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 quantity in the subset can be expressed in terms of the others. The SI units, or Systeme International d'unites which consists of the metre, kilogram, second, ampere, Kelvin, mole and candela are base units.

This article is about the type of unit of measure. For the concept in number theory, see Fundamental unit (number theory).

A base unit is one that has been explicitly so designated; a secondary unit for the same quantity is a derived unit. For example, when used with the International System of Units, the gram is a derived unit, not a base unit.

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.

International System of Units

In the International System of Units, there are seven base units: kilogram, metre, candela, second, ampere, kelvin, and mole.

Natural units

Main article: Natural units

A set of fundamental dimensions of physical quantity is a minimal set of units such that every physical quantity can be expressed in terms of this set. The traditional fundamental dimensions of physical quantity are mass, length, time, charge, and temperature, but in principle, other fundamental quantities could be used. Electric current could be used instead of charge or speed could be used instead of length. Some physicists have not recognized temperature as a fundamental dimension of physical quantity since it simply expresses the energy per particle per degree of freedom which can be expressed in terms of energy (or mass, length, and time). In addition, some physicists recognize electric charge as a separate fundamental dimension of physical quantity, even if it has been expressed in terms of mass, length, and time in unit systems such as the electrostatic cgs system. There are also physicists who have cast doubt on the very existence of incompatible fundamental quantities[1].

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. The ontological issue is whether these fundamental constants really exist as dimensional or dimensionless quantities. This is equivalent to treating length as the same commensurable physical material as time or understanding electric charge as a combination of quantities of mass, length, and time which may seem less natural than thinking of temperature as measuring the same material as energy (which is expressible in terms of mass, length, and time).

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, ε₀.

• One could eliminate either the metre or the second by setting c to unity (or to any other fixed dimensionless number).
• One could then eliminate the kilogram by setting ħ to a dimensionless number.
• One could then further eliminate the ampere by setting either the vacuum permittivity ε₀ (alternatively, the Coulomb constant k_e = 1/(4πε₀)) or the elementary charge e to a dimensionless number.
• One could eliminate the mole as a base unit by setting the Avogadro constant N_A to 1. This is natural as it is a technical scaling constant.
• 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 is a technical scaling constant and could be set to a fixed dimensionless number.
• Similarly, one could eliminate the candela, as that is defined in terms of other physical quantities via a technical scaling constant, K_cd.
• That leaves one base dimension and an associated base unit, but there are several 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.

Using natural units leaves every physical quantity expressed as a dimensionless number, which is noted by physicists disputing the existence of incompatible fundamental physical quantities.[1][2][3]

See also

Characteristic units Dimensional analysis Natural units

References

1. Michael Duff (2015). "How fundamental are fundamental constants?". Contemporary Physics. 56 (1): 35–47. arXiv:1412.2040. doi:10.1080/00107514.2014.980093 (inactive 2020-01-22).
2. Jackson, John David (1998). "Appendix on Units and Dimensions" (PDF). 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.
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 (3): 102–109. Bibcode:1935AmJPh...3..102B. doi:10.1119/1.1992945. Archived from the original (PDF) on 23 September 2015. 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.