Section: 1 | International System of Units (SI) |

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INTERNATIONAL SYSTEM OF UNITS (SI)

The International Bureau of Weights and Measures (BIPM) was established by Article 1 of the Metre Convention, which was signed on May 20, 1875. BIPM is charged with providing the basis for a single, coherent system of measurements to be used throughout the world and operates under the authority of the International Committee of Weights and Measures (CIPM). In 1960, the 11th General Conference on Weights and Measures (CGPM) formally defined and established the International System of Units (SI). Since then, the SI has been periodically updated to take into account advances in science and the need for measurements in new domains.

In 2018, the 26th CGPM (2019) decided that the SI would be based on the fixed numerical values of a set of seven defining constants from which the definitions of the seven base units of the SI would be deduced. This major change was made to allow anchoring of the SI to specific experimental realizations of measurable quantities and to remove the base units from dependence on physical artifacts. The change has been taking place over a number of years, and its completion was made possible by realization of the base unit kilogram separate from its previous physical artifact as stored at BIPM in Sevres, France. This discussion of the the newly constituted SI is based on more complete documentation in References 1 and 2. Because of the importance of the SI in science, much of the discussion below is taken verbatim from these references.

The core of the SI is the seven base units for physical quantities as shown in Table 1.

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TABLE 1. SI Base Units

Base quantity	Name	Symbol
length	meter	m
mass	kilogram	kg
time	second	s
electric current	ampere	A
thermodynamic temperature	kelvin	K
amount of substance	mole	mol
luminous intensity	candela	cd

As of May 20, 2019, the SI is the system of units in which the base units are now defined by the seven fundamental constants given in Table 2. More complete discussions of these constants and their experimental realizations can be found in References 1 and 2.

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TABLE 2. Fundamental Constants Used to Define the SI

Fundamental constant	Symbol	Value
Unperturbed ground-state hyperfine transition frequency of the cesium 133 atom ¹³³Cs	Δν_Cs	9 192 631 770 Hz
Speed of light in vacuum	c	299 792 458 m s^-1
Planck constant	h	6.626 070 15 × 10^-34 J s
Elementary charge	e	1.602 176 634 × 10^-19 C
Boltzmann constant	k	1.380 649 × 10^-23 J K^-1
Avogadro constant	N_A	6.022 140 76 × 10²³ mol^-1
Luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz	K_cd	683 lm W^-1

The hertz, joule, coulomb, lumen, and watt are related to the units second, meter, kilogram, ampere, kelvin, mole, and candela as follows in Table 3. (Note sr is steradian.)

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TABLE 3. Relationship of Important Units to the Basic SI Units

Unit name	Symbol	Relationship
hertz	Hz	s^-1
joule	J	kg m² s^-2
coulomb	C	A s
lumen	lm	cd m² m^-2 = cd sr
watt	W	kg m² s^-3

Table 4 provides the definitions of the base quantities. The defintions replace older definitions as discussed in detail in References 1 and 2. These definitions specify the exact numerical value of each constant when its value is expressed in the corresponding SI unit. By fixing the exact numerical value, the unit becomes defined because the product of the numerical value and the unit has to equal the value of the constant, which is invariant. The defining constants have been chosen such that, when taken together, their units cover all of the units of the SI. In general, there is no one-to-one correspondence between the defining constants and the SI base units, except for the cesium frequency Δν_Cs and the Avogadro constant N_A. Any SI unit is a product of powers of these seven constants and a dimensionless factor.

TABLE 4. Definitions of the SI Base Units as Adopted by BIPM in 2019

ampere:	The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602 176 634 × 10^-19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of Δν_Cs.
candela:	The candela, symbol cd, is the SI unit of luminous intensity in a given direction. It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540 × 10¹² Hz, K_cd, to be 683, when expressed in the unit lm W^-1, which is equal to cd sr W^-1, or cd sr kg^-1 m^-1 s³, where the kilogram, meter, and second are defined in terms of h, c, and Δν_Cs.
kelvin:	The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380 649 × 10^-23 when expressed in the unit J K^-1, which is equal to kg m² s^-2 K^-1, where the kilogram, meter, and second are defined in terms of h, c, and Δν_Cs.
kilogram:	The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 15 × 10^-34 when expressed in the unit J s, which is equal to kg m² s^-1, where the meter and second are defined in terms of c and Δν_Cs.
meter:	The meter, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit m s^-1, where the second is defined in terms of the cesium frequency Δν_Cs.
mole:	The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.022 140 76 × 10²³ elementary entities. This number is the fixed numerical value of the Avogadro constant, N_A, when expressed in the unit mol^-1 and is called the Avogadro number. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particles, or specified group of particles.
second:	The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the cesium frequency Δν_Cs, the unperturbed ground-state hyperfine transition frequency of the cesium 133 atom (¹³³Cs), to be 9 192 631 770 when expressed in the unit Hz, which is equal to s^-1.

