International System of Units

The International System of Units, internationally known by the abbreviation SI (from its official French name, Système international d'unités), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official status in nearly every country in the world, employed in science, technology, industry, and everyday commerce. The International Bureau of Weights and Measures (abbreviated BIPM from French: Bureau international des poids et mesures) coordinates the SI.

The seven SI base units
Symbol	Name	Quantity
s	second	time
m	metre	length
kg	kilogram	mass
A	ampere	electric current
K	kelvin	thermodynamic temperature
mol	mole	amount of substance
cd	candela	luminous intensity

The SI comprises a coherent system of units of measurement starting with seven base units, which are the second (symbol: s, the unit of time), metre (m, length), kilogram (kg, mass), ampere (A, electric current), kelvin (K, thermodynamic temperature), mole (mol, amount of substance), and candela (cd, luminous intensity). The system can accommodate coherent units for an unlimited number of additional quantities. These are called coherent derived units, which can always be represented as products of powers of the base units. Twenty-two coherent derived units have been provided with special names and symbols.

The 7 base units and the 22 coherent derived units with special names and symbols may be used in combination to express other coherent derived units. Since the sizes of coherent units will be convenient for only some applications and not for others, the SI provides 24 prefixes which, when added to the name and symbol of a coherent unit produce 24 additional (non-coherent) SI units for the same quantity; these non-coherent units are always decimal (i.e. power-of-ten) multiples and sub-multiples of the coherent unit.

The current way of defining the SI is a result of a decades-long move towards definitions of the units that do not depend on artefacts made as physical realisations. A consequence is that as science and technologies develop, new and potentially superior realisations may be introduced without the need to redefine the unit. One problem with artefacts is that they can be lost, damaged, or changed; another is that they introduce uncertainties that cannot be reduced by advancements in science and technology.

The original motivation for the development of the SI was the diversity of units that had sprung up within the centimetre–gram–second (CGS) systems (specifically the inconsistency between the systems of electrostatic units and electromagnetic units) and the lack of coordination between the various disciplines that used them. The General Conference on Weights and Measures (French: Conférence générale des poids et mesures – CGPM), which was established by the Metre Convention of 1875, brought together many international organisations to establish the definitions and standards of a new system and to standardise the rules for writing and presenting measurements. The system was published in 1960 as a result of an initiative that began in 1948, and is based on the metre–kilogram–second system of units (MKS) combined with ideas from the development of the CGS system.

Definition

The International System of Units consists of a set of seven defining constants with seven corresponding base units, derived units, and a set of decimal-based multipliers that are used as prefixes.^[1]^: 125

SI defining constants

SI defining constants
Symbol	Defining constant	Exact value and units
$Δ ν Cs$	hyperfine transition frequency of ¹³³Cs	9192631770 Hz
$c$	speed of light	299792458 m/s
$h$	Planck constant	6.62607015×10⁻³⁴ J⋅s
$e$	elementary charge	1.602176634×10⁻¹⁹ C
$k$	Boltzmann constant	1.380649×10⁻²³ J/K
$N A$	Avogadro constant	6.02214076×10²³ mol⁻¹
$K cd$	luminous efficacy of 540 THz radiation	683 lm/W

The seven defining constants are the most fundamental feature of the definition of the system of units. Each defining constant consists of an exact numerical value and units.^[1]^: 125 The defining constants are the speed of light in vacuum $c$ , the hyperfine transition frequency of caesium $Δ ν Cs$ , the Planck constant $h$ , the elementary charge $e$ , the Boltzmann constant $k$ , the Avogadro constant $N A$ , and the luminous efficacy $K cd$ . The nature of the defining constants ranges from fundamental constants of nature such as $c$ to the purely technical constant $K cd$ . The values assigned to these constants were fixed to ensure continuity with previous definitions of the base units.^[1]^: 128

SI base units

The SI selects seven units to serve as base units, corresponding to seven base physical quantities. They are the second for time, metre for length, kilogram for mass, ampere for electric current, kelvin for thermodynamic temperature, mole for amount of substance, and candela for luminous intensity.^[1] The base units are defined in terms of the defining constants. For example, the kilogram is defined by taking the Planck constant $h$ to be 6.62607015×10⁻³⁴ J⋅s, giving the expression in terms of the defining constants^[1]^: 131

1 kg = ⁠(299792458)²(6.62607015×10⁻³⁴)(9192631770)⁠⁠

h

Δ ν Cs

c

²⁠.

