# International System of Units

SI defining constants
Symbol Defining constant Exact value
ΔνCs hyperfine transition frequency of Cs 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
Kcd luminous efficacy of 540 THz radiation 683 lm/W
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 International System of Units, known by the international abbreviation SI[a] in all languages[1]: 125 [2]: iii [3] and sometimes pleonastically as the SI system,[b] is the modern form[1]: 117 [6][7] of the metric system[g] and the world's most widely used system of measurement.[1]: 123 [9][10] Established and maintained[11] by the General Conference on Weights and Measures[j] (CGPM[k]), it is the only system of measurement with an official status[m] in nearly every country in the world,[n] employed in science, technology, industry, and everyday commerce.

The SI comprises a

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.[p] Twenty-two coherent derived units have been provided with special names and symbols.[q]

The seven base units and the 22 coherent derived units with special names and symbols may be used in combination to express other coherent derived units.[r] Since the sizes of coherent units will be convenient for only some applications and not for others, the SI provides twenty-four prefixes which, when added to the name and symbol of a coherent unit[s] produce twenty-four 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.[t][u] The SI is intended to be an evolving system; units and prefixes are created and unit definitions are modified through international agreement as the technology of measurement progresses and the precision of measurements improves.

Since 2019, the magnitudes of all SI units have been defined by declaring that seven defining constants have certain exact numerical values when expressed in terms of their SI units. These defining constants are the speed of light in vacuum c, the hyperfine transition frequency of caesium ΔνCs, the Planck constant h, the elementary charge e, the Boltzmann constant k, the Avogadro constant NA, and the luminous efficacy Kcd. The nature of the defining constants ranges from fundamental constants of nature such as c to the purely technical constant Kcd. Prior to 2019, h, e, k, and NA were not defined a priori but were rather very precisely measured quantities. In 2019, their values were fixed by definition to their best estimates at the time, ensuring continuity with previous definitions of the base units.

The current way of defining the SI is a result of a decades-long move towards increasingly abstract and idealised formulation in which the realisations of the units are separated conceptually from the definitions. A consequence is that as science and technologies develop, new and 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 last artefact used by the SI was the International Prototype of the Kilogram, a cylinder of platinum–iridium.

The original motivation for the development of the SI was the diversity of units that had sprung up within the

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, so it is based on the metre–kilogram–second system of units
(MKS) rather than any variant of the CGS.

## Introduction

US customary
systems as of 2019.

The International System of Units, or SI,

system of units established in 1960 and periodically updated since then. The SI has an official status in most countries,[x] including the United States,[y] Canada, and the United Kingdom, although these three countries are amongst a handful of nations that, to various degrees, also continue to use their customary systems. Nevertheless, with this nearly universal level of acceptance, the SI "has been used around the world as the preferred system of units, the basic language for science, technology, industry and trade."[1]
: 123

The only other types of measurement system that still have widespread use across the world are the Imperial and US customary measurement systems,[z] and they are legally defined in terms of the SI.[aa] There are other, less widespread systems of measurement that are occasionally used in particular regions of the world. In addition, there are many individual non-SI units that don't belong to any comprehensive system of units, but that are nevertheless still regularly used in particular fields and regions. Both of these categories of unit are also typically defined legally in terms of SI units.[ab]

### Controlling body

The SI was established and is maintained by the

International Committee for Weights and Measures (CIPM[ac]), which, in turn, reports to the CGPM. See below
for more details.

All the decisions and recommendations concerning units are collected in a brochure called The International System of Units (SI),[ad] which is published by the International Bureau of Weights and Measures (BIPM[ae]) and periodically updated.

### Overview of the units

#### SI base units

The SI selects seven units to serve as base units, corresponding to seven base physical quantities.[af][ag] They are the second, with the symbol s, which is the SI unit of the physical quantity of time; the metre, symbol m, the SI unit of length; kilogram (kg, the unit of mass); ampere (A, electric current); kelvin (K, thermodynamic temperature); mole (mol, amount of substance); and candela (cd, luminous intensity).[1] 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.

#### SI 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.[ah] The base and coherent derived units of the SI together form a coherent system of units (the set of coherent SI units).[ai] Twenty-two coherent derived units have been provided with special names and symbols.[q] The seven base units and the 22 derived units with special names and symbols may be used in combination to express other derived units,[r] which are adopted to facilitate measurement of diverse quantities.

