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The International System of Units (SI)

    The recommended practical system of units of measurement is the International System of Units (Système International d'Unités), with the international abbreviation SI.

    The SI is defined by the SI Brochure, which is published by the BIPM.

    In a landmark decision, the BIPM's Member States voted on 16 November 2018 to revise the SI, changing the world's definition of the kilogram, the ampere, the kelvin and the mole.

    This decision, made at the 26th meeting of the General Conference on Weights and Measures (CGPM), means that from 20 May 2019 all SI units are defined in terms of constants that describe the natural world. This will assure the future stability of the SI and open the opportunity for the use of new technologies, including quantum technologies, to implement the definitions.

    The seven defining constants of the SI are:

    • the caesium hyperfine frequency DeltanuCs;
    • the speed of light in vacuum c;
    • the Planck constant h;
    • the elementary charge e;
    • the Boltzmann constant k;
    • the Avogadro constant NA; and
    • the luminous efficacy of a defined visible radiation Kcd.

    The SI was previously defined in terms of seven base units and derived units defined as products of powers of the base units. The seven base units were chosen for historical reasons, and were, by convention, regarded as dimensionally independent: the metre, the kilogram, the second, the ampere, the kelvin, the mole, and the candela. This role for the base units continues in the present SI even though the SI itself is now defined in terms of the defining constants above.

    The definition of the SI units is established in terms of a set of seven defining constants. The complete system of units can be derived from the fixed values of these defining constants, expressed in the units of the SI. These seven defining constants are the most fundamental feature of the definition of the entire system of units.

    The seven defining constants of the SI and the seven corresponding units they define:

    Defining constant Symbol Numerical value Unit
    hyperfine transition frequency of Cs DeltanuCs 9 192 631 770 Hz
    speed of light in vacuum c 299 792 458 m s–1
    Planck constant h 6.626 070 15 x 10–34 J s
    elementary charge e 1.602 176 634 x 10–19 C
    Boltzmann constant k 1.380 649 x 10–23 J K–1
    Avogadro constant NA 6.022 140 76 x 1023 mol–1
    luminous efficacy Kcd 683 lm W–1
    1

    These particular constants were chosen after having been identified as being the best choice, taking into account the previous definition of the SI, which was based on seven base units, and progress in science.

    The definitions below 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, since the product of the numerical value and the unit has to equal the value of the constant, which is postulated to be invariant. The seven constants are chosen in such a way that any unit of the SI can be written either through a defining constant itself or through products or quotients of defining constants.

    The International System of Units, the SI, is the system of units in which

    • the unperturbed ground state hyperfine transition frequency of the caesium 133 atom DeltanuCs is 9 192 631 770 Hz,
    • the speed of light in vacuum c is 299 792 458 m/s,
    • the Planck constant h is 6.626 070 15 x 10–34 J s,
    • the elementary charge e is 1.602 176 634 x 10–19 C,
    • the Boltzmann constant k is 1.380 649 x 10–23 J/K,
    • the Avogadro constant NA is 6.022 140 76 x 1023 mol–1,
    • the luminous efficacy of monochromatic radiation of frequency 540 x 1012 Hz, Kcd, is 683 lm/W.

    where the hertz, joule, coulomb, lumen, and watt, with unit symbols Hz, J, C, lm, and W, respectively, are related to the units second, metre, kilogram, ampere, kelvin, mole, and candela, with unit symbols s, m, kg, A, K, mol, and cd, respectively, according to Hz = s–1, J = kg m2 s–2, C = A s, lm = cd m2 m–2 = cd sr, and W = kg m2 s–3.

    The seven constants are chosen in such a way that any unit of the SI can be written either through a defining constant itself or through products or quotients of defining constants.

    The numerical values of the seven defining constants have no uncertainty.

    The SI base units:

    Base quantity
    Base unit
    Name Typical symbol Name Symbol
    time t second s
    length l, x, r, etc. metre m
    mass m kilogram kg
    electric current I, i ampere A
    thermodynamic temperature T kelvin K
    amount of substance n mole mol
    luminous intensity Iv candela cd

Definitions

Starting from the definition of the SI in terms of fixed numerical values of the defining constants, definitions of each of the seven base units are deduced by using, as appropriate, one or more of these defining constants to give the following set of definitions:

The second

    The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency DeltanuCs, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s–1.