All units in the SI can be expressed in terms of the base units, and the base units serve as a preferred set for expressing or analysing the relationships between units. The choice of which and even how many quantities to use as base quantities is not fundamental or even unique – it is a matter of convention.^[1]^: 126

SI base units^[1]^: 136
Unit name	Unit symbol	Dimension symbol	Quantity name	Typical symbols	Definition
second	s	${\mathsf {T}}$	time	$t$	The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.
metre	m	${\mathsf {L}}$	length	$l$ , $x$ , $r$	The distance travelled by light in vacuum in ⁠1/299792458⁠ second.
kilogram ^{[n 1]}	kg	${\mathsf {M}}$	mass	$m$	The kilogram is defined by setting the Planck constant $h$ to 6.62607015×10⁻³⁴ J⋅s (J = kg⋅m²⋅s⁻²), given the definitions of the metre and the second.^[2]
ampere	A	${\mathsf {I}}$	electric current	$I,\;i$	The flow of ⁠1/1.602176634×10⁻¹⁹⁠ times the elementary charge $e$ per second, which is approximately 6.2415090744×10¹⁸ elementary charges per second.
kelvin	K	${\mathsf {\Theta }}$	thermodynamic temperature	$T$	The kelvin is defined by setting the fixed numerical value of the Boltzmann constant $k$ to 1.380649×10⁻²³ J⋅K⁻¹, (J = kg⋅m²⋅s⁻²), given the definition of the kilogram, the metre, and the second.
mole	mol	${\mathsf {N}}$	amount of substance	$n$	The amount of substance of 6.02214076×10²³ elementary entities.^{[n 2]} This number is the fixed numerical value of the Avogadro constant, $N A$ , when expressed in the unit mol⁻¹.
candela	cd	${\mathsf {J}}$	luminous intensity	$I_{\rm {v}}$	The candela 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.
Notes ↑ Despite the prefix "kilo-", the kilogram is the coherent base unit of mass, and is used in the definitions of derived units. Nonetheless, prefixes for the unit of mass are determined as if the gram were the base unit. ↑ When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

Derived units

The system allows for an unlimited number of additional units, called derived units, which can always be represented as products of powers of the base units, possibly with a nontrivial numeric multiplier. When that multiplier is one, the unit is called a coherent derived unit. For example, the coherent derived SI unit of velocity is the metre per second, with the symbol m/s.^[1]^: 139 The base and coherent derived units of the SI together form a coherent system of units (the set of coherent SI units). A useful property of a coherent system is that when the numerical values of physical quantities are expressed in terms of the units of the system, then the equations between the numerical values have exactly the same form, including numerical factors, as the corresponding equations between the physical quantities.^[3]^: 6

Twenty-two coherent derived units have been provided with special names and symbols as shown in the table below. The radian and steradian have no base units but are treated as derived units for historical reasons.^[1]^: 137

The 22 SI derived units with special names and symbols^[1]^: 137
Name	Symbol	Quantity	In SI base units	In other SI units
radian^{[nc 1]}	rad	plane angle		1
steradian^{[nc 1]}	sr	solid angle		1
hertz	Hz	frequency	s⁻¹
newton	N	force	kg⋅m⋅s⁻²
pascal	Pa	pressure, stress	kg⋅m⁻¹⋅s⁻²	N/m²
joule	J	energy, work, amount of heat	kg⋅m²⋅s⁻²	N⋅m
watt	W	power, radiant flux	kg⋅m²⋅s⁻³	J/s
coulomb	C	electric charge	s⋅A
volt	V	electric potential difference^[a]	kg⋅m²⋅s⁻³⋅A⁻¹	W/A
ohm	Ω	electrical resistance	kg⋅m²⋅s⁻³⋅A⁻²	V/A
siemens	S	electrical conductance	kg⁻¹⋅m⁻²⋅s³⋅A²	A/V
farad	F	capacitance	kg⁻¹⋅m⁻²⋅s⁴⋅A²	C/V
henry	H	inductance	kg⋅m²⋅s⁻²⋅A⁻²	Wb/A
tesla	T	magnetic flux density	kg⋅s⁻²⋅A⁻¹	Wb/m²
weber	Wb	magnetic flux	kg⋅m²⋅s⁻²⋅A⁻¹	V⋅s
degree Celsius	°C	Celsius temperature	K
lumen	lm	luminous flux	cd⋅sr^{[nc 2]}	cd⋅sr
lux	lx	illuminance	cd⋅sr⋅m⁻²^{[nc 2]}	lm/m²
becquerel	Bq	activity referred to a radionuclide	s⁻¹
gray	Gy	absorbed dose, kerma	m²⋅s⁻²	J/kg
sievert	Sv	dose equivalent	m²⋅s⁻²	J/kg
katal	kat	catalytic activity	mol⋅s⁻¹
Notes 1 2 The radian and steradian are defined as dimensionless derived units. 1 2 In photometry, the steradian is usually retained in expressions for units.

The derived units in the SI are formed by powers, products, or quotients of the base units and are unlimited in number.^[1]^: 138^[4]^: 14, 16

Derived units apply to some derived quantities, which may by definition be expressed in terms of base quantities, and thus are not independent; for example, electrical conductance is the inverse of electrical resistance, with the consequence that the siemens is the inverse of the ohm, and similarly, the ohm and siemens can be replaced with a ratio of an ampere and a volt, because those quantities bear a defined relationship to each other.^[b] Other useful derived quantities can be specified in terms of the SI base and derived units that have no named units in the SI, such as acceleration, which has the SI unit m/s².^[1]^: 139

A combination of base and derived units may be used to express a derived unit. For example, the SI unit of force is the newton (N), the SI unit of pressure is the pascal (Pa) – and the pascal can be defined as one newton per square metre (N/m²).^[5]

Prefixes

Like all metric systems, the SI uses metric prefixes to systematically construct, for the same physical quantity, a set of units that are decimal multiples of each other over a wide range. For example, driving distances are normally given in kilometres (symbol km) rather than in metres. Here the metric prefix 'kilo-' (symbol 'k') stands for a factor of 1000; thus, 1 km = 1000 m.

The SI provides twenty-four metric prefixes that signify decimal powers ranging from 10⁻³⁰ to 10³⁰, the most recent being adopted in 2022.^[1]^{: 143–144}^[6]^[7]^[8] Most prefixes correspond to integer powers of 1000; the only ones that do not are those for 10, 1/10, 100, and 1/100. The conversion between different SI units for one and the same physical quantity is always through a power of ten. This is why the SI (and metric systems more generally) is called decimal systems of measurement units.^[9]

The grouping formed by a prefix symbol attached to a unit symbol (e.g. 'km', 'cm') constitutes a new inseparable unit symbol. This new symbol can be raised to a positive or negative power. It can also be combined with other unit symbols to form compound unit symbols.^[1]^: 143 For example, g/cm³ is an SI unit of density, where cm³ is to be interpreted as (cm)³.

Prefixes are added to unit names to produce multiples and submultiples of the original unit. All of these are integer powers of ten, and above a hundred or below a hundredth all are integer powers of a thousand. For example, kilo- denotes a multiple of a thousand and milli- denotes a multiple of a thousandth, so there are one thousand millimetres to the metre and one thousand metres to the kilometre. The prefixes are never combined, so for example a millionth of a metre is a micrometre, not a millimillimetre. Multiples of the kilogram are named as if the gram were the base unit, so a millionth of a kilogram is a milligram, not a microkilogram.^[10]^: 122^[11]^: 14

The BIPM specifies 24 prefixes for the International System of Units (SI):

SI prefixes
v
t
e
Prefix		Base 10	Decimal	Adoption ^{[nb 1]}
Name	Symbol	Base 10	Decimal	Adoption ^{[nb 1]}
quetta	Q	10³⁰	1000000000000000000000000000000	2022^[12]
ronna	R	10²⁷	1000000000000000000000000000	2022^[12]
yotta	Y	10²⁴	1000000000000000000000000	1991
zetta	Z	10²¹	1000000000000000000000	1991
exa	E	10¹⁸	1000000000000000000	1975^[13]
peta	P	10¹⁵	1000000000000000	1975^[13]
tera	T	10¹²	1000000000000	1960
giga	G	10⁹	1000000000	1960
mega	M	10⁶	1000000	1873
kilo	k	10³	1000	1795
hecto	h	10²	100
deca	da	10¹	10
—	—	10⁰	1	—
deci	d	10⁻¹	0.1	1795
centi	c	10⁻²	0.01
milli	m	10⁻³	0.001
micro	μ	10⁻⁶	0.000001	1873
nano	n	10⁻⁹	0.000000001	1960
pico	p	10⁻¹²	0.000000000001	1960
femto	f	10⁻¹⁵	0.000000000000001	1964
atto	a	10⁻¹⁸	0.000000000000000001	1964
zepto	z	10⁻²¹	0.000000000000000000001	1991
yocto	y	10⁻²⁴	0.000000000000000000000001	1991
ronto	r	10⁻²⁷	0.000000000000000000000000001	2022^[12]
quecto	q	10⁻³⁰	0.000000000000000000000000000001	2022^[12]
Notes ↑ Prefixes adopted before 1960 already existed before SI. The introduction of the centimetre–gram–second system of units was in 1873.

Coherent and non-coherent SI units

The base units and the derived units formed as the product of powers of the base units with a numerical factor of one form a coherent system of units. Every physical quantity has exactly one coherent SI unit. For example, 1 m/s = (1 m) / (1 s) is the coherent derived unit for velocity.^[1]^: 139 With the exception of the kilogram (for which the prefix kilo- is required for a coherent unit), when prefixes are used with the coherent SI units, the resulting units are no longer coherent, because the prefix introduces a numerical factor other than one.^[1]^: 137 For example, the metre, kilometre, centimetre, nanometre, etc. are all SI units of length, though only the metre is a coherent SI unit. The complete set of SI units consists of both the coherent set and the multiples and sub-multiples of coherent units formed by using the SI prefixes.^[1]^: 138

The kilogram is the only coherent SI unit whose name and symbol include a prefix. For historical reasons, the names and symbols for multiples and sub-multiples of the unit of mass are formed as if the gram were the base unit. Prefix names and symbols are attached to the unit name gram and the unit symbol g respectively. For example, 10⁻⁶ kg is written milligram and mg, not microkilogram and μkg.^[1]^: 144

The same coherent SI unit may be used for different physical quantities. For example, the joule per kelvin (symbol J/K) is the coherent SI unit for two distinct quantities, heat capacity and entropy, and the ampere is the coherent SI unit for both electric current and magnetomotive force.^[1]^: 140

Furthermore, the same coherent SI unit may be a base unit in one context, but a coherent derived unit in another. For example, the ampere is a base unit when it is a unit of electric current, but a coherent derived unit when it is a unit of magnetomotive force.^[1]^: 140

Examples of coherent derived units in terms of base units^[4]^: 17
Name	Symbol	Derived quantity	Typical symbol
square metre	m²	area	$A$
cubic metre	m³	volume	$V$
metre per second	m/s	speed, velocity	$v$
metre per second squared	m/s²	acceleration	$a$
reciprocal metre	m⁻¹	wavenumber	$σ$ , $ṽ$
reciprocal metre	m⁻¹	vergence (optics)	$V$ , 1/ $f$
kilogram per cubic metre	kg/m³	density	$ρ$
kilogram per square metre	kg/m²	surface density	$ρ A$
cubic metre per kilogram	m³/kg	specific volume	$v$
ampere per square metre	A/m²	current density	$j$
ampere per metre	A/m	magnetic field strength	$H$
mole per cubic metre	mol/m³	concentration	$c$
kilogram per cubic metre	kg/m³	mass concentration	$ρ$ , $γ$
candela per square metre	cd/m²	luminance	$L v$