#### Why SI kept the distinction between base and derived units

Prior to its redefinition in 2019, the SI was defined through the seven base units from which the derived units were constructed as products of powers of the base units. After the redefinition, the SI is defined by fixing the numerical values of seven defining constants. This has the effect that the distinction between the base units and derived units is, in principle, not needed, since all units, base as well as derived, may be constructed directly from the defining constants. Nevertheless, the distinction is retained because 'it is useful and historically well established', and also because the ISO/IEC 80000 series of standards[aj] specifies base and derived quantities that necessarily have the corresponding SI units.[1]: 129

#### SI metric prefixes and the decimal nature of the SI

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, while the coherent unit of length is the metre,[ak] the SI provides a full range of smaller and larger units of length, any of which may be more convenient for any given application – 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.[al]

The current version of 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]: 143–4 [17][18] Most prefixes correspond to integer powers of 1000; the only ones that do not are those for 10, 1/10, 100, and 1/100.

In general, given any coherent unit with a separate name and symbol,[am] one forms a new unit by simply adding an appropriate metric prefix to the name of the coherent unit (and a corresponding prefix symbol to the coherent unit's symbol).[an] Since the metric prefix signifies a particular power of ten, the new unit is always a power-of-ten multiple or sub-multiple of the coherent unit. Thus, the conversion between different SI units for one and the same physical quantity is always through a power of ten.[ao] This is why the SI (and metric systems more generally) are called decimal systems of measurement units.[19][ap]

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 and can 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.

#### Coherent and non-coherent SI units

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  The one exception is the kilogram, the only coherent SI unit whose name and symbol, for historical reasons, include a prefix.[an]

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.

Pg
/km3, etc. are all SI units of density, but of these, only kg/m3 is a coherent SI unit.

Moreover, the metre is the only coherent SI unit of length. Every physical quantity has exactly one coherent SI unit, although this unit may be expressible in different forms by using some of the special names and symbols.[1]: 140  For example, the coherent SI unit of linear momentum may be written as either kg⋅m/s or as N⋅s, and both forms are in use (e.g. compare respectively here[20]:205 and here[21]:135).

On the other hand, several different quantities may share the same coherent SI unit. For example, the joule per kelvin (symbol J/K) is the coherent SI unit for two distinct quantities: heat capacity and entropy; another example is the ampere, which is the coherent SI unit for both electric current and magnetomotive force. This is why it is important not to use the unit alone to specify the quantity.[aq]

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  As perhaps a more familiar example, consider rainfall, defined as volume of rain (measured in m3) that fell per unit area (measured in m2). Since m3/m2 = m, it follows that the coherent derived SI unit of rainfall is the metre, even though the metre is also the base SI unit of length.[ar]

#### Permitted non-SI units

There is a special group of units that are called "non-SI units that are accepted for use with the SI".[1]: 145  See Non-SI units mentioned in the SI for a full list. Most of these, in order to be converted to the corresponding SI unit, require conversion factors that are not powers of ten. Some common examples of such units are the customary units of time, namely the minute (conversion factor of 60 s/min, since 1 min = 60 s), the hour (3600 s), and the day (86400 s); the degree (for measuring plane angles, = π/180 rad); and the electronvolt (a unit of energy, 1 eV = 1.602176634×10−19 J).

#### New units

The SI is intended to be an evolving system; units[as] and prefixes are created and unit definitions are modified through international agreement as the technology of measurement progresses and the precision of measurements improves.

### Defining magnitudes of units

Since 2019, the magnitudes of all SI units have been defined in an abstract way, which is conceptually separated from any practical realisation of them.[1]: 126 [at] Namely, the SI units are defined by declaring that seven defining constants[1]: 125–9  have certain exact numerical values when expressed in terms of their SI units. Probably the most widely known of these constants is the speed of light in vacuum, c, which in the SI by definition has the exact value of c = 299792458 m/s. The other six constants are ΔνCs, the hyperfine transition frequency of caesium; h, the Planck constant; e, the elementary charge; k, the Boltzmann constant; NA, the Avogadro constant; and Kcd, the luminous efficacy of monochromatic radiation of frequency 540×1012 Hz.[au] The nature of the defining constants ranges from fundamental constants of nature such as c to the purely technical constant Kcd.[1]: 128–9  Prior to 2019, h, e, k, and NA were not defined a priori but were rather very precisely measured quantities. In 2019, their values were fixed by definition to their best estimates at the time, ensuring continuity with previous definitions of the base units.

