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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 , , and , are dimensionless. By contrast, examples of quantities with dimensions are length, time, and speed, which are measured in dimensional units, such as meter, second and meter/second. Dimensionless quantities are often obtained as products or ratios of quantities that are not dimensionless, but whose dimensions cancel in the mathematical operation. This is the case, for instance, with the engineering strain, a measure of deformation. It is defined as change in length, divided by initial length, but because these quantities both have dimensions ''L'' (length), the result is a dimensionless quantity. == Properties == All pure numbers are dimensionless quantities. 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 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), ‰ (= 0.001), 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. When a quantity is the ratio of two other quantities, each of the same dimension, the defined quantity is dimensionless and has the same value regardless of the units used to calculate the two composing quantities. 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'' is always 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 one switched from SI to CGS, 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. This assumption that the laws of physics are not contingent upon a specific unit system is the basis for the Buckingham π theorem, as discussed in a later section. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dimensionless quantity」の詳細全文を読む スポンサード リンク
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