Dimensionless

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. [http://www.bipm.fr/utils/common/pdf/CCU16.pdf [http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=15588029&query_hl=3]
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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'' = ''n'' − ''k'' 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 {Link without Title}

Time T {Link without Title}

Mass M {Link without Title}

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 (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):

in optical materials

s due to Density differences

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of solids

distribution

driven by Buoyancy {Link without Title}

and Bond Number

Coefficient Of Static Friction : Friction of solid bodies at rest

Coefficient Of Kinetic Friction : Friction of solid bodies in translational motion

Damköhler Numbers : Reaction time scales vs transport phenomena

Darcy Friction Factor : Fluid flow

of Viscoelastic fluids

Drag Coefficient : Flow resistance

Eckert Number : Convective heat transfer

) forces in Geophysics

Eötvös Number : determination of bubble/drop shape

(pressure forces vs. inertia forces)

Fanning Friction Factor : Fluid flow in pipes {Link without Title}

in Chaos Theory {Link without Title}

transfer

at a slit {Link without Title}

and surface behaviour

Gain : Amount of output compared to input

Galilei Number : gravity driven viscous flow

flow

Knudsen Number : Continuum approximation in fluids

Laplace Number : Free convection with immiscible fluids

Lewis Number : Ratio of mass diffusivity and thermal diffusivity

Lockhart-Martinelli Parameter : flow of wet gases {Link without Title}

at given Angle Of Attack

Courant-Friedrich-Levy Number : Non-hydrostatic dynamics {Link without Title}

dynamics

to the leakage of such lines from the fluid {Link without Title}

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with forced convection

Ohnesorge Number : Atomization of liquids

Péclet Number : Competition between viscous and Brownian forces

Peel Number : Adhesion of microstructures with substrate {Link without Title}

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

or Turbulent )

on flow stability {Link without Title}

in flowing systems {Link without Title}

Sherwood Number : Mass transfer with forced convection

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Strouhal Number : Continuous and pulsating flow {Link without Title}

Stokes Number : Dynamics of particles

Strouhal Number : Oscillatory flows

and K_b in Quantitative Analysis

gas {Link without Title}

Weber Number : Characterization of multiphase flow with strongly curved surfaces

flows {Link without Title}

Womersley Number : continuous and pulsating flows {Link without Title}

DIMENSIONLESS PHYSICAL CONSTANTS

Certain Physical Constant s, 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 Constant s include:

$\alpha$ , the Fine Structure Constant and the electromagnetic Coupling Constant

$\beta$ , the ratio of the Rest Mass of the Proton to that of the Electron

more generally, the mass of all Fundamental Particles relative to that of the electron

the strong Coupling Constant

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Dimensionless