As far as realisations, what are believed to be the current best practical realisations of units are described in the mises en pratique,[av] which are also published by the BIPM.[24] The abstract nature of the definitions of units is what makes it possible to improve and change the mises en pratique as science and technology develop without having to change the actual definitions themselves.[ay]

In a sense, this way of defining the SI units is no more abstract than the way derived units are traditionally defined in terms of the base units. Consider a particular derived unit, for example, the joule, the unit of energy. Its definition in terms of the base units is kgm2/s2. Even if the practical realisations of the metre, kilogram, and second are available, a practical realisation of the joule would require some sort of reference to the underlying physical definition of work or energy—some actual physical procedure for realising the energy in the amount of one joule such that it can be compared to other instances of energy (such as the energy content of gasoline put into a car or of electricity delivered to a household).

The situation with the defining constants and all of the SI units is analogous. In fact, purely mathematically speaking, the SI units are defined as if we declared that it is the defining constant's units that are now the base units, with all other SI units being derived units. To make this clearer, first note that each defining constant can be taken as determining the magnitude of that defining constant's unit of measurement;[1]: 128  for example, the definition of c defines the unit m/s as 1 m/s = c/299792458 ('the speed of one metre per second is equal to one 299792458th of the speed of light'). In this way, the defining constants directly define the following seven units:

(conversion constant between the physical power carried by electromagnetic radiation and the intrinsic ability of that same radiation to produce visual perception of brightness in humans).

Further, one can show, using dimensional analysis, that every coherent SI unit (whether base or derived) can be written as a unique product of powers of the units of the SI defining constants (in complete analogy to the fact that every coherent derived SI unit can be written as a unique product of powers of the base SI units). For example, the kilogram can be written as kg = (Hz)(J⋅s)/(m/s)2.[az] Thus, the kilogram is defined in terms of the three defining constants ΔνCs, c, and h because, on the one hand, these three defining constants respectively define the units Hz, m/s, and J⋅s,[ba] while, on the other hand, the kilogram can be written in terms of these three units, namely, kg = (Hz)(J⋅s)/(m/s)2.[bb] While the question of how to actually realise the kilogram in practice would, at this point, still be open, that is not really different from the fact that the question of how to actually realise the joule in practice is still in principle open even once one has achieved the practical realisations of the metre, kilogram, and second.

### Specifying fundamental constants vs. other methods of definition

The current way of defining the SI is the result of a decades-long move towards increasingly abstract and idealised formulation in which the realisations of the units are separated conceptually from the definitions.[1]: 126

The great advantage of doing it this way is that as science and technologies develop, new and superior realisations may be introduced without the need to redefine the units.[aw] Units can now be realised with an accuracy that is ultimately limited only by the quantum structure of nature and our technical abilities but not by the definitions themselves.[ax] Any valid equation of physics relating the defining constants to a unit can be used to realise the unit, thus creating opportunities for innovation... with increasing accuracy as technology proceeds.'[1]: 122  In practice, the CIPM Consultative Committees provide so-called "mises en pratique" (practical techniques),[24] which are the descriptions of what are currently believed to be best experimental realisations of the units.[28]

This system lacks the conceptual simplicity of using artefacts (referred to as prototypes) as realisations of units to define those units: with prototypes, the definition and the realisation are one and the same.

revision of the definition of the base units, put into effect on 20 May 2019.[35] This was the biggest change in the SI since it was first formally defined and established in 1960, and it resulted in the definitions described above.[36]

In the past, there were also various other approaches to the definitions of some of the SI units. One made use of a specific physical state of a specific substance (the triple point of water, which was used in the definition of the kelvin[37]: 113–4 ); others referred to idealised experimental prescriptions[1]: 125  (as in the case of the former SI definition of the ampere[37]: 113  and the former SI definition (originally enacted in 1979) of the candela[37]: 115 ).

In the future, the set of defining constants used by the SI may be modified as more stable constants are found, or if it turns out that other constants can be more precisely measured.[bk]

### History

The original motivation for the development of the SI was the diversity of units that had sprung up within the

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.