Examples of derived units that include units with special names^[4]^: 18
Name	Symbol	Quantity	In SI base units
pascal-second	Pa⋅s	dynamic viscosity	m⁻¹⋅kg⋅s⁻¹
newton-metre	N⋅m	moment of force	m²⋅kg⋅s⁻²
newton per metre	N/m	surface tension	kg⋅s⁻²
radian per second	rad/s	angular velocity, angular frequency	s⁻¹
radian per second squared	rad/s²	angular acceleration	s⁻²
watt per square metre	W/m²	heat flux density, irradiance	kg⋅s⁻³
joule per kelvin	J/K	entropy, heat capacity	m²⋅kg⋅s⁻²⋅K⁻¹
joule per kilogram-kelvin	J/(kg⋅K)	specific heat capacity, specific entropy	m²⋅s⁻²⋅K⁻¹
joule per kilogram	J/kg	specific energy	m²⋅s⁻²
watt per metre-kelvin	W/(m⋅K)	thermal conductivity	m⋅kg⋅s⁻³⋅K⁻¹
joule per cubic metre	J/m³	energy density	m⁻¹⋅kg⋅s⁻²
volt per metre	V/m	electric field strength	m⋅kg⋅s⁻³⋅A⁻¹
coulomb per cubic metre	C/m³	electric charge density	m⁻³⋅s⋅A
coulomb per square metre	C/m²	surface charge density, electric flux density, electric displacement	m⁻²⋅s⋅A
farad per metre	F/m	permittivity	m⁻³⋅kg⁻¹⋅s⁴⋅A²
henry per metre	H/m	permeability	m⋅kg⋅s⁻²⋅A⁻²
joule per mole	J/mol	molar energy	m²⋅kg⋅s⁻²⋅mol⁻¹
joule per mole-kelvin	J/(mol⋅K)	molar entropy, molar heat capacity	m²⋅kg⋅s⁻²⋅K⁻¹⋅mol⁻¹
coulomb per kilogram	C/kg	exposure (X- and γ-rays)	kg⁻¹⋅s⋅A
gray per second	Gy/s	absorbed dose rate	m²⋅s⁻³
watt per steradian	W/sr	radiant intensity	m²⋅kg⋅s⁻³
watt per square metre-steradian	W/(m²⋅sr)	radiance	kg⋅s⁻³
katal per cubic metre	kat/m³	catalytic activity concentration	m⁻³⋅s⁻¹⋅mol

Lexicographic conventions

Example of lexical conventions. In the expression of acceleration due to gravity, a space separates the value and the units, both the 'm' and the 's' are lowercase because neither the metre nor the second are named after people, and exponentiation is represented with a superscript '2'.

Unit names

The SI standard is that unit names are treated as common nouns of the context language.^[1]^: 148 This means they are typeset in the same character set as other common nouns (e.g. Latin alphabet in English,

[2] Despite the prefix "kilo-", the kilogram is the coherent base unit of mass, and is used in the definitions of derived units. Nonetheless, prefixes for the unit of mass are determined as if the gram were the base unit.

[4] When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

[:0-6] 1 2 The radian and steradian are defined as dimensionless derived units.

[keep-sr-8] 1 2 In photometry, the steradian is usually retained in expressions for units.

[18] Prefixes adopted before 1960 already existed before SI. The introduction of the centimetre–gram–second system of units was in 1873.

[1]

[n 1]

[2]

[n 2]

[3]

[nc 1]

[a]

[nc 2]

[4]

[b]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[nb 1]

[12]

[13]