# International System of Units

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The International System of Units, abbreviated SI from its French language name, Le Système International d'Unités, is a comprehensive set of units of measurement. Aside from its dominance in science, the United States Omnibus Trade and Comptetitiveness Act of 1988 states "the metric system of measurement is the preferred system of weights and measures for United States trade and commerce".[1]

The SI is based on the original metric system developed in France in the 1790s. In October 1960 the 11th international "General Conference on Weights and Measures" met in Paris and renamed the Metric System (MKSA) of units (based on the six base units: meter, kilogram, second, ampere, kelvin and candela—in 1971 mole was added as seventh base unit) to the "International System of Units." The 11th Conference also established the abbreviation "SI" as the official abbreviation, to be used in all languages. Adoption of the abbreviation SI, especially outside scientific circles, is slow. The terms "metric system" or "MKSA units" are still frequently being used.

For more background and a bibliography for the SI units see the SI units page on the NIST website.[2]

## Base units

The SI is founded on seven SI base units for seven base quantities assumed to be mutually independent:

SI base units
Name Symbol Quantity Definition
metre m length The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.
kilogram kg mass The kilogram is equal to the mass of the international prototype of the kilogram.
second s time The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.
ampere A electrical current The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length.
kelvin K temperature The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
mole mol amount of substance The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12
candela cd luminous intensity The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

### Prefixes

To allow for ease of discussion of quantities orders of magnitude different from the base units, prefixes may be used to form decimal multiples and submultiples of units. The SI prefixes with their meanings and symbols are:

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

It is important to note that the kilogram is the only SI unit with a prefix as part of its name and symbol. Because multiple prefixes may not be used, in the case of the kilogram the prefix names are used with the unit name "gram" and the prefix symbols are used with the unit symbol "g." With this exception, any SI prefix may be used with any SI unit, including the degree Celsius and its symbol °C.[3]

## Derived units

Other quantities, called derived quantities, are defined in terms of the seven base quantities via a system of quantity equations. The SI derived units for these derived quantities are obtained from these equations and the seven SI base units.

### Dimensionless derived units

There are two dimensionless derived units, for plane angle and solid angle:

Dimensionless SI units
Name Symbol Quantity Definition
radian rad angle The unit of angle is the angle subtended at the centre of a circle by an arc of the circumference equal in length to the radius of the circle. There are ${\displaystyle 2\pi }$ radians in a circle.
steradian sr solid angle The unit of solid angle is the solid angle subtended at the centre of a sphere of radius r by a portion of the surface of the sphere having an area r2. There are ${\displaystyle 4\pi }$ steradians on a sphere.

### Named derived units

Twenty other derived units have specific names; most are named after pioneering researchers in the fields in which they are used. These are:

Named units derived from SI base units
Name Symbol Quantity Expression in terms of other units Expression in terms of SI base units
hertz Hz Frequency 1/s s−1
newton N Force, Weight m∙kg/s2 m∙kg∙s−2
joule J Energy, Work, Heat N∙m m2∙kg∙s−2
watt W Power, Radiant flux J/s m2∙kg∙s−3
pascal Pa Pressure, Stress N/m2 m−1∙kg∙s−2
lumen lm Luminous flux cd∙sr cd
lux lx Illuminance lm/m2 m−2∙cd
coulomb C Electric charge or flux s∙A s∙A
volt V Electrical potential difference, Electromotive force W/A = J/C m2∙kg∙s−3∙A−1
ohm Ω Electric resistance, Impedance, Reactance V/A m2∙kg∙s−3∙A−2
farad F Electric capacitance C/V m−2∙kg−1∙s4∙A2
weber Wb Magnetic flux J/A m2∙kg∙s−2∙A−1
tesla T Magnetic flux density, magnetic induction V∙s/m2 = Wb/m2 kg∙s−2∙A−1
henry H Inductance V∙s/A = Wb/A m2∙kg∙s−2∙A−2
siemens S Electric conductance 1/Ω m−2∙kg−1∙s3∙A2
becquerel Bq Radioactivity (decays per unit time) 1/s 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 s−1∙mol
degree Celsius °C Thermodynamic temperature T°C = TK − 273.15

### Other derived units

Some derived units are named after the basic units from which they are derived, sometimes including the dimension. Other derived units have names which are a mix of base unit names and derived unit names. Some are listed below:

Compound units derived from basic SI units
Name Symbol Quantity Expression in terms
of SI base units
square metre m2 area m2
cubic metre m3 volume m3
metre per second m·s−1 speed, velocity m·s−1
metre per second squared m·s−2 acceleration m·s−2
metre per second cubed m·s−3 jerk m·s−3
reciprocal metre m−1 wavenumber m−1
kilogram per cubic metre kg·m−3 Density, mass density kg·m−3
cubic metre per kilogram kg−1·m3 specific volume kg−1·m3
mole per cubic metre m−3·mol amount (-of-substance) concentration m−3·mol
cubic metre per mole m3·mol−1 molar volume m3·mol−1
square metre per second m2·s−1 kinematic viscosity, diffusion coefficient m2·s−1
ampere per square metre A·m−2 electric current density A·m−2
ampere per metre A·m−1 magnetic field strength A·m−1
candela per square metre cd·m−2 luminance cd·m−2

Compound units derived from SI units
Name Symbol Quantity Expression in terms
of SI base units
newton second N·s momentum, impulse kg·m·s−1
newton metre second N·m·s angular momentum kg·m2·s−1
newton metre N·m torque, moment of force kg·m2·s−2
joule per kelvin J·K−1 heat capacity, entropy kg·m2·s−2·K−1
joule per kelvin mole J·K−1·mol−1 molar heat capacity, molar entropy kg·m2·s−2·K−1·mol−1
joule per kilogram kelvin J·K−1·kg−1 specific heat capacity, specific entropy m2·s−2·K−1
joule per mole J·mol−1 molar energy kg·m2·s−2·mol−1
joule per kilogram J·kg−1 specific energy m2·s−2
joule per cubic metre J·m−3 energy density kg·m−1·s−2
newton per metre N·m−1 = J·m−2 surface tension kg·s−2
watt per square metre W·m−2 heat flux density, irradiance kg·s−3
watt per metre kelvin W·m−1·K−1 thermal conductivity kg·m·s−3·K−1
pascal second Pa·s = N·s·m−2 dynamic viscosity kg·m−1·s−1
coulomb per cubic metre C·m−3 electric charge density m−3·s·A
siemens per metre S·m−1 conductivity kg−1·m−3·s3·A2
siemens square metre per mole S·m2·mol−1 molar conductivity kg-1·s3·mol−1·A2
farad per metre F·m−1 permittivity kg−1·m−3·s4·A2
henry per metre H·m−1 permeability kg·m·s−2·A−2
volt per metre V·m−1 electric field strength kg·m·s−3·A−1
coulomb per kilogram C·kg−1 exposure (X and gamma rays) kg−1·s·A
gray per second Gy·s−1 absorbed dose rate m2·s−3

## Non-SI units accepted for use

The 2006 edition of the International System of Units, published by the International Bureau of Weights and Measures (BIPM) includes non-SI units that are accepted for use with the International System because they are widely used in everyday life.[4] Their use is expected to continue indefinitely, and each has an exact definition in terms of an SI unit. The values in the table below were extracted from Tables 6 and 8 of the 2006 Edition:

Non-SI units accepted for use with the International System of 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
area hectare ha 1 ha = 1 hm2 = 104m2
volume litre L or l 1 L = 1 dm3 = 103 cm3 = 10−3 m3
mass tonne t 1 t = 103 kg
plane angle degree ° 1 ° = (${\displaystyle \pi }$/180) rad
minute ' 1 ' = (1/60)° = (${\displaystyle \pi }$/10800) rad
second " 1 " = (1/60)' = (${\displaystyle \pi }$/648000) rad
pressure bar bar 1 bar = 0.1 MPa = 100 kPa = 105 Pa
millimetre of mercury mmHg 1 mmHg ≈ 133.322 Pa
speed knot kn 1 kn = (1852/3600) m/s