Adopted in 1889, use of the MKS system of units succeeded the centimetre–gram–second system of units (CGS) in commerce and engineering. The metre and kilogram system served as the basis for the development of the International System of Units (abbreviated SI), which now serves as the international standard. Because of this, the standards of the CGS system were gradually replaced with metric standards incorporated from the MKS system.[38]

In 1901, Giovanni Giorgi proposed to the Associazione elettrotecnica italiana [it] (AEI) that this system, extended with a fourth unit to be taken from the units of electromagnetism, be used as an international system.[39] This system was strongly promoted by electrical engineer

George A. Campbell.[40]

The International System was published in 1960, based on the MKS units, as a result of an initiative that began in 1948.

## Controlling authority

The SI is regulated and continually developed by three international organisations that were established in 1875 under the terms of the Metre Convention. They are the General Conference on Weights and Measures (CGPM[k]), the International Committee for Weights and Measures (CIPM[ac]), and the International Bureau of Weights and Measures (BIPM[ae]). The ultimate authority rests with the CGPM, which is a plenary body through which its Member States[bl] act together on matters related to measurement science and measurement standards; it usually convenes every four years.[12] The CGPM elects the CIPM, which is an 18-person committee of eminent scientists. The CIPM operates based on the advice of a number of its Consultative Committees, which bring together the world's experts in their specified fields as advisers on scientific and technical matters.[41][bm] One of these committees is the Consultative Committee for Units (CCU), which is responsible for matters related to the development of the International System of Units (SI), preparation of successive editions of the SI brochure, and advice to the CIPM on matters concerning units of measurement.[42] It is the CCU which considers in detail all new scientific and technological developments related to the definition of units and the SI. In practice, when it comes to the definition of the SI, the CGPM simply formally approves the recommendations of the CIPM, which, in turn, follows the advice of the CCU.

The CCU has the following as members:[43][44] national laboratories of the Member States of the CGPM charged with establishing national standards;[bn] relevant intergovernmental organisations and international bodies;[bo] international commissions or committees;[bp] scientific unions;[bq] personal members;[br] and, as an ex officio member of all Consultative Committees, the Director of the BIPM.

All the decisions and recommendations concerning units are collected in a brochure called The International System of Units (SI),[1][ad] which is published by the BIPM and periodically updated.

## Units and prefixes

The International System of Units consists of a set of

coherent system of units, which is based on a system of quantities in such a way that the equations between the numerical values expressed in coherent units have exactly the same form, including numerical factors, as the corresponding equations between the quantities. For example, 1 N = 1 kg × 1 m/s2 says that one newton is the force required to accelerate a mass of one kilogram at one metre per second squared
, as related through the principle of coherence to the equation relating the corresponding quantities: F = m × a.

Derived units apply to derived quantities, which may by definition be expressed in terms of base quantities, and thus are not independent; for example,

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.[bt]
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 is defined in SI units as m/s2.

### Base units

The SI base units are the building blocks of the system and all the other units are derived from them.

SI base units[2]: 6 [47][48]
Unit name Unit symbol Dimension symbol Quantity name Typical symbols Definition
second
[n 1]
s T time ${\displaystyle t}$ The duration of 9192631770 periods of the radiation corresponding to the transition between the two
caesium-133
atom.
metre m L length ${\displaystyle l}$, ${\displaystyle h}$, ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle r}$, etc.[n 2] The distance travelled by light in a vacuum in 1/299792458 seconds.
kilogram
[n 3]
kg M mass ${\displaystyle m}$ The kilogram is defined by setting the Planck constant h exactly to 6.62607015×10−34 J⋅s (J = kg⋅m2⋅s−2), given the definitions of the metre and the second.[35]
ampere A I electric current ${\displaystyle I,\;i}$ The flow of exactly 1/1.602176634×10−19 times the elementary charge e per second.

Equalling approximately 6.2415090744×1018 elementary charges per second.

kelvin K Θ thermodynamic
temperature
${\displaystyle T}$ 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 N amount of substance ${\displaystyle n}$ The amount of substance of exactly 6.02214076×1023 elementary entities.[n 4] This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit mol−1.
candela cd J luminous intensity ${\displaystyle I_{v}}$ The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 5.4×1014 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.
Notes
1. ^ Within the context of the SI, the second is the coherent base unit of time, and is used in the definitions of derived units. The name "second" historically arose as being the 2nd-level sexagesimal division (1602) of some quantity, the hour in this case, which the SI classifies as an "accepted" unit along with its first-level sexagesimal division the minute.
2. ^ Symbols for length vary greatly with context. Problems involving intuitive three-dimensional quantities often use ${\displaystyle l}$, ${\displaystyle w}$, and ${\displaystyle h}$ for length, distance, and height, respectively. More generally, physicists tend to set up the coordinate system of a given problem so that one axis lies conveniently parallel to the length being measured. Length is then often denoted either by some constant (e.g. ${\displaystyle a}$, ${\displaystyle b}$) along said axis, or by the same symbol as the axis itself (e.g. ${\displaystyle x}$, ${\displaystyle y}$, or ${\displaystyle r}$ for horizontal, vertical, and radial axes, respectively).
3. ^ 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.

