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.
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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 = n − k 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:
According to the πtheorem, 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
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 rottentogathered 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}).
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  Ratio of yield stress to viscous stress^{[5]}  
Biot number  Bi  surface vs. volume conductivity of solids  
Bodenstein number  residencetime 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  μ_{s}  friction of solid bodies at rest  
Coefficient of kinetic friction  μ_{k}  friction of solid bodies in translational motion  
Colburn j factor  dimensionless heat transfer coefficient  
CourantFriedrichLevy number  ν  numerical solutions of hyperbolic PDEs ^{[8]}  
Damkohler number  Da  reaction time scales vs. transport phenomena  
Darcy friction factor  C_{f} 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  C_{d}  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)  
fnumber  f  optics, photography  
Foppl–von Karman number  thinshell 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  gravitydriven viscous flow  
Golden ratio  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  K_{C}  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  countercurrent twophase flow  
Laplace number  La  free convection within immiscible fluids  
Lewis number  Le  ratio of mass diffusivity and thermal diffusivity  
Lift coefficient  C_{L}  lift available from an airfoil at a given angle of attack  
LockhartMartinelli 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  R_{m}  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  advection–diffusion 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  N_{p}  power consumption by agitators  
Prandtl number  Pr  convection heat transfer (thickness of thermal and momentum boundary layers)  
Pressure coefficient  C_{P}  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  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  C_{rr}  Vehicle dynamics  
Rossby number  R_{o}  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 (K_{f} and K_{b})  
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]} 
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:
In dimensional analysis, a dimensionless quantity (or more precisely, a quantity with the dimensions of 1) is a quantity without any physical units and thus a pure number. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all the units cancel out.
"out of every 10 apples I gather, 1 is rotten."  the rottentogathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity.
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  Field of application 

Abbe number  optics (dispersion in optical materials) 
Albedo  climatology, astronomy (reflectivity of surfaces or bodies) 
Archimedes number  motion of fluids due to density differences 
Bagnold number  flow of grain, sand, etc. [1] 
Biot number  surface vs. volume conductivity of solids 
Bodenstein number  residencetime distribution 
Bond number  capillary action driven by buoyancy [2] 
Brinkman number  heat transfer by conduction from the wall to a viscous fluid 
Brownell Katz number  combination of capillary number and Bond number 
Capillary number  fluid flow influenced by surface tension 
Coefficient of static friction  friction of solid bodies at rest 
Coefficient of kinetic friction  friction of solid bodies in translational motion 
Colburn j factor  dimensionless heat transfer coefficient 
CourantFriedrichLevy number  numerical solutions of hyperbolic PDEs[3] 
Damköhler numbers  reaction time scales vs. transport phenomena 
Darcy friction factor  fluid flow 
Dean number  vortices in curved ducts 
Deborah number  rheology of viscoelastic fluids 
Decibel  ratio of two intensities of sound 
Drag coefficient  flow resistance 
e  mathematics 
Eckert number  convective heat transfer 
Ekman number  geophysics (frictional (viscous) forces) 
Elasticity (economics)  widely used to measure how demand or supply responds to price changes 
Eötvös number  determination of bubble/drop shape 
Euler number  hydrodynamics (pressure forces vs. inertia forces) 
Fanning friction factor  fluid flow in pipes [4] 
Feigenbaum constants  chaos theory (period doubling) [5] 
Fine structure constant  quantum electrodynamics (QED) 
Foppl–von Karman number  thinshell buckling 
Fourier number  heat transfer 
Fresnel number  slit diffraction [6] 
Froude number  wave and surface behaviour 
Gain  electronics (signal output to signal input) 
Galilei number  gravitydriven viscous flow 
Graetz number  heat flow 
Grashof number  free convection 
Hatta number  adsorption enhancement due to chemical reaction 
Hagen number  forced convection 
Karlovitz number  turbulent combustion 
Knudsen number  continuum approximation in fluids 
Kt/V  medicine 
Laplace number  free convection within immiscible fluids 
Lewis number  ratio of mass diffusivity and thermal diffusivity 
LockhartMartinelli parameter  flow of wet gases [7] 
Lift coefficient  lift available from an airfoil at a given angle of attack 
Mach number  gas dynamics 
Magnetic Reynolds number  magnetohydrodynamics 
Manning roughness coefficient  open channel flow (flow driven by gravity) [8]PDF (109 KiB) 
Marangoni number  Marangoni flow due to thermal surface tension deviations 
Morton number  determination of bubble/drop shape 
Nusselt number  heat transfer with forced convection 
Ohnesorge number  atomization of liquids, Marangoni flow 
Péclet number  advection–diffusion problems 
Peel number  adhesion of microstructures with substrate [9] 
Pi  mathematics (ratio of a circle's circumference to its diameter) 
Poisson's ratio  elasticity (load in transverse and longitudinal direction) 
Power factor  electronics (real power to apparent power) 
Power number  power consumption by agitators 
Prandtl number  forced and free convection 
Pressure coefficient  pressure experienced at a point on an airfoil 
Radian  measurement of angles 
Rayleigh number  buoyancy and viscous forces in free convection 
Refractive index  electromagnetism, optics 
Reynolds number  flow behavior (inertia vs. viscosity) 
Richardson number  effect of buoyancy on flow stability [10] 
Rockwell scale  mechanical hardness 
Rossby number  inertial forces in geophysics 
Schmidt number  fluid dynamics (mass transfer and diffusion) [11] 
Sherwood number  mass transfer with forced convection 
Sommerfeld number  boundary lubrication [12] 
Stanton number  heat transfer in forced convection 
Stefan number  heat transfer during phase change 
Stokes number  particle dynamics 
Strain  materials science, elasticity 
Strouhal number  continuous and pulsating flow [13] 
Taylor number  rotating fluid flows 
van 't Hoff factor  quantitative analysis (K_{f} and K_{b}) 
Weaver flame speed number  laminar burning velocity relative to hydrogen gas [14] 
Weber number  multiphase flow with strongly curved surfaces 
Weissenberg number  viscoelastic flows [15] 
Womersley number  continuous and pulsating flows [16] 
