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Dimensionless quantities, or quantities of dimension one, [1] are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. [2] [3] Typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined units. For instance, alcohol by volume (ABV) represents a volumetric ratio; its value remains independent of the specific units of volume used, such as in milliliters per milliliter (mL/mL).
The number one is recognized as a dimensionless base quantity. [4] Radians serve as dimensionless units for angular measurements, derived from the universal ratio of 2π times the radius of a circle being equal to its circumference. [5]
Dimensionless quantities play a crucial role serving as parameters in differential equations in various technical disciplines. In calculus, concepts like the unitless ratios in limits or derivatives often involve dimensionless quantities. In differential geometry, the use of dimensionless parameters is evident in geometric relationships and transformations. Physics relies on dimensionless numbers like the Reynolds number in fluid dynamics, [6] the fine-structure constant in quantum mechanics, [7] and the Lorentz factor in relativity. [8] In chemistry, state properties and ratios such as mole fractions concentration ratios are dimensionless. [9]
Quantities having dimension one, dimensionless quantities, regularly occur in sciences, and are formally treated within the field of dimensional analysis. In the 19th century, French mathematician Joseph Fourier and Scottish physicist James Clerk Maxwell led significant developments in the modern concepts of dimension and unit. Later work by British physicists Osborne Reynolds and Lord Rayleigh contributed to the understanding of dimensionless numbers in physics. Building on Rayleigh's method of dimensional analysis, Edgar Buckingham proved the π theorem (independently of French mathematician Joseph 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 of fluid mechanics and heat transfer. Measuring logarithm of ratios as levels in the (derived) unit decibel (dB) finds widespread use nowadays.
There have been periodic proposals to "patch" the SI system to reduce confusion regarding physical dimensions. For example, a 2017 op-ed in Nature [11] argued for formalizing the radian 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 the Strouhal number, and for mathematically distinct entities that happen to have the same units, like torque (a vector product) versus energy (a scalar product). In another instance in the early 2000s, the International 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]
The Buckingham π 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 an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked by Boyle'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 the functional dependence between a certain number (say, n) of variables can be reduced by the number (say, k) of independent dimensions occurring in those variables to give a set of p = n − k independent, dimensionless quantities. For the purposes of the experimenter, different systems that share the same description by dimensionless quantity 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 express countable quantities. [17] [18] The concept is formalized as quantity number of entities (symbol N) in ISO 80000-1. [19] Examples include number of particles and population size. In mathematics, the "number of elements" in a set is termed cardinality . Countable nouns is a related linguistics concept. Counting numbers, such as number of bits, can be compounded with units of frequency (inverse second) to derive units of count rate, such as bits per second. Count data is a related concept in statistics. The concept may be generalized by allowing non-integer numbers to account for fractions of a full item, e.g., number of turns equal to one half.
Dimensionless quantities can be obtained as ratios of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation. [19] [20] Examples of quotients of dimension one include calculating slopes or some unit conversion factors. Another set of examples is mass fractions or mole fractions, often written using parts-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 or mol/mol). For example, alcohol by volume, which characterizes the concentration of ethanol in an alcoholic beverage, could be written as mL / 100 mL.
Other common proportions are percentages % (= 0.01), ‰ (= 0.001). Some angle units such as turn, radian, and steradian are defined as ratios of quantities of the same kind. In statistics the coefficient of variation is the ratio of the standard deviation to the mean and is used to measure the dispersion in the data.
It has been argued that quantities defined as ratios Q = A/B having equal dimensions in numerator and denominator are actually only unitless quantities and still have physical dimension defined as dim Q = dim A × dim B−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 the speed of light in vacuum, the universal gravitational constant, the Planck constant, the Coulomb constant, and the Boltzmann constant can be normalized to 1 if appropriate units for time, length, mass, charge, and temperature are chosen. The resulting system of units is known as the natural units, specifically regarding these five constants, Planck units. However, not all physical 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]
In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measurement and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to conversion of units from one dimensional unit to another, which can be used to evaluate scientific formulae.
