International System of Units

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".

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.

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	quetta-	ronna-	yotta-	zetta-	exa-	peta-	tera-	giga-	mega-	kilo-	hecto-	deca-
Symbol	Q	R	Y	Z	E	P	T	G	M	k	h	da
Factor	1030	1027	1024	1021	1018	1015	1012	109	106	103	102	101
Name	deci-	centi-	milli-	micro-	nano-	pico-	femto-	atto-	zepto-	yocto-	ronto-	quecto-
Symbol	d	c	m	µ	n	p	f	a	z	y	r	q
Factor	10-1	10-2	10-3	10-6	10-9	10-12	10-15	10-18	10-21	10-24	10-27	10-30

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.

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 $2\pi$ 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 $4\pi$ 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
radian per second	rad·s−1	angular velocity	s−1
reciprocal metre	m−1	wavenumber	m−1
kilogram metre per second	kg·m·s−1	momentum	kg·m·s−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. 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 ° = ($\pi$/180) rad
	minute	'	1 ' = (1/60)° = ($\pi$/10800) rad
	second	"	1 " = (1/60)' = ($\pi$/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, 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. 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" . 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 $\textstyle {\frac {\mathrm {m} }{\mathrm {s} }}.$ 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. 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 of America, 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".

• 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. NIST maintains a web site with the current numerical values of the physical constants in SI units.