# Dimensionless quantity

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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.

## Examples

"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 of "radians" the length that is compared is the length of the radius of the circle. When using the unit of "degrees" the length that is compared is 1/360 of the circumference of the circle.

Dimensionless quantities are widely used in the fields of mathematics, physics, engineering, and economics but also in everyday life. Whenever one measures 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. A quantity Q is defined as the product of that dimensionless number n (the number of tick marks) and the unit U (the standard):

${\displaystyle \mathrm {Q} \ {\stackrel {\mathrm {def} }{=}}\ n\mathrm {U} \ }$

But, ultimately, people always work with dimensionless numbers in reading measuring instruments and manipulating (changing or calculating with) even dimensionful quantities.

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

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 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.
• 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 quantity. For the purposes of the experimenter, different systems which share the same description by dimensionless quantity 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 a very 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 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 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. [5]
Biot number surface vs. volume conductivity of solids
Bodenstein number residence-time distribution
Bond number capillary action driven by buoyancy [6]
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
Courant-Friedrich-Levy number numerical solutions of hyperbolic PDEs[7]
Courtin number torque on rotating fluids
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 [8]
Feigenbaum constants chaos theory (period doubling) [9]
Fine structure constant quantum electrodynamics (QED)
Foppl–von Karman number thin-shell buckling
Fourier number heat transfer
Fresnel number slit diffraction [10]
Froude number wave and surface behaviour
Gain electronics (signal output to signal input)
Galilei number gravity-driven 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
Lockhart-Martinelli parameter flow of wet gases [11]
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) [12]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
Peel number adhesion of microstructures with substrate [13]
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
Rayleigh number buoyancy and viscous forces in free convection
Refractive index electromagnetism, optics
Reynolds number flow behavior (inertia vs. viscosity)
Relative density hydrometers, material comparisons
Richardson number effect of buoyancy on flow stability [14]
Rockwell scale mechanical hardness
Rossby number inertial forces in geophysics
Schmidt number fluid dynamics (mass transfer and diffusion) [15]
Sherwood number mass transfer with forced convection
Sommerfeld number boundary lubrication [16]
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 [17]
Taylor number rotating fluid flows
van 't Hoff factor quantitative analysis (Kf and Kb)
Weaver flame speed number laminar burning velocity relative to hydrogen gas [18]
Weber number multiphase flow with strongly curved surfaces
Weissenberg number viscoelastic flows [19]
Womersley number continuous and pulsating flows [20]

## Dimensionless physical constants

Certain physical constants, such as the speed of light in a vacuum, 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 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 fundamental physical constants include: