Base unit (measurement)

A base unit (also referred to as a fundamental unit) is a unit adopted for measurement of a base quantity. A base quantity is one of a conventionally chosen subset of physical quantities, where no subset quantity can be expressed in terms of the others. In the International System of Units, there are seven base units: kilogram, metre, candela, second, ampere, kelvin, and mole.

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 magnitude or quantity. Many of these quantities are related to each other by various physical laws, and as a result the units of a quantities can be generally be expressed as a product of powers of other units; for example, momentum is mass multiplied by velocity, while velocity is measured in distance divided by time. These relationships are discussed in dimensional analysis. Those that can be expressed in this fashion in terms of the base units are called derived units.

Natural units

There are other relationships between physical quantities that 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 explicit 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 unit of time or that of distance. Similar considerations apply to the Planck constant, h, which relates energy (with dimension expressible in terms of mass, length and time) to frequency (with dimension expressible in terms of time). In theoretical physics it is customary to use such units (natural units) in which c = 1 and ħ = 1. A similar choice can be applied to the vacuum permittivity or permittivity of free space, ε₀.

• One could eliminate any of the metre and second by setting c to unity (or to any other fixed dimensionless number).
• One could then eliminate the kilogram by setting h to a dimensionless number.
• One could then further eliminate the ampere by setting either the permittivity of free space ε₀ (equivalently, the Coulomb constant k_e = 1/(4πε₀)) or the elementary charge e to a dimensionless number.
• One could similarly eliminate the mole as a base 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 k_B 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 leaves one base dimension and an associated base unit, but we still have plenty of fundamental constants left to eliminate that too – for instance one could use G, the gravitational constant, m_e, the electron rest mass, or Λ, the cosmological constant.

A widely used choice, in particular for theoretical physics, is given by the system of Planck units, which are defined by setting ħ = c = G = k_B = k_e = 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.^[1][2][3]

See also

Characteristic units Dimensional analysis Natural units

References

1. M. J. Duff, L. B. Okun and G. Veneziano, Trialogue on the number of fundamental constants, JHEP 0203, 023 (2002) preprint pdf.
2. Jackson, John David (1998). "Appendix on Units and Dimensions". Classical Electrodynamics (PDF). 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.
3. Birge, Raymond T. (1935). "On the establishment of fundamental and derived units, with special reference to electric units. Part I." (PDF). 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.