Dimensionless quantity

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In dimensional analysis, a dimensionless quantity is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and in everyday life (such as in counting). Numerous well-known quantities, such as π, e, and φ, are dimensionless.

Dimensionless quantities are often defined as products or ratios of quantities that are not dimensionless, but whose dimensions cancel out when their powers are multiplied. This is the case, for instance, with the engineering strain, a measure of deformation. It is defined as change in length over initial length but, since these quantities both have dimensions L (length), the result is a dimensionless quantity.

A dimensionless quantity is not always a ratio; for instance, the number of people N in a room is a dimensionless quantity.

Contents

Properties

• Even though a dimensionless quantity has no physical dimension associated with it, it can still have dimensionless units. 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 (for example, to distinguish a mass ratio from a volume ratio). The quantity may also be given as a ratio of two different units that have the same dimension (for instance, light years over meters). 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  % (= 0.01), ppt (= 10−3), ppm (= 10−6), ppb (= 10−9), and angle units (radians, grad, degrees). Units of amount such as the dozen and the gross are also dimensionless.
• The -dimensionless- ratio of two quantities with the same dimensions has the same value regardless of the units used to calculate them. For instance, if body A exerts a force of magnitude F on body B, and B exerts a force of magnitude f on A, then the ratio F/f will always be equal to -1, regardless of the actual units used to measure F and f. This is a fundamental property of dimensionless proportions and follows from the assumption that the laws of physics are independent of the system of units used in their expression. In this case, if the ratio F/f was not always equal to -1, but changed if we switched from SI to CGS, for instance, that would mean that Newton's Third Law's truth or falsity would depend on the system of units used, which would contradict this fundamental hypothesis. The assumption that the laws of physics are not contingent upon a specific unit system is also closely related to the Buckingham π theorem. A formulation of this theorem is that any physical law can be expressed as an identity (always true equation) 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.