SI derived units

Derived units are units that may be expressed in terms of base units by means of the mathematical symbols of multiplication and division (and, in the case of °C, subtraction). Certain derived units have been given special names and symbols, and these special names and symbols may themselves be used in combination with those for base and other derived units to express the units of other quantities. Table 5 lists some examples of derived units expressed directly in terms of base units.

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TABLE 5. Examples of SI Derived Units

Physical quantity	Name	Symbol
area	square meter	m²
volume	cubic meter	m³
speed, velocity	meter per second	m s^-1
acceleration	meter per second squared	m s^-2
wave number	reciprocal meter	m^-1
density, mass density	kilogram per cubic meter	kg m^-3
specific volume	cubic meter per kilogram	m³ kg^-1
current density	ampere per square meter	A m^-2
magnetic field strength	ampere per meter	A m^-1
concentration (of amount of substance)	mole per cubic meter	mol m^-3
luminance	candela per square meter	cd m^-2
refractive index	(the number) one	1^a

^aThe symbol “1” (for the number "one") is generally omitted in combination with a numerical value.

For convenience, certain derived units, which are listed in Table 6, have been given special names and symbols. These names and symbols may themselves be used to express other derived units. The special names and symbols are a compact form for the expression of units that are used frequently. The final column shows how the SI units concerned may be expressed in terms of SI base units. In this column, factors such as m⁰, kg⁰ …, which are all equal to 1, are not shown explicitly.

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TABLE 6. SI Derived Units with Special Names and Symbols

Physical quantity	Name	Symbol	Other SI units	SI base units
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plane angle	radian^(a)	rad	m · m^-1 = 1^(b)
solid angle	steradian^(a)	sr^(c)	m² · m^-2 = 1^(b)
frequency	hertz	Hz	s^-1
force	newton	N	m · kg · s^-2
pressure, stress	pascal	Pa	N m^-2	m^-1 · kg · s^-2
energy, work, quantity of heat	joule	J	N · m	m² · kg · s^-2
power, radiant flux	watt	W	J s^-1	m² · kg · s^-3
electric charge, quantity of electricity	coulomb	C	s · A
electric potential difference, electromotive force	volt	V	W A^-1	m² · kg · s^-3 · A^-1
capacitance	farad	F	C V^-1	m^-2 · kg^-1 · s⁴ · A²
electric resistance	ohm	Ω	V A^-1	m² · kg · s^-3 · A^-2
electric conductance	siemens	S	A V^-1	m^-2 · kg^-1 · s³ · A²
magnetic flux	weber	Wb	V · s	m² · kg · s^-2 · A^-1
magnetic flux density	tesla	T	Wb m^-2	kg · s^-2 · A^-1
inductance	henry	H	Wb A^-1	m² · kg · s^-2 · A^-2
Celsius temperature	degree Celsius^(d)	°C		K
luminous flux	lumen	lm	cd · sr^(c)	m² · m^–2 · cd = cd
illuminance	lux	lx	lm m^-2	m² · m^–4 · cd = m^–2 · cd
activity (of a radionuclide)	becquerel	Bq		s^-1

^(a)The radian and steradian may be used with advantage in expressions for derived units to distinguish between quantities of different nature but the same dimension. Some examples of their use in forming derived units are given in the next table.
^(b)In practice, the symbols rad and sr are used where appropriate, but the derived unit “1” is generally omitted in combination with a numerical value.
^(c)In photometry, the name steradian and the symbol sr are usually retained in expressions for units.
^(d)It is common practice to express a thermodynamic temperature, symbol T, in terms of its difference from the reference temperature T₀ = 273.15 K. The numerical value of a Celsius temperature t expressed in degrees Celsius is given by t/°C = T/K-273.15. The unit °C may be used in combination with SI prefixes, e.g., millidegree Celsius, m°C. Note that there should never be a space between the ° sign and the letter C, and that the symbol for kelvin is K, not °K.

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INTERNATIONAL SYSTEM OF UNITS (SI)

TABLE 1. SI Base Units

TABLE 2. Fundamental Constants Used to Define the SI

TABLE 3. Relationship of Important Units to the Basic SI Units

TABLE 4. Definitions of the SI Base Units as Adopted by BIPM in 2019

SI derived units

TABLE 5. Examples of SI Derived Units

TABLE 6. SI Derived Units with Special Names and Symbols

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