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. Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... Number is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. ... In number and more generally in algebra, a ratio is the linear relationship between two quantities of the same unit. ... Quantity is a general term used to refer to any type of quantitative property or attribute, such as mass, length, or time. ...

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. Euclid, detail from The School of Athens by Raphael. ... A Superconductor demonstrating the Meissner Effect Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Engineering applies scientific and technical knowledge to solve human problems. ...

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] The Comité international des poids et mesures or The International Committee for Weights and Measures (CIPM) consists of eighteen persons from Member States of the Metre Convention. ...

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.

CGS is an acronym for centimetre-gram-second. ... ‹ The template below has been proposed for deletion. ... The statcoulomb (statC) or franklin (Fr) or electrostatic unit of charge (esu) is the physical unit for electrical charge used in the centimetre-gram-second (cgs) electrostatic system of units. ... Permittivity is a physical quantity that describes how an electric field affects and is affected by a medium. ... The International System of Units (abbreviated SI from the French language name Système International dUnités) is the modern form of the metric system. ... In physics, Planck units are physical units of measurement originally proposed by Max Planck. ... In physics, Planck units are physical units of measurement originally proposed by Max Planck. ... Atomic units (au) are a convenient system of units of measurement used in atomic physics, particularly for describing the properties of electrons. ...

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. The Buckingham Ï€ theorem is a key theorem in dimensional analysis. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...

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. Electric power is the amount of work done by an electric current in a unit time. ... Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ... The pitch drop experiment at the University of Queensland. ... For the geometric term, see diameter. ... Speed (symbol: v) is the rate of motion, or equivalently the rate of change of position, expressed as distance d moved per unit of time t. ...

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 = 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)

The Reynolds number is the most important dimensionless number in fluid dynamics and provides a criterion for determining dynamic similitude. ... The power number Np (also known as Newton number) is a dimensionless number relating the resistance force to the inertia force. ...

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
• 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
• 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
• Lewis number: Ratio of mass diffusivity and thermal diffusivity
• 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]
• Stokes number: Dynamics of particles
• Strouhal number: Oscillatory flows
• van't Hoff factor: K_f and K_b in quantitative analysis
• 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]