### Derived units

The derived units in the SI are formed by powers, products, or quotients of the base units and are potentially unlimited in number.[37]: 103 [2]: 14, 16  Derived units are associated with derived quantities; for example, velocity is a quantity that is derived from the base quantities of time and length, and thus the SI derived unit is metre per second (symbol m/s). The dimensions of derived units can be expressed in terms of the dimensions of the base units.

Combinations of base and derived units may be used to express other derived units. 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).[49]

SI derived units with special names and symbols[2]: 15
Name Symbol Quantity In SI base units In other SI units
steradian[N 1] sr solid angle m2/m2 1
hertz Hz frequency s−1
newton N force, weight kg⋅m⋅s−2
pascal Pa
stress
kg⋅m−1⋅s−2 N/m2
joule J
work, heat
kg⋅m2⋅s−2 N⋅m = Pa⋅m3
watt W power, radiant flux kg⋅m2⋅s−3 J/s
coulomb C electric charge s⋅A
volt V electric potential, voltage, emf kg⋅m2⋅s−3⋅A−1 W/A = J/C
farad F capacitance kg−1⋅m−2⋅s4⋅A2 C/V = C2/J
ohm
Ω
reactance
kg⋅m2⋅s−3⋅A−2 V/A = J⋅s/C2
siemens S
electrical conductance
kg−1⋅m−2⋅s3⋅A2 Ω−1
weber Wb magnetic flux kg⋅m2⋅s−2⋅A−1 V⋅s
tesla T
magnetic flux density
kg⋅s−2⋅A−1 Wb/m2
henry H inductance kg⋅m2⋅s−2⋅A−2 Wb/A
degree Celsius
°C temperature relative to 273.15 K K
lumen lm luminous flux cd⋅sr cd⋅sr
lux lx illuminance cd⋅sr⋅m−2 lm/m2
becquerel Bq activity referred to a radionuclide (decays per unit time) s−1
gray Gy
)
m2⋅s−2 J/kg
sievert Sv
)
m2⋅s−2 J/kg
katal kat
catalytic activity
mol⋅s−1
Notes
1. ^ a b The radian and steradian are defined as dimensionless derived units.
Examples of coherent derived units in terms of base units[2]: 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,
vergence (optics) V, 1/f
kilogram per cubic metre kg/m3 density ρ
kilogram per square metre kg/m2
surface density
ρA
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 Lv
Examples of derived units that include units with special names[2]: 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
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 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
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
katal per cubic metre kat/m3
catalytic activity concentration
m−3⋅s−1⋅mol

### Prefixes

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.[37]: 122 [50]: 14  When prefixes are used to form multiples and submultiples of SI base and derived units, the resulting units are no longer coherent.[37]
: 7

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

Prefix Base 10 Decimal English word Adoption
[nb 1][52]
Name Symbol Short scale Long scale
quetta Q 1030 1000000000000000000000000000000 nonillion quintillion 2022
ronna R 1027 1000000000000000000000000000 octillion quadrilliard 2022
yotta Y 1024 1000000000000000000000000 septillion quadrillion 1991
zetta Z 1021 1000000000000000000000 sextillion trilliard 1991
exa E 1018 1000000000000000000 quintillion trillion 1975
peta P 1015 1000000000000000 quadrillion billiard 1975
tera T 1012 1000000000000 trillion billion 1960
giga G 109 1000000000 billion milliard 1960
mega M 106 1000000 million 1873
kilo k 103 1000 thousand 1795
hecto h 102 100 hundred 1795
deca da 101 10 ten 1795
100 1 one
deci d 10−1 0.1 tenth 1795
centi c 10−2 0.01 hundredth 1795
milli m 10−3 0.001 thousandth 1795
micro μ 10−6 0.000001 millionth 1873
nano n 10−9 0.000000001 billionth milliardth 1960
pico p 10−12 0.000000000001 trillionth billionth 1960
femto f 10−15 0.000000000000001 quadrillionth billiardth 1964
atto a 10−18 0.000000000000000001 quintillionth trillionth 1964
zepto z 10−21 0.000000000000000000001 sextillionth trilliardth 1991
yocto y 10−24 0.000000000000000000000001 septillionth quadrillionth 1991
ronto r 10−27 0.000000000000000000000000001 octillionth quadrilliardth 2022
quecto q 10−30 0.000000000000000000000000000001 nonillionth quintillionth 2022
Notes
1. ^ Prefixes adopted before 1960 already existed before SI. The introduction of the CGS system was in 1873.

