In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units. Such a number is typically defined as a product or ratio of quantities which have units of identical dimension, in such a way that the corresponding units can be converted to identical units and then cancel.

For example: "one out of every 10 apples I gather is rotten." -- the rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1, which is a dimensionless quantity. Another more typical example in physics and engineering is the measure of plane angles with the unit of "radian". An angle measured this way is the length of arc lying on a circle (with center being the vertex of the angle) swept out by the angle to the length of the radius of the circle. The units of the ratio is length divided by length which is dimensionless.

Dimensionless numbers are widely used in the fields of mathematics, physics, and engineering but also in everyday life. Whenever one measures anything, any physical quantity, they are measuring that physical quantity against a like dimensioned standard. Whenever one commonly measures a length with a ruler or tape measure, they are counting tick marks on the standard of length they are using, which is a dimensionless number. When they attach that dimensionless number (the number of tick marks) to the units that the standard represents, they conceptually are referring to a dimensionful quantity. But, ultimately, we always work with dimensionless numbers in measuring and manipulating even dimensionful quantities.

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

Properties

• A dimensionless number 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.
• A dimensionless number has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the metric measurement system or the imperial measurement system.
• However, a physical quantity may be dimensionless in one system of units and not dimensionless in another system of units. For example, in the nonrationalized cgs system of units, the unit of electric charge (the statcoulomb) is defined in such a way so that the permittivity of free space ε0 = 1/(4π) whereas in the rationalized SI system, it is ε0 = 8.85419×10-12 F/m. In systems of natural units (e.g. Planck units or atomic units), the physical units are defined in such a way that several fundamental constants are made dimensionless and set to 1 (thus removing these scaling factors from equations). While this is convenient in some contexts, abolishing of all or most units and dimensions makes practical physical calculations more error prone.

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 numbers. For the purposes of the experimenter, different systems which share the same description by dimensionless numbers are equivalent.

Example

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 the most important dimensionless number; it describes the fluid flow regime)
• Power number (describes the stirrer and also involves the density of the fluid)

List of dimensionless numbers

There are infinitely many dimensionless numbers. Some of those that are used most often have been given names, as in the following list of examples (in alphabetical order, indicating their field of use):

• Abbe number: Dispersion in optical materials
• Archimedes number: Motion of fluids due to density differences
• Bagnold number: Flow of grain [5]
• Biot number: Surface vs volume conductivity of solids
• Bodenstein number: Residence-time distribution
• Bond number: Capillary action driven by buoyancy [6]
• Brownell Katz number: Combination of capillary number and Bond number
• Capillary number: Fluid flow influenced by surface tension
• Damköhler numbers: Reaction time scales vs transport phenomena
• Darcy friction factor: Fluid flow
• Deborah number: Rheology of viscoelastic fluids
• Drag coefficient: Flow resistance
• Eckert number: Convective heat transfer
• Ekman number: Frictional (viscous) forces in geophysics
• Euler number: Hydrodynamics (pressure forces vs. inertia forces)
• Fanning friction factor: Fluid flow in pipes [7]
• Feigenbaum's delta: Period doubling in chaos theory [8]
• Fourier number: Heat transfer
• Fresnel number: Diffraction at a slit [9]
• Froude number: Wave and surface behaviour
• Graetz number: Heat flow
• Grashof number: Free convection
• Hagen number: Forced convection
• Knudsen number: Continuum approximation in fluids
• Laplace number: Free convection with immiscible fluids
• Lockhart-Martinelli parameter: flow of wet gases [10]
• Lift coefficient: amount of lift available from given airfoil at given angle of attack
• Courant-Friedrich-Levy number: Non-hydrostatic dynamics [11]
• Mach number: Gas dynamics
• Magnetic Reynolds number: used to compare the transport of magnetic lines of force in a conducting fluid to the leakage of such lines from the fluid [12]
• Manning roughness coefficient: see Manning equation [13]
• Nusselt number: Heat transfer with forced convection
• Ohnesorge number: Atomization of liquids
• Péclet number: Competition between viscous and Brownian forces
• Peel number: Adhesion of microstructures with substrate [14]
• Pressure coefficient: Coefficient of pressure experienced at a point on an airfoil
• Poisson's ratio: Load in transverse and longitudinal direction
• Power factor : ratio of real power to apparent power
• Power number: Power consumption by agitators
• Prandtl number: Forced and free convection
• Rayleigh number: Buoyancy and viscous forces in free convection
• Reynolds number: Characterizing flow behaviour (laminar or turbulent)
• Richardson number: Effect of buoyancy on flow stability [15]
• Rockwell scale: Mechanical hardness
• Rossby number: Inertial forces in geophysics
• Schmidt number: mass transfer, and diffusion in flowing systems [16]
• Sherwood number: Mass transfer with forced convection
• Sommerfeld number: Boundary lubrication [17]
• Strouhal number: Continuous and pulsating flow [18]
• Coefficient of static friction: Friction of solid bodies at rest
• Coefficient of kinetic friction: Friction of solid bodies in translational motion
• Stokes number: Dynamics of particles
• Strouhal number: Oscillatory flows
• Weaver flame speed number: laminar burning velocity relative to hydrogen gas [19]
• Weber number: Characterization of multiphase flow with strongly curved surfaces
• Weissenberg number: Viscoelastic flows [20]
• Womersley number: continuous and pulsating flows [21]

Dimensionless physical constants

The system of natural units chooses its base units in such a way as to eliminate a few physical constants such as the speed of light by choosing units that express these physical constants as 1 in terms of the natural units. However, the dimensionless physical constants cannot be eliminated in any system of units, and are measured experimentally. These are often called fundamental physical constants.

These include:

• the fine structure constant
• the electromagnetic coupling constant
• the strong coupling constant