrac{\partial{F}}{\partial{X}} - 1$	materials science, elasticity
Strouhal number	St or Sr	$St = {\omega L\over v}$	nondimensional frequency, continuous, and pulsating flow
Stuart number	N	$\mathit{N} = \frac {B^2 L_{c} \sigma}{\rho U} = \frac{\mathit{Ha}^2}{\mathit{Re}}$	magnetohydrodynamics and fluid dynamics (ratio of electromagnetic to inertial forces)
Taylor number	Ta	$Ta = \frac{4\Omega^2 R^4}{\nu^2}$	rotating fluid flows
Ursell number	U	$U = \frac{H\, \lambda^2}{h^3}$	nonlinearity of surface gravity waves on a shallow fluid layer
Vadasz number	Va	$Va = \frac{\phi Pr}{Da}$	governs the effects of porosity $\phi$, the Prandtl number and the Darcy number on flow in a porous medium
van 't Hoff factor	i	$i = 1 + \alpha (n - 1)$	quantitative analysis (Kf and Kb)
Wallis parameter	J*	$\alpha = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2}$	nondimensional superficial velocity in multiphase flows
Weaver flame speed number			laminar burning velocity relative to hydrogen gas
Weber number	We	$We = \frac{\rho v^2 l}{\sigma}$	multiphase flow with strongly curved surfaces
Weissenberg number	Wi	$Wi = \dot{\gamma} \lambda$	viscoelastic flows
Womersley number	$\alpha$	$\alpha = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2}$	continuous and pulsating flows

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. However, not all physical constants can be eliminated in any system of units; the values of the remaining ones must be determined experimentally. Resulting constants include:

• α, the fine structure constant, the coupling constant for the electromagnetic interaction: α ≈ 1/137;
• μ or β, the proton-to-electron mass ratio, the rest mass of the proton divided by that of the electron. More generally, the rest masses of all elementary particles relative to that of the electron: μ ≈ 1836;
• α_s, the coupling constant for the strong force;
• α_G, the gravitational coupling constant: α_G ≈ 1.75×10⁻⁴⁵.

See also

• Similitude (model)
• Orders of magnitude (numbers)
• Dimensional analysis
• Dimensionless physical constant
• Normalization (statistics) and standardized moment, the analogous concepts in statistics
• Buckingham π theorem

External links

• John Baez, "How Many Fundamental Constants Are There?"
• Huba, J. D., 2007, NRL Plasma Formulary: Dimensionless Numbers of Fluid Mechanics. Naval Research Laboratory. p. 23, 24, 25
• Sheppard, Mike, 2007, "Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants."