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Dimensionless quantities, or quantities of dimension one,[1] are quantitiesimplicitly defined in a manner that prevents their aggregation intounits of measurement.[2][3] Typically expressed asratios that align with another system, these quantities do not necessitate explicitly definedunits. For instance,alcohol by volume (ABV) represents avolumetric ratio; its value remains independent of the specificunits of volume used, such as inmilliliters per milliliter (mL/mL).
Thenumber one is recognized as a dimensionlessbase quantity.[4]Radians serve as dimensionless units forangular measurements, derived from the universal ratio of 2π times theradius of a circle being equal to its circumference.[5]
Dimensionless quantities play a crucial role serving asparameters indifferential equations in various technical disciplines. Incalculus, concepts like the unitless ratios inlimits orderivatives often involve dimensionless quantities. Indifferential geometry, the use of dimensionless parameters is evident in geometric relationships and transformations. Physics relies on dimensionless numbers like theReynolds number influid dynamics,[6] thefine-structure constant inquantum mechanics,[7] and theLorentz factor inrelativity.[8] Inchemistry,state properties and ratios such asmole fractionsconcentration ratios are dimensionless.[9]
Quantities having dimension one,dimensionless quantities, regularly occur in sciences, and are formally treated within the field ofdimensional analysis. In the 19th century, French mathematicianJoseph Fourier and Scottish physicistJames Clerk Maxwell led significant developments in the modern concepts ofdimension andunit. Later work by British physicistsOsborne Reynolds andLord Rayleigh contributed to the understanding of dimensionless numbers in physics. Building on Rayleigh's method of dimensional analysis,Edgar Buckingham proved theπ theorem (independently of French mathematicianJoseph Bertrand's previous work) to formalize the nature of these quantities.[10]
Numerous dimensionless numbers, mostly ratios, were coined in the early 1900s, particularly in the areas offluid mechanics andheat transfer. Measuring logarithm of ratios aslevels in the (derived) unitdecibel (dB) finds widespread use nowadays.
There have been periodic proposals to "patch" the SI system to reduce confusion regarding physical dimensions. For example, a 2017op-ed inNature[11] argued for formalizing theradian as a physical unit. The idea was rebutted[12] on the grounds that such a change would raise inconsistencies for both established dimensionless groups, like theStrouhal number, and for mathematically distinct entities that happen to have the same units, liketorque (avector product) versus energy (ascalar product). In another instance in the early 2000s, theInternational Committee for Weights and Measures discussed naming the unit of 1 as the "uno", but the idea of just introducing a new SI name for 1 was dropped.[13][14][15]
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The Buckinghamπ theorem[16] indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as anidentity involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked byBoyle's Law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold.
Another consequence of the theorem is that thefunctional dependence between a certain number (say,n) ofvariables can be reduced by the number (say,k) ofindependentdimensions occurring in those variables to give a set ofp =n −k independent, dimensionlessquantities. For the purposes of the experimenter, different systems that share the same description by dimensionlessquantity are equivalent.
| Number of entities | |
|---|---|
Common symbols | N |
| SI unit | Unitless |
| Dimension | 1 |
Integer numbers may represent dimensionless quantities. They can represent discrete quantities, which can also be dimensionless. More specifically,counting numbers can be used to expresscountable quantities.[17][18]The concept is formalized as quantitynumber of entities (symbolN) inISO 80000-1.[19]Examples includenumber of particles andpopulation size. In mathematics, the "number of elements" in a set is termedcardinality.Countable nouns is a related linguistics concept.Counting numbers, such as number ofbits, can be compounded with units of frequency (inverse second) to derive units of count rate, such asbits per second.Count data is a related concept in statistics.The concept may be generalized by allowingnon-integer numbers to account for fractions of a full item, e.g.,number of turns equal to one half.
Dimensionless quantities can be obtained asratios of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation.[19][20] Examples of quotients of dimension one include calculatingslopes or someunit conversion factors. Another set of examples ismass fractions ormole fractions, often written usingparts-per notation such as ppm (= 10−6), ppb (= 10−9), and ppt (= 10−12), or perhaps confusingly as ratios of two identical units (kg/kg ormol/mol). For example,alcohol by volume, which characterizes the concentration ofethanol in analcoholic beverage, could be written asmL / 100 mL.
Other common proportions are percentages% (= 0.01), ‰ (= 0.001). Some angle units such asturn,radian, andsteradian are defined as ratios of quantities of the same kind. Instatistics thecoefficient of variation is the ratio of thestandard deviation to themean and is used to measure thedispersion in thedata.
It has been argued that quantities defined as ratiosQ =A/B having equal dimensions in numerator and denominator are actually onlyunitless quantities and still have physical dimension defined asdimQ = dimA × dimB−1.[21]For example,moisture content may be defined as a ratio of volumes (volumetric moisture, m3⋅m−3, dimension L3⋅L−3) or as a ratio of masses (gravimetric moisture, units kg⋅kg−1, dimension M⋅M−1); both would be unitless quantities, but of different dimension.
Certain universal dimensioned physical constants, such as thespeed of light in vacuum, theuniversal gravitational constant, thePlanck constant, theCoulomb constant, and theBoltzmann constant can be normalized to 1 if appropriate units fortime,length,mass,charge, andtemperature are chosen. The resultingsystem of units is known as thenatural units, specifically regarding these five constants,Planck units. However, not allphysical constants can be normalized in this fashion. For example, the values of the following constants are independent of the system of units, cannot be defined, and can only be determined experimentally:[22]
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