International System of Units

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

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 that are ampere, kelvin, second, metre, kilogram, candela, mole, 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 derived from invariant constants of nature, such as the speed of light and the triple point of water, which can be observed and measured with great accuracy, and one physical artefact. The artefact is the international prototype kilogram, certified in 1889, and consisting of a cylinder of platinum-iridium, which nominally has the same mass as one litre of water at the freezing point. Its stability has been a matter of significant concern, culminating in a proposed revision of the definition of the base units entirely in terms of constants of nature, expected to be put into effect in 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 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 been adopted by all countries except the United States, Liberia and Myanmar.[2]

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. When Maxwell first introduced the concept of a coherent system, he identified three quantities that could be used as base units: mass, length and time. Giorgi later identified the need for an electrical base unit, for which the unit of electric current was chosen for SI. Another three base units (for temperature, amount of substance and luminous intensity) were added later.

SI base units [4]:23[5][6]
Unit
name
Unit
symbol
Dimension
symbol
Quantity
name
Definition [n 1]
metre m L length
  • Prior (1793): 1/10000000 of the meridian through Paris between the North Pole and the Equator.FG
  • Interim (1960): 1650763.73 wavelengths in a vacuum of the radiation corresponding to the transition between the 2p10 and 5d5 quantum levels of the krypton-86 atom.
  • Current (1983): The distance travelled by light in vacuum in 1/299792458 second.
kilogram[n 2] kg M mass
  • Prior (1793): The grave was defined as being the mass (then called weight) of one litre of pure water at its freezing point.FG
  • Current (1889): The mass of a small squat cylinder of ~47 cubic centimetres of platinum-iridium alloy kept in the Pavillon de Breteuil[citation needed], France. Also, in practice, any of numerous official replicas of it.[Note 3][7]
second s T time
  • Prior: 1/86400 of a day of 24 hours of 60 minutes of 60 seconds
  • Interim (1956): 1/31556925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time.
  • Current (1967): 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.
ampere A I electric current
  • Prior (1881): A tenth of the electromagnetic CGS unit of current. The [CGS] electromagnetic unit of current is that current, flowing in an arc 1 cm long of a circle 1 cm in radius, that creates a field of one oersted at the centre.[8] IEC
  • Current (1946): The constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2×10−7 newtons per metre of length.
kelvin K Θ thermodynamic temperature
mole mol N amount of substance
  • Prior (1900): A stoichiometric quantity which is the equivalent mass in grams of Avogadro's number of molecules of a substance.ICAW
  • Current (1967): The amount of substance of a system which contains as many elementary entities[n 4] as there are atoms in 0.012 kilogram of carbon-12.
candela cd J luminous intensity
  • Prior (1946): The value of the new candle is such that the brightness of the full radiator at the temperature of solidification of platinum is 60 new candles per square centimetre.
  • Current (1979): 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.
Note: both old and new definitions are approximately the luminous intensity of a whale blubber candle burning modestly bright, in the late 19th century called a "candlepower" or a "candle".
Notes
  1. ^ Interim definitions are given here only when there has been a significant difference in the definition.
  2. ^ Despite the prefix "kilo-", the kilogram is the base unit of mass. The kilogram, not the gram, is the coherent unit and is used in the definitions of derived units. Nonetheless, prefixes are determined as if the gram were the base unit of mass.
  3. ^ In 1954 the unit of thermodynamic temperature was known as the "degree Kelvin" (symbol °K; "Kelvin" spelt with an upper-case "K"). It was renamed the "kelvin" (symbol "K"; "kelvin" spelt with a lower case "k") in 1967.
  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.

The Prior definitions of the various base units in the above table were made by the following authorities:

All other definitions result from resolutions by either CGPM or the CIPM and are catalogued in the SI Brochure.

The early metric systems defined a unit of weight as a base unit, while the SI defines an analogous unit of mass. In everyday use, these are mostly interchangeable, but in scientific contexts the difference matters. Mass, strictly the inertial mass, represents a quantity of matter. It relates the acceleration of a body to the applied force via Newton's law, F = m × a: force equals mass times acceleration. A force of 1 N (newton) applied to a mass of 1 kg will accelerate it at 1 m/s2. This is true whether the object is floating in space or in a gravity field e.g. at the Earth's surface. Weight is the force exerted on a body by a gravitational field, and hence its weight depends on the strength of the gravitational field. Weight of a 1 kg mass at the Earth's surface is m × g; mass times the acceleration due to gravity, which is 9.81 newtons at the Earth's surface and is about 3.5 newtons at the surface of Mars. Since the acceleration due to gravity is local and varies by location and altitude on the Earth, weight is unsuitable for precision measurements of a property of a body, and this makes a unit of weight unsuitable as a base unit.

