Dimensionless Quantities

Units for dimensionless quantities, also called quantities of dimension one, “1”:

Certain quantities are defined as the ratio of two quantities of the same kind, and are thus dimensionless, or have a dimension that may be expressed by the number one.
The coherent SI unit of all such dimensionless quantities, or quantities of dimension one, is the number one, since the unit must be the ratio of two identical SI units.
The values of all such quantities are simply expressed as numbers, and the unit one is not explicitly shown. Examples of such quantities are refractive index, relative permeability, and friction factor.

In a few cases, however, a special name is given to the unit one, in order to facilitate the identification of the quantity involved. This is the case for the radian and the steradian. The radian and steradian have been identified as special names for the coherent derived unit one, to be used to express values of plane angle and solid angle, respectively.

Another class of dimensionless quantities are numbers that represent a count, such as:

degeneracy (number of energy levels),
results of measurements using counters such as photons,
and partition function in statistical thermodynamics (number of thermally accessible states).

All of these counting quantities are also described as being dimensionless, or of dimension one, and are taken to have the SI unit one, although the unit of counting quantities cannot be described as a derived unit expressed in terms of the base units of the SI. For such quantities, the unit one may instead be regarded as a further base unit.