Dimensionless quantity
Dimensionless quantities, also known as quantities of dimension one[1] are 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 derivation remains independent of the specific units of volume used; any common unit may be applied. Notably, ABV is never expressed as milliliters per milliliter, underscoring its dimensionless nature.
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]
Integers[edit]
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.
Ratios, proportions, and angles[edit]
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]