A physical quantity is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a value, which is the algebraic multiplication of a numerical value and a unit of measurement. For example, the physical quantity mass, symbol m, can be quantified as m=n kg, where n is the numerical value and kg is the unit symbol. Quantities that are vectors have, besides numerical value and unit, direction or orientation in space.
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement.
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius. The unit was formerly an SI supplementary unit and is currently a dimensionless SI derived unit, defined in the SI as 1 rad = 1 and expressed in terms of the SI base unit metre (m) as rad = m/m. Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.
The mole (symbol mol) is a unit of measurement, the base unit in the International System of Units (SI) for amount of substance, a quantity proportional to the number of elementary entities of a substance. One mole contains exactly 6.02214076×1023 elementary entities (approximately 602 sextillion or 602 billion times a trillion), which can be atoms, molecules, ions, or other particles. The number of particles in a mole is the Avogadro number (symbol N0) and the numerical value of the Avogadro constant (symbol NA) expressed in mol-1. The value was chosen on the basis of the historical definition of the mole as the amount of substance that corresponds to the number of atoms in 12 grams of 12C, which made the mass of a mole of a compound expressed in grams, numerically equal to the average molecular mass of the compound expressed in daltons. With the 2019 redefinition of the SI base units, the numerical equivalence is now only approximate but may be assumed for all practical purposes.
The dalton or unified atomic mass unit is a unit of mass defined as 1/12 of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state and at rest. It is a non-SI unit accepted for use with SI. The atomic mass constant, denoted mu, is defined identically, giving mu = 1/12m(12C) = 1 Da.
In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalisation of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters π1, π2, ..., πp constructed from the original variables, where k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.
The Boltzmann constant is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the gas constant, and in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy and heat capacity. It is named after the Austrian scientist Ludwig Boltzmann.
A base unit of measurement is a unit of measurement adopted for a base quantity. A base quantity is one of a conventionally chosen subset of physical quantities, where no quantity in the subset can be expressed in terms of the others. The SI base units, or Systéme International d'unités, consists of the metre, kilogram, second, ampere, kelvin, mole and candela.
The atomic units are a system of natural units of measurement that is especially convenient for calculations in atomic physics and related scientific fields, such as computational chemistry and atomic spectroscopy. They were originally suggested and named by the physicist Douglas Hartree. Atomic units are often abbreviated "a.u." or "au", not to be confused with similar abbreviations used for astronomical units, arbitrary units, and absorbance units in other contexts.
A geometrized unit system or geometrodynamic unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, c, and the gravitational constant, G, are set equal to unity.
In physics, a dimensionless physical constant is a physical constant that is dimensionless, i.e. a pure number having no units attached and having a numerical value that is independent of whatever system of units may be used.
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit of measurement. Mass, time, distance, heat, and angle are among the familiar examples of quantitative properties.
Similitude is a concept applicable to the testing of engineering models. A model is said to have similitude with the real application if the two share geometric similarity, kinematic similarity and dynamic similarity. Similarity and similitude are interchangeable in this context. The term dynamic similitude is often used as a catch-all because it implies that geometric and kinematic similitude have already been met.
Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units. The term "cgs units" is ambiguous and therefore to be avoided if possible: there are several variants of cgs with conflicting definitions of electromagnetic quantities and units.
Nondimensionalization is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term scaling is used interchangeably with nondimensionalization, in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units.
The International System of Quantities (ISQ) is a standard system of quantities used in physics and in modern science in general. It includes basic quantities such as length and mass and the relationships between those quantities. This system underlies the International System of Units (SI) but does not itself determine the units of measurement used for the quantities.
Quantity calculus is the formal method for describing the mathematical relations between abstract physical quantities.
A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations directly relating the quantities. It is a system in which every quantity has a unique unit, or one that does not use conversion factors.