    This definition implies the exact relation DeltanuCs = 9 192 631 770 Hz. Inverting this relation gives an expression for the unit second in terms of the defining constant DeltanuCs:

    or

    The effect of this definition is that the second is equal to the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the unperturbed ground state of the 133Cs atom.

The metre

    The metre, 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 caesium frequency DeltanuCs.

    This definition implies the exact relation c = 299 792 458 m s–1. Inverting this relation gives an exact expression for the metre in terms of the defining constants c and DeltanuCs:

    The effect of this definition is that one metre is the length of the path travelled by light in vacuum during a time interval with duration of 1/299 792 458 of a second.

The 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 x 10–34 when expressed in the unit J s, which is equal to kg m2 s–1, where the metre and the second are defined in terms of c and DeltanuCs.

    This definition implies the exact relation h = 6.626 070 15 x 10–34 kg m2 s–1. Inverting this relation gives an exact expression for the kilogram in terms of the three defining constants h, DeltanuCs and c:

    which is equal to

    The effect of this definition is to define the unit kg m2 s–1 (the unit of both the physical quantities action and angular momentum). Together with the definitions of the second and the metre this leads to a definition of the unit of mass expressed in terms of the Planck constant h.

The 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 x 10–19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of DeltanuCs.

    This definition implies the exact relation e = 1.602 176 634 x 10–19 A s. Inverting this relation gives an exact expression for the unit ampere in terms of the defining constants e and DeltanuCs:

    which is equal to

    The effect of this definition is that one ampere is the electric current corresponding to the flow of 1/(1.602 176 634 x 10–19) elementary charges per second.

The 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 x 10–23 when expressed in the unit J K–1, which is equal to kg m2 s–2 K–1, where the kilogram, metre and second are defined in terms of h, c and DeltanuCs.

    This definition implies the exact relation k = 1.380 649 x 10–23 kg m2 s–2 K–1. Inverting this relation gives an exact expression for the kelvin in terms of the defining constants k, h and DeltanuCs:

    which is equal to

    The effect of this definition is that one kelvin is equal to the change of thermodynamic temperature that results in a change of thermal energy k T by 1.380 649 x 10–23 J.

The mole

    The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.022 140 76 x 1023 elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, 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 particle or specified group of particles.

    This definition implies the exact relation NA = 6.022 140 76 x 1023 mol–1. Inverting this relation gives an exact expression for the mole in terms of the defining constant NA:

    The effect of this definition is that the mole is the amount of substance of a system that contains 6.022 140 76 x 1023 specified elementary entities.

The 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 x 1012 Hz, Kcd, 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–2 s3, where the kilogram, metre and second are defined in terms of h, c and DeltanuCs.

    This definition implies the exact relation Kcd = 683 cd sr kg–1 m–2 s3 for monochromatic radiation of frequency nu = 540 x 1012 Hz. Inverting this relation gives an exact expression for the candela in terms of the defining constants Kcd, h and DeltanuCs:

    which is equal to

    The effect of this definition is that one candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 Hz and has a radiant intensity in that direction of (1/683) W/sr.

All other SI units can be derived from these, by multiplying together different powers of the base units.


Decimal multiples and submultiples of SI units can be written using the SI prefixes listed in the table below:


Factor Name Symbol Multiplying Factor
1024
yotta
Y
1 000 000 000 000 000 000 000 000
1021
zetta
Z
1 000 000 000 000 000 000 000
1018
exa
E
1 000 000 000 000 000 000
1015
peta
P
1 000 000 000 000 000
1012
tera
T
1 000 000 000 000
109
giga
G
1 000 000 000
106
mega
M
1 000 000
103
kilo
k
1 000
102
hecto
h
100
101
deca
da
10
10–1
deci
d
0.1
10–2
centi
c
0.01
10–3
milli
m
0.001
10–6
micro
µ
0.000 001
10–9
nano
n
0.000 000 001
10–12
pico
p
0.000 000 000 001
10–15
femto
f
0.000 000 000 000 001
10–18
atto
a
0.000 000 000 000 000 001
10–21
zepto
z
0.000 000 000 000 000 000 001
10–24
yocto
y
0.000 000 000 000 000 000 000 001


For full details please refer to Chapter 3 of the SI Brochure.