In physics and optics, the Abbe number, also known as the V-number or constringence of a transparent material is a measure of the materials dispersion (variation of refractive index with wavelength). ... Dispersion of a light beam in a prism. ... An Archimedes number, named after the ancient Greek scientist Archimedes, to determine the motion of fluids due to density differences, is a dimensionless number in the form where: g = gravitational acceleration (9. ... A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ... Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ... The word grain has several meanings, most being descriptive of a small piece or particle. ... The Biot number (Bi) is a dimensionless number used in unsteady-state (or transient) heat transfer calculations. ... Electrical conductivity is a measure of how well a material accommodates the transport of electric charge. ... The Damköhler numbers (Da) are dimensionless numbers used in chemical engineering to relate chemical reaction timescale to other phenomena occurring in a system. ... In fluid mechanics, the Bond number expresses the ratio of gravitational forces to surface tension forces: where is the density, the acceleration due to gravity, a representative length scale (typically the radius of a drop), and the strength of the surface tension. ... Capillary action or capillarity (also known as capillary motion) is the ability of a narrow tube to draw a liquid upwards against the force of gravity. ... In physics, buoyancy is an upward force on an object immersed in a fluid (i. ... The capillary number represents the relative effect of viscous forces and surface tension acting across an interface between a liquid and a gas, or between two immiscible liquids. ... In fluid mechanics, the Bond number expresses the ratio of gravitational forces to surface tension forces: where is the density, the acceleration due to gravity, a representative length scale (typically the radius of a drop), and the strength of the surface tension. ... The capillary number represents the relative effect of viscous forces and surface tension acting across an interface between a liquid and a gas, or between two immiscible liquids. ... In physics, surface tension is an effect within the surface layer of a liquid that causes the layer to behave as an elastic sheet. ... The coefficient of static friction is a physical concept that determines how much force is required before an inert object, of a given material, at rest on another known substance, can be put into motion. ... The Damköhler numbers (Da) are dimensionless numbers used in chemical engineering to relate chemical reaction timescale to other phenomena occurring in a system. ... The Darcy friction factor is a dimensionless number used in internal flow calculations. ... The Deborah number is a dimensionless number, used in rheology to characterize how fluid a material is. ... Rheology is the study of the deformation and flow of matter. ... A viscoelastic material is one in which: hysteresis is seen in the stress-strain curve. ... The drag coefficient (Cd or Cx) is a number that describes a characteristic amount of aerodynamic drag caused by fluid flow, used in the drag equation. ... The Eckert number is a dimensionless number used in flow calculations. ... The Ekman number, named for V. Walfrid Ekman, is a dimensionless number used in describing geophysical phenomena in the oceans and atmosphere. ... The pitch drop experiment at the University of Queensland. ... Geophysics, the study of the earth by quantitative physical methods, especially by seismic reflection and refraction, geodesy, gravity, magnetic, electrical, electromagnetic, and radioactivity methods. ... The Euler number or cavitation number is a dimensionless number used in flow calculations. ... Hydrodynamics is fluid dynamics applied to liquids, such as water, alcohol, oil, and blood. ... The fanning friction factor is a dimensionless number used in fluid flow calculations. ... The Feigenbaum constants are two mathematical constants named after the mathematician Mitchell Feigenbaum. ... A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory deals with the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... The Fourier number (Fo) is a dimensionless number that characterises heat conduction. ... In physics, heat is defined as energy in transit. ... The Fresnel number F, named after the physicist Augustin-Jean Fresnel, is a dimensionsless number occuring in optics, diffraction theory in particular. ... Diffraction is the bending and spreading of waves when they meet an obstruction. ... In fluid dynamics, the Froude number (named after William Froude) is the reciprocal of the square root of the Richardson number. ... The concept wave is related to a disturbance that propagates through space, often transferring energy. ... The Graetz number (Gz) is a dimensionless number that characterises laminar flow in a conduit. ... In physics, heat is defined as energy in transit. ... The Grashof number is a dimensionless number in fluid dynamics which approximates the ratio of the buoyancy force to the viscous force acting on a fluid. ... Convection is the transfer of heat by currents within a fluid. ... The Hagen number is a dimensionless number used in forced flow calculations. ... Convection is the transfer of heat by currents within a fluid. ... The Knudsen number (Kn) is the ratio of the molecular mean free path length to a representative physical length scale. ... The Laplace number (La) is a dimensionless number used in the characterisation of free surface fluid dynamics. ... The Lewis number is a dimensionless number approximating the ratio of mass diffusivity and thermal diffusivity, and is used to characterize fluid flows in where there are simultaneous heat and mass transfer by convection. ... The Lockhart-Martinelli parameter is a dimensionless number used in internal two-phase fluid flow calculations. ... The coefficient of lift is a number associated with a particular shape of an aerofoil, and is incorporated in the lift equation to predict the lift force generated by a wing using this particular cross section. ... An airfoil (in American English, or aerofoil in British English) is the shape of a wing or blade (of a propeller or ships screw) as seen in cross-section. ... In this diagram, the black arrow represents the direction of the wind. ... In mathematics, the Courant–Friedrichs–Lewy condition (CFL condition) is a condition for certain algorithms for solving partial differential equations to be numerically stable. ... Mach number (Ma) (pronounced mack in British English and mock in American English) is defined as a ratio of the speed of an object or flow relative to the speed of sound in the medium through which it is travelling. ... A gas is one of the four main phases of matter (after solid and liquid, and followed by plasma), that subsequently appear as a solid material is subjected to increasingly higher temperatures. ... The Reynolds number is the ratio of inertial forces (vsρ) to viscous forces (μ/L) and is used for determining whether a flow will be laminar or turbulent. ... Magnetohydrodynamics (MHD) (magnetofluiddynamics or hydromagnetics), is the academic discipline which studies the dynamics of electrically conducting fluids. ... The Manning formula is an empirical formula for open channel flow, or flow driven by gravity. ... The Nusselt number is a dimensionless number which measures the enhancement of heat transfer from a surface which occurs in a real situation, compared to the heat transfer that would be measured if only conduction could occur. ... Heat transfer is the study of the energy transfer via either conduction, convection, or radiation. ... In physics, the Péclet number is a dimensionless number relating the forced convection of a system to its heat conduction. ... The pressure coefficient is a dimensionless number used in aerodynamics. ... An airfoil (in American English, or aerofoil in British English) is the shape of a wing or blade (of a propeller or ships screw) as seen in cross-section. ... When a sample of material is stretched in one direction, it tends to get thinner in the other two directions. ... The power factor of an AC electric power system is defined as the ratio of the real power to the apparent power. ... The power number Np (also known as Newton number) is a dimensionless number relating the resistance force to the inertia force. ... Prandtl Number is a dimensionless number approximating the ratio of momentum diffusivity and thermal diffusivity, It is defined as: where is the kinematic viscosity and α is the thermal diffusivity. ... In fluid mechanics, the Rayleigh number for a fluid is a dimensionless number associated with the heat transfer within the fluid. ... The Reynolds number is the most important dimensionless number in fluid dynamics and provides a criterion for determining dynamic similitude. ... laminar and turbulent water flow over the hull of a submarine In fluid dynamics, laminar flow is a flow regime characterized by high momentum diffusion, low momentum convection, and pressure and velocity independence from time. ... Turbulent flow around an obstacle; the flow further away is laminar Laminar and turbulent water flow over the hull of a submarine Turbulence creating a vortex on an airplane wing In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by low-momentum diffusion, high momentum convection, and... The Richardson number is named after Lewis Fry Richardson (1881 - 1953). ... In physics, buoyancy is an upward force on an object immersed in a fluid (i. ... The Rockwell scale characterises the indentation hardness of materials through the depth of penetration of an indenter, loaded on a material sample and compared to the penetration in some reference material. ... In materials science, hardness is the characteristic of a solid material expressing its resistance to permanent deformation. ... The Rossby number, named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing geophysical phenomena in the oceans and atmosphere. ... Geophysics, the study of the earth by quantitative physical methods, especially by seismic reflection and refraction, geodesy, gravity, magnetic, electrical, electromagnetic, and radioactivity methods. ... The Schmidt number is a dimensionless number approximating the ratio of momentum diffusivity (viscosity) and mass diffusivity, and is used to characterize fluid flows in where there are simultaneous momentum and mass diffusion convection processes. ... Schematic drawing of the effects of diffusion through a semipermeable membrane. ... The Sherwood number (Sh) is a dimensionless number used in mass-transfer operation. ... Lubrication occurs when opposing surfaces are completely separated by a lubricant film. ... In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms. ... The Stokes number is a dimensionless number corresponding to the behavior of particles suspended in a fluid flow. ... In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms. ... In physical chemistry, the van t Hoff factor i is the number of moles of solute actually in a solution in water, per mole of solid solute added. ... Freezing-point depression is the difference between the freezing points of a pure solvent and a solution mixed with a solute. ... Boiling-point elevation is a colligative property that states that a solution will have a higher boiling point than that of a pure solvent. ... Quantitative analysis has different meanings in different contexts. ... General Name, Symbol, Number hydrogen, H, 1 Chemical series nonmetals Group, Period, Block 1, 1, s Appearance colorless Atomic mass 1. ... The Weber number is a dimensionless quantity in fluid mechanics that is often useful in analysing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces. ... The Weissenberg number is a dimensionless number used in the study of viscoelastic flows. ... A viscoelastic material is one in which: hysteresis is seen in the stress-strain curve. ...

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. In physics, Planck units are physical units of measurement originally proposed by Max Planck. ... In science, a physical constant is a physical quantity whose numerical value does not change. ... In physics, fundamental physical constants are physical constants that are independent of systems of units and are in general dimensionless numbers. ...

These include:

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

The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ...

See also

• Similitude (model)
• Orders of magnitude (numbers)

A full scale X-43 Wind tunnel test. ... This list compares various sizes of positive numbers, including counts of things, dimensionless numbers and probabilities. ...

External links

• http://ichmt.me.metu.edu.tr/dimensionless/ - Biographies of 16 scientists with dimensionless numbers of heat and mass transfer named after them
• How Many Fundamental Constants Are There? by John Baez