### Non-SI units accepted for use with SI

Many non-SI units continue to be used in the scientific, technical, and commercial literature. Some units are deeply embedded in history and culture, and their use has not been entirely replaced by their SI alternatives. The CIPM recognised and acknowledged such traditions by compiling a list of

non-SI units accepted for use with SI:[37]

Some units of time, angle, and legacy non-SI units have a long history of use. Most societies have used the solar day and its non-decimal subdivisions as a basis of time and, unlike the foot or the pound, these were the same regardless of where they were being measured. The radian, being 1/2π of a revolution, has mathematical advantages but is rarely used for navigation. Further, the units used in navigation around the world are similar. The tonne, litre, and hectare were adopted by the CGPM in 1879 and have been retained as units that may be used alongside SI units, having been given unique symbols. The catalogued units are given below:

Non-SI units accepted for use with SI units
Quantity Name Symbol Value in SI units
time minute min 1 min = 60 s
hour h 1 h = 60 min = 3600 s
day d 1 d = 24 h = 86400 s
length astronomical unit au 1 au = 149597870700 m
plane and
phase angle
degree ° 1° = π/180 rad
arcminute 1′ = 1/60° = π/10800 rad
arcsecond 1″ = 1/60′ = π/648000 rad
area hectare ha 1 ha = 1 hm2 = 104 m2
volume litre l, L 1 l = 1 L = 1 dm3 = 103 cm3 = 10−3 m3
mass tonne (metric ton) t 1 t = 1 Mg = 103 kg
dalton Da 1 Da = 1.660539040(20)×10−27 kg
energy electronvolt eV 1 eV = 1.602176634×10−19 J
logarithmic
ratio quantities
neper Np In using these units it is important that the nature of the quantity be specified and that any reference value used be specified.
bel B
decibel dB

These units are used in combination with SI units in common units such as the kilowatt-hour (1 kW⋅h = 3.6 MJ).

### Common notions of the metric units

The basic units of the metric system, as originally defined, represented common quantities or relationships in nature. They still do – the modern precisely defined quantities are refinements of definition and methodology, but still with the same magnitudes. In cases where laboratory precision may not be required or available, or where approximations are good enough, the original definitions may suffice.[bu]

• A second is 1/60 of a minute, which is 1/60 of an hour, which is 1/24 of a day, so a second is 1/86400 of a day (the use of base 60 dates back to Babylonian times); a second is the time it takes a dense object to freely fall 4.9 metres from rest.[bv]
• The length of the equator is close to 40000000 m (more precisely 40075014.2 m).[53] In fact, the dimensions of our planet were used by the French Academy in the original definition of the metre.[54]
• The metre is close to the length of a pendulum that has a period of 2 seconds;[bw] most dining tabletops are about 0.75 metres high;[55] a very tall human (basketball forward) is about 2 metres tall.[56]
• The kilogram is the mass of a litre of cold water; a cubic centimetre or millilitre of water has a mass of one gram; a 1-euro coin weighs 7.5 g;[57] a Sacagawea US 1-dollar coin weighs 8.1 g;[58] a UK 50-pence coin weighs 8.0 g.[59]
• A candela is about the luminous intensity of a moderately bright candle, or 1 candle power; a 60 W tungsten-filament incandescent light bulb has a luminous intensity of about 64 candelas.[bx]
• A mole of a substance has a mass that is its molecular mass expressed in units of grams; the mass of a mole of carbon is 12.0 g, and the mass of a mole of table salt is 58.4 g.
• Since all gases have the same volume per mole at a given temperature and pressure far from their points of liquefaction and solidification (see Perfect gas), and air is about 1/5 oxygen (molecular mass 32) and 4/5 nitrogen (molecular mass 28), the density of any near-perfect gas relative to air can be obtained to a good approximation by dividing its molecular mass by 29 (because 4/5 × 28 + 1/5 × 32 = 28.8 ≈ 29). For example, carbon monoxide (molecular mass 28) has almost the same density as air.
• A temperature difference of one kelvin is the same as one degree Celsius: 1/100 of the temperature differential between the freezing and boiling points of water at sea level; the absolute temperature in kelvins is the temperature in degrees Celsius plus about 273; human body temperature is about 37 °C or 310 K.
• A 60 W incandescent light bulb rated at 120 V (US mains voltage) consumes 0.5 A at this voltage. A 60 W bulb rated at 230 V (European mains voltage) consumes 0.26 A at this voltage.[by]