## SI writing style

• Symbols for units are written in lower case, except for symbols derived from the name of a person. For example, the unit of pressure is named after Blaise Pascal, so its symbol is written "Pa" whereas the unit itself is written "pascal".
• The one exception is the litre, whose original symbol "l" is unsuitably similar to the numeral "1" or the uppercase letter "i" (depending on the typographic font used), at least in many English-speaking countries. The American National Institute of Standards and Technology recommends that "L" be used instead, a usage which is common in the U.S., Canada, Australia, and New Zealand (but not elsewhere). This has been accepted as an alternative by the CGPM since 1979. The cursive "ℓ" is occasionally seen, especially in Japan and Greece, but this is not currently recommended by any standards body. For more information, see Litre.
• The SI rule for pluralising units is that symbols of units are not pluralised[4], for example "25 kg" (not "25 kgs").
• The American National Institute of Standards and Technology has defined guidelines for using the SI units in its own publications and for other users of the SI[5]. These guidelines give guidance on pluralizing unit names: the plural is formed by using normal English grammar rules, for example, "henries" is the plural of "henry". The units lux, hertz, and siemens are exceptions from this rule: they remain the same in singular and plural. Note that this rule only applies to the full names of units, not to their symbols.
• Symbols do not have an appended period/full stop (.) unless at the end of a sentence.
• Symbols are written in upright Roman type (m for metres, L for litres), so as to differentiate from the italic type used for mathematical variables (m for mass, l for length).
• A space separates the number and the symbol, e.g. "2.21 kg", "7.3×102 m2", "22 °C" [6]. Exceptions are the symbols for plane angular degrees, minutes and seconds (°, ′ and ″), which are placed immediately after the number with no intervening space.
• Spaces may be used to group decimal digits in threes, e.g. "1 000 000" or "342 142" (in contrast to the commas or dots used in other systems, e.g. "1,000,000" or "1.000.000"). This is presumably to reduce confusion because a comma is used as a decimal in many countries while others use a period. In print, the space used for this purpose is typically narrower than that between words.
• The 10th resolution of CGPM in 2003 declared that "the symbol for the decimal marker shall be either the point on the line or the comma on the line". In practice, the decimal point is used in English, and the comma in most other European languages.
• Symbols for derived units formed from multiple units by multiplication are joined with a space or centre dot (·), e.g. "N m" or "N·m".
• Symbols formed by division of two units are joined with a solidus (⁄), or given as a negative exponent. For example, the "metre per second" can be written "m/s", "m s−1", "m·s−1" or ${\displaystyle \textstyle {\frac {\mathrm {m} }{\mathrm {s} }}.}$ A solidus should not be used if the result is ambiguous, i.e. "kg·m−1·s−2" is preferable to "kg/m·s2". (Taylor (§ 6.1.6) specifically calls for the use of a solidus.[5] Many computer users will type the / character provided on American computer keyboards, which in turn produces the Unicode character U+002F, which is named solidus. Taylor does not offer suggestions about which mark should be used when more sophisticated typesetting options are available.)
• In countries using ideographic writing systems such as Chinese and Japanese, often the full symbol of the unit, including prefixes, is placed in one square. (See the "Letterlike Symbols" Unicode subrange.)

### Spelling variations

• Several nations, notably the United States, typically use the spellings "meter" and "liter" instead of "metre" and "litre" in keeping with standard American English spelling, which also corresponds to the official spelling used in several other languages, such as German, Dutch, Swedish, etc. In addition, the official U.S. spelling for the SI prefix "deca" is "deka".[3]
• In some English-speaking countries, the unit "ampere" is often shortened to "amp" (singular) or "amps" (plural).

## Numerical values

The numerical values of the principal physical constants can be found in a NIST summary.[7] NIST maintains a web site with the current numerical values of the physical constants in SI units.[8]

## References

1. See the official documentation by BN Taylor listed on the Bibliography page.
2. International system of units (SI). The NIST reference on constants units and uncertainty. National Institute of Standards and Technology (2000). Retrieved on 2011-03-28.
3. Taylor, Barry N. (December 2003). The NIST Reference on Constants, Units, and Uncertainty. National Institute of Standards and Technology. Retrieved on 1 March 2007.
4. Bureau International des Poids et Mesures (2006). The International System of Units (SI). 8th ed.. Retrieved on 14 July 2006.
5. Taylor, B.N. (1995). NIST Special Publication 811: Guide for the Use of the International System of Units (SI). National Institute of Standards and Technology. Retrieved on 9 June 2006.
6. Taylor, B.N. (1995). NIST Special Publication 811: Guide for the Use of the International System of Units (SI). National Institute of Standards and Technology. Retrieved on 1 March 2007.
7. See Table L in PJ Mohr, BN Taylor, and DB Newell (2008). "CODATA recommended values of the fundamental physical constants: 2006". Reviews of Modern Physics vol. 80: p. 633 ff.
8. CODATA internationally recommended values of the fundamental physical constants. The NIST reference on constants units and uncertainty. NIST. Retrieved on 2011-03-28.