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In the language of measurement, quantities are quantifiable aspects of the world, such as time, distance, velocity, mass, temperature, energy, and weight, and units are used to describe their measure. Many of these quantities are related to each other by various physical laws, and as a result the units of some of the quantities can be expressed as products (or ratios) of powers of other units (e.g., momentum is mass times velocity and velocity is measured in distance divided by time). These relationships are discussed in dimensional analysis. Those that cannot be so expressed can be regarded as "fundamental" in this sense.
There are other relationships between physical quantities which can be expressed by means of fundamental constants, and to some extent it is an arbitrary decision whether to retain the fundamental constant as a quantity with dimensions or simply to define it as unity or a fixed dimensionless number, and reduce the number of fundamental constants by one.
For instance, time and distance are related to each other by the speed of light, c, which is a fundamental constant. It is possible to use this relationship to eliminate either the fundamental unit of time or that of distance. Similar considerations apply to Planck's constant, h, which relates energy (with dimensions of mass, length and time) to frequency (dimensions of time). In theoretical physics it is customary to use such units (natural units) in which c = 1 and ħ = 1.
Slightly different considerations apply to the so-called permittivity of free space, which historically has been regarded as a separate physical constant in some systems of measurement but not in others.
In theory, a system of fundamental quantities (or sometimes fundamental dimensions) would be such that every other physical quantity (or dimension of physical quantity) can be generated from them as algebraic combinations.
Reduction of the number of fundamental quantities and units
- One could eliminate any two of the meter, kilogram and second by setting c and h to unity or to a fixed dimensionless number.
- One could then eliminate the ampere either by setting the permittivity of free space to a fixed dimensionless number or by setting the electronic charge to such a number.
- One could similarly eliminate the mole as a fundamental unit by reference to Avogadro's number.
- One could eliminate the kelvin as it can be argued that temperature simply expresses the energy per particle per degree of freedom which can be expressed in terms of energy (or mass, length, and time). Another way of saying this is that Boltzmann's constant could be expressed as a fixed dimensionless number.
- Similarly, one could eliminate the candela as that is defined in terms of other physical quantities.
- That just leaves one fundamental dimension and one fundamental unit, but we still have plenty of fundamental constants left to eliminate that too - for instance one could use G, the gravitational constant, or me, the electron rest mass.
A widely used choice is the so-called Planck units, which are defined by setting ħ = c = G = 1.
That leaves every physical quantity expressed simply as a dimensionless number, so it is not surprising that there are also physicists who have cast doubt on the very existence of incompatible fundamental quantities. 
- M. J. Duff, L. B. Okun and G. Veneziano, Trialogue on the number of fundamental constants, JHEP 0203, 023 (2002) preprint pdf.
- Jackson, John David (1998). "Appendix on Units and Dimensions". Classical Electrodynamics. John Wiley and Sons. p. 775. Retrieved 13 January 2014.
The arbitrariness in the number of fundamental units and in the dimensions of any physical quantity in terms of those units has been emphasized by Abraham, Plank, Bridgman, Birge, and others.
- Birge, Raymond T. (1935). "On the establishment of fundamental and derived units, with special reference to electric units. Part I.". American Journal of Physics 3: 102–109. Bibcode:1935AmJPh...3..102B. doi:10.1119/1.1992945. Retrieved 13 January 2014.
Because, however, of the arbitrary character of dimensions, as presented so ably by Bridgman, the choice and number of fundamental units are arbitrary.