International System of Units

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

The International System of Units (SI, abbreviated from the French Système international (d'unités)) is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, which are the second, metre, kilogram, ampere, kelvin, mole, candela, and a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system also specifies names for 22 derived units, such as lumen and watt, for other common physical quantities.

The base units are defined in terms of invariant constants of nature, such as the speed of light in vacuum and the charge of the electron, which can be observed and measured with great accuracy. Seven constants are used in various combinations to define the seven base units. Prior to 2019, artefacts were used instead of some of these constants, the last being the International Prototype of the Kilogram, a cylinder of platinum-iridium. Concern regarding its stability led to a revision of the definition of the base units entirely in terms of constants of nature, which was put into effect on 20 May 2019.[1]

Derived units may be defined in terms of base units or other derived units. They are adopted to facilitate measurement of diverse quantities. 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. The most recent derived unit, the katal, was defined in 1999.

The reliability of the SI depends not only on the precise measurement of standards for the base units in terms of various physical constants of nature, but also on precise definition of those constants. The set of underlying constants is modified as more stable constants are found, or may be more precisely measured. For example, in 1983 the metre was redefined as the distance that light propagates in vacuum in a given fraction of a second, thus making the value of the speed of light in terms of the defined units exact.

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

Since then, the SI has officially been adopted by all countries except the United States, Liberia, and Myanmar.[2] Both Myanmar and Liberia make substantial use of SI units, as do the scientific, military, and medical communities in the US. Countries such as the United Kingdom, Canada, and certain islands in the Caribbean have partially metricated, currently employing a mixture of SI, imperial, and US Customary units. For instance, road signs in the United Kingdom continue to use miles whilst produce in Canada and the United Kingdom continue to, in certain context, be advertised in pounds rather than kilograms. The incomplete processes of metrication in Canada and the United Kingdom illustrate the complex status of metrication internationally beyond the three countries (US, Myanmar, and Liberia) commonly cited as not having adopted the SI.

Units and prefixes

The International System of Units consists of a set of base units, derived units, and a set of decimal-based multipliers that are used as prefixes.[3]:103–106 The units, excluding prefixed units,[Note 1] form a 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 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.[Note 2] Other useful derived quantities can be specified in terms of the SI base and derived units that have no named units in the SI system, 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[4]:23[5][6]
Unit
name
Unit
symbol
Dimension
symbol
Quantity
name
Definition
second
[n 1]
s T time 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 L length The distance travelled by light in vacuum in 1/299792458 second.
kilogram
[n 2]
kg M mass 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.[1]
ampere A I electric current The flow of 1/1.602176634×10−19 times the elementary charge e per second.
kelvin K Θ thermodynamic
temperature
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
The amount of substance of exactly 6.02214076×1023 elementary entities.[n 3] 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.
candela cd J luminous
intensity
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. ^ 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.
  3. ^ 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.[3]:103[4]:9 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).[7]

SI derived units with special names and symbols[4]:25
Name Symbol Quantity In SI base units In other SI units
radiannote 1 rad plane angle m/m 1
steradiannote 1 sr solid angle m2/m2 1
hertz Hz frequency s−1
newton N force, weight kg⋅m⋅s−2
pascal Pa pressure, stress kg⋅m−1⋅s−2 N/m2
joule J energy, 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 or quantity of electricity s⋅A
volt V voltage (electrical potential), emf kg⋅m2⋅s−3⋅A−1 W/A = J/C
farad F capacitance kg−1⋅m−2⋅s4⋅A2 C/V
ohm Ω resistance, impedance, reactance kg⋅m2⋅s−3⋅A−2 V/A
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 m−2⋅cd lm/m2
becquerel Bq radioactivity (decays per unit time) s−1
gray Gy absorbed dose (of ionising radiation) m2⋅s−2 J/kg
sievert Sv equivalent dose (of ionising radiation) m2⋅s−2 J/kg
katal kat catalytic activity mol⋅s−1
Notes
1. The radian and steradian are defined as dimensionless derived units.
Examples of coherent derived units in terms of base units[4]:24
SI derived unit 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,
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[4]:26
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 s−1
radian per second squared rad/s2 angular acceleration s−2
watt per square metre W/m2 heat flux density kg⋅s−3
joule per kelvin J/K heat capacity, entropy 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 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 heat capacity, molar entropy m2⋅kg⋅s−2⋅K−1⋅mol−1
coulomb per kilogram C/kg exposure 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

Prefixes

Prefixes are added to unit names to produce multiples and sub-multiples 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.[3]:122[8]:14 When prefixes are used to form multiples and submultiples of SI base and derived units, the resulting units are no longer coherent.[3]:7

The BIPM specifies 20 prefixes for the International System of Units (SI):

Prefix Base 1000 Base 10 Decimal English word Adoption[nb 1]
Name Symbol Short scale Long scale
yotta Y  10008  1024 1000000000000000000000000  septillion  quadrillion 1991
zetta Z  10007  1021 1000000000000000000000  sextillion  trilliard 1991
exa E  10006  1018 1000000000000000000  quintillion  trillion 1975
peta P  10005  1015 1000000000000000  quadrillion  billiard 1975
tera T  10004  1012 1000000000000  trillion  billion 1960
giga G  10003  109 1000000000  billion  milliard 1960
mega M  10002  106 1000000  million 1873
kilo k  10001  103 1000  thousand 1795
hecto h  10002/3  102 100  hundred 1795
deca da  10001/3  101 10  ten 1795
 10000  100 1  one
deci d  1000−1/3  10−1 0.1  tenth 1795
centi c  1000−2/3   10−2 0.01  hundredth 1795
milli m  1000−1  10−3 0.001  thousandth 1795
micro μ  1000−2  10−6 0.000001  millionth 1873
nano n  1000−3  10−9 0.000000001  billionth  milliardth 1960
pico p  1000−4  10−12 0.000000000001  trillionth  billionth 1960
femto f  1000−5  10−15 0.000000000000001  quadrillionth  billiardth 1964
atto a  1000−6  10−18 0.000000000000000001  quintillionth  trillionth 1964
zepto z  1000−7  10−21 0.000000000000000000001  sextillionth  trilliardth 1991
yocto y  1000−8  10−24  0.000000000000000000000001  septillionth  quadrillionth 1991
  1. ^ Prefixes adopted before 1960 already existed before SI. 1873 was the introduction of the CGS system.

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:[3]

While not an SI-unit, the litre may be used with SI units. It is equivalent to (10 cm)3 = (1 dm)3 = 10−3 m3

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/ 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 the 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
minute 1′ = (1/60)° = (π/10800) rad
second 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 = 1000 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

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.[Note 3]

  • 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.
  • The length of the equator is close to 40,000,000 metres (more precisely 40,075,014.2 metres). In fact, the dimensions of our planet were used by the French Academy in the original definition of the metre.
  • The metre is close to the length of a pendulum that has a period of 2 seconds; most dining tabletops are about 0.75 metres high; a very tall human (basketball forward) is about 2 metres tall.
  • 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, 7.5 g; a Sacagawea US 1-dollar coin, 8.1 g; a UK 50-pence coin, 8.0 g.
  • 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 candela.
  • 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.
  • 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 consumes 0.5 amperes at 120 V (US mains voltage) and about 0.25 amperes at 240 V (European mains voltage).