Derived units

The derived units in the SI are formed by powers, products or quotients of the base units and are unlimited in number.[3]:103[4]:3 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).[9]

Named SI derived units[4]:3
Namenote 1 Symbol Quantity In other SI units In SI base units
radiannote 2 rad angle (m⋅m−1)
steradiannote 2 sr solid angle (m2⋅m−2)
hertz Hz frequency s−1
newton N force, weight kg⋅m⋅s−2
pascal Pa pressure, stress N/m2 kg⋅m−1⋅s−2
joule J energy, work, heat N⋅m = Pa⋅m3 kg⋅m2⋅s−2
watt W power, radiant flux J/s kg⋅m2⋅s−3
coulomb C electric charge or quantity of electricity s⋅A
volt V voltage (electrical potential), emf W/A kg⋅m2⋅s−3⋅A−1
farad F capacitance C/V kg−1⋅m−2⋅s4⋅A2
ohm Ω resistance, impedance, reactance V/A kg⋅m2⋅s−3⋅A−2
siemens S electrical conductance Ω−1 kg−1⋅m−2⋅s3⋅A2
weber Wb magnetic flux V⋅s kg⋅m2⋅s−2⋅A−1
tesla T magnetic flux density Wb/m2 kg⋅s−2⋅A−1
henry H inductance Wb/A kg⋅m2⋅s−2⋅A−2
degree Celsius °C temperature relative to 273.15 K K
lumen lm luminous flux cd⋅sr cd
lux lx illuminance lm/m2 m−2⋅cd
becquerel Bq radioactivity (decays per unit time) s−1
gray Gy absorbed dose (of ionizing radiation) J/kg m2⋅s−2
sievert Sv equivalent dose (of ionizing radiation) J/kg m2⋅s−2
katal kat catalytic activity mol⋅s−1
Notes
1. The table is ordered so that a derived unit is listed after the units upon which its definition depends.
2. The radian and steradian are defined as dimensionless derived units.

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[10]: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 twenty 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, which are grouped as follows:[3]:123–129[10]:7–11[Note 4]

The litre is classed as a non-SI unit accepted for use with the SI.
Being one thousandth of a cubic metre, the litre is not a coherent unit of measure with respect to SI.
  • Non-SI units accepted for use with the SI (Table 6):
    Certain units of time, angle, and legacy non-SI metric units have a long history of consistent 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 it is cumbersome for navigation[citation needed], and, as with time, the units used in navigation are largely consistent around the world. 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
    minute, hour, day, degree of arc, minute of arc, second of arc, hectare, litre, tonne, astronomical unit
    Some of the units listed in Table 7 and Table 8 are also accepted for use with the SI.
  • Non-SI units whose values in SI units must be obtained experimentally (Table 7):
    Physicists often use units of measure that are based on natural phenomena, particularly when the quantities associated with these phenomena are many orders of magnitude greater than or less than the equivalent SI unit. The most common ones have been catalogued in the SI Brochure together with consistent symbols and accepted values, but with the caveat that their values in SI units need to be measured.
    electronvolt (symbol eV), and dalton/unified atomic mass unit (Da or u)
Sphygmomanometer – the traditional device that measures blood pressure using mercury in a manometer. Pressures are recorded in "millimetres of mercury" – a non-SI unit
  • Other non-SI units (Table 8):
    A number of non-SI units that had never been formally sanctioned by the CGPM have continued to be used across the globe in many spheres including health care and navigation. As with the units of measure in Tables 6 and 7, these have been catalogued by the CIPM in the SI Brochure to ensure consistent usage, but with the recommendation that authors who use them should define them wherever they are used.
    bar, millimetre of mercury, ångström, nautical mile, barn, knot, neper, bel and decibel
    The neper, bel and decibel have been accepted for use with the SI by the CIPM.
    In the interests of standardising health-related units of measure used in the nuclear industry, the 12th CGPM (1964) accepted the continued use of the curie (symbol Ci) as a non-SI unit of activity for radionuclides;[3]:152 the SI derived units becquerel, sievert and gray were adopted in later years. Similarly, the millimetre of mercury (symbol mmHg) was retained for measuring blood pressure.[3]:127
  • Non-SI units associated with the CGS and the CGS-Gaussian system of units (Table 9):
    The SI manual also catalogues a number of legacy units of measure that are used in specific fields such as geodesy and geophysics or are found in the literature, particularly in classical and relativistic electrodynamics where they have certain advantages. The units that are catalogued are:
    erg, dyne, poise, stokes, stilb, phot, gal, maxwell, gauss, and oersted.

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 5]

  • 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; a second is the time it takes a dense object to freely fall 4.9 metres from rest.
  • The metre is close to the length of a pendulum that has a period of 2 seconds; most dining tabletops are about 0.75 metre 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 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 0.26 amperes at 230 V (European mains voltage).
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