Dimensionless quantity

In dimensional analysis, a dimensionless quantity (or more precisely, a quantity with the dimensions of 1) is a quantity without any physical units and thus a pure number. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all the units cancel out.

Examples
"out of every 10 apples I gather, 1 is rotten." -- the rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity. Another more typical example in physics and engineering is the measure of plane angles. Angles are typically measured as the ratio of the length of an arc lying on a circle (with its center being the vertex of the angle) swept out by the angle, compared to some other length. The ratio (length divided by length) is dimensionless. When using the unit of "radians" the length that is compared is the length of the radius of the circle. When using the unit of "degrees" the length that is compared is 1/360 of the circumference of the circle.

Dimensionless quantities are widely used in the fields of mathematics, physics, engineering, and economics but also in everyday life. Whenever one measures any physical quantity, they are measuring that physical quantity against a like dimensioned standard. Whenever one commonly measures a length with a ruler or tape measure, they are counting tick marks on the standard of length they are using, which is a dimensionless number. When they attach that dimensionless number (the number of tick marks) to the units that the standard represents, they conceptually are referring to a dimensionful quantity. A quantity Q is defined as the product of that dimensionless number n (the number of tick marks) and the unit U (the standard):
 * $$ \mathrm{Q} \ \stackrel{\mathrm{def}}{=}\ n \mathrm{U} \ $$

But, ultimately, people always work with dimensionless numbers in reading measuring instruments and manipulating (changing or calculating with) even dimensionful quantities.

In case of dimensionless quantities the unit U is a quotient of like dimensioned quantities that can be reduced to a number (kg/kg = 1, μg/g = 1-6). Dimensionless quantities can also carry dimensionless units like % (=0.01), ppt (=10-3), ppm (=10-6), ppb (=10-9).

The CIPM Consultative Committee for Units toyed with the idea of defining the unit of 1 as the 'uno', but the idea was dropped.  

Properties

 * A dimensionless quantity has no physical unit associated with it. However, 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.
 * A dimensionless proportion has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the SI system of units or the imperial system of units. This doesn't hold for all dimensionless quantities; it is guaranteed to hold only for proportions.

Buckingham π theorem
According to the Buckingham π theorem of dimensional analysis, the functional dependence between a certain number (e.g., n) of variables can be reduced by the number (e.g., k) of independent dimensions occurring in those variables to give a set of p = n &minus; k independent, dimensionless quantity. For the purposes of the experimenter, different systems which share the same description by dimensionless quantity are equivalent.

Example
The power consumption of a stirrer with a particular geometry is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n = 5 variables representing our example.

Those n = 5 variables are built up from k = 3 dimensions which are:
 * Length: L (m)
 * Time: T (s)
 * Mass: M (kg)

According to the π-theorem, the n = 5 variables can be reduced by the k = 3 dimensions to form p = n &minus; k = 5 &minus; 3 = 2 independent dimensionless numbers which are in case of the stirrer
 * Reynolds number (This is a very important dimensionless number; it describes the fluid flow regime)
 * Power number (describes the stirrer and also involves the density of the fluid)

List of dimensionless quantities
There are infinitely many dimensionless quantities and they are often called numbers. Some of those that are used most often have been given names, as in the following list of examples (alphabetical order):

Dimensionless physical constants
Certain physical constants, such as the speed of light in a vacuum, are normalized to 1 if the units for time, length, mass, charge, and temperature are chosen appropriately. The resulting system of units is known as Planck units. However, a handful of dimensionless physical constants cannot be eliminated in any system of units; their values must be determined experimentally. The resulting fundamental physical constants include:
 * $$\alpha$$, the fine structure constant and the electromagnetic coupling constant
 * $$\beta$$, the ratio of the rest mass of the proton to that of the electron
 * more generally, the masses of all fundamental particles relative to that of the electron
 * the strong Coupling constant
 * the gravitational coupling constant