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.[1] 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.