## Lexicographic conventions

### Unit names

According to the SI Brochure,

headings and publication titles. As a nontrivial application of this rule, the SI Brochure notes[1]: 148  that the name of the unit with the symbol °C is correctly spelled as 'degree Celsius': the first letter of the name of the unit, 'd', is in lowercase, while the modifier 'Celsius' is capitalized because it is a proper name.[bz][1]
: 148

The English spelling and even names for certain SI units and metric prefixes depend on the variety of English used.

US English uses the spelling deka-, meter, and liter, whilst International English uses deca-, metre, and litre. Additionally, the name of the unit whose symbol is t and which is defined according to 1 t = 103 kg is 'metric ton' US English but 'tonne' in International English.[2]
: iii

### Unit symbols and the values of quantities

Symbols of SI units are intended to be unique and universal, independent of the context language.[37]: 130–135  The SI Brochure has specific rules for writing them.[37]: 130–135  The guideline produced by the National Institute of Standards and Technology (NIST)[61] clarifies language-specific details for American English that were left unclear by the SI Brochure, but is otherwise identical to the SI Brochure.[62]

#### General rules

General rules[ca] for writing SI units and quantities apply to text that is either handwritten or produced using an automated process:

• The value of a quantity is written as a number followed by a space (representing a multiplication sign) and a unit symbol; e.g., 2.21 kg, 7.3×102 m2, 22 K. This rule explicitly includes the percent sign (%)[37]: 134  and the symbol for degrees Celsius (°C).[37]: 133  Exceptions are the symbols for plane angular degrees, minutes, and seconds (°, ′, and ″, respectively), which are placed immediately after the number with no intervening space.
• Symbols are mathematical entities, not abbreviations, and as such do not have an appended period/full stop (.), unless the rules of grammar demand one for another reason, such as denoting the end of a sentence.
• A prefix is part of the unit, and its symbol is prepended to a unit symbol without a separator (e.g., k in km, M in MPa, G in GHz, μ in μg). Compound prefixes are not allowed. A prefixed unit is atomic in expressions (e.g., km2 is equivalent to (km)2).
• Unit symbols are written using roman (upright) type, regardless of the type used in the surrounding text.
• Symbols for derived units formed by multiplication are joined with a
centre dot
(⋅) or a non-breaking space; e.g., N⋅m or N m.
• Symbols for derived units formed by division are joined with a
exponent
. E.g., the "metre per second" can be written m/s, m s−1, m⋅s−1, or m/s. In cases where a solidus is followed by a centre dot (or space), or more than one solidus is present, parentheses must be used to avoid ambiguity; e.g., kg/(m⋅s2), kg⋅m−1⋅s−2, and (kg/m)/s2 are acceptable, but kg/m/s2 and kg/m⋅s2 are ambiguous and unacceptable.
superscript
'2'.

#### Printing SI symbols

The rules covering printing of quantities and units are part of ISO 80000-1:2009.[64]

Further rules[ca] are specified in respect of production of text using printing presses, word processors, typewriters, and the like.

## International System of Quantities

SI Brochure

The CGPM publishes a brochure that defines and presents the SI.[37] Its official version is in French, in line with the Metre Convention.[37]: 102  It leaves some scope for local variations, particularly regarding unit names and terms in different languages.[cb][2]

The writing and maintenance of the CGPM brochure is carried out by one of the committees of the

International Committee for Weights and Measures
(CIPM). The definitions of the terms "quantity", "unit", "dimension" etc. that are used in the SI Brochure are those given in the
International vocabulary of metrology.[65]

The quantities and equations that provide the context in which the SI units are defined are now referred to as the International System of Quantities (ISQ). The ISQ is based on the

ISO 80000-1,[67]
and has largely been revised in 2019–2020 with the remainder being under review.

## Realisation of units

standard uncertainty of 2×10−8 or less, held by Achim Leistner[68]