Dimensionless

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is applicable. It is thus a bare number, and is therefore also known as a quantity of dimension one. Dimensionless quantities are widely used in many fields, such as mathematics, physics, engineering, and economics. Numerous well-known quantities, such as π, e, and φ, are dimensionless. By contrast, examples of quantities with dimensions are length, time, and speed, which are measured in dimensional units, such as metre, second and metre/second.

Dimensionless quantities are often obtained as products or ratios of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation. An example of such a ratio is engineering strain, a measure of physical deformation. It is defined as a change in length divided by the initial length. Since both quantities have the dimension length, their ratio is dimensionless. Another example is alcohol by volume, which characterizes the concentration of ethanol in an alcoholic beverage.

All pure numbers are dimensionless. A dimensionless quantity may have dimensionless units, even though it has no physical dimension associated with it. For example, to show the quantity being measured, for example mass fraction or mole fraction, it is sometimes helpful to use the same units in both the numerator and denominator (kg/kg or mol/mol). The quantity may also be given as a ratio of two different units that have the same dimension (for instance, light years over metres). This may be the case when calculating slopes in graphs, or when making unit conversions. Such notation does not indicate the presence of physical dimensions, and is purely a notational convention. Other common dimensionless units are percentages % (= 0.01),  ‰ (= 0.001); parts-per notation ppm (= 10⁻⁶), ppb (= 10⁻⁹), ppt (= 10⁻¹²); angle units (degrees, radians, grad); dalton and mole. Units of number such as the dozen, gross, and googol are also dimensionless.

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