# SI derived unit

Last updated

SI derived units are units of measurement derived from the seven base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriate power of exponentiation (see: Buckingham π theorem). Some are dimensionless, as when the units cancel out in ratios of like quantities.

## Contents

The SI has special names for 22 of these derived units (for example, hertz, the SI unit of measurement of frequency), but the rest merely reflect their derivation: for example, the square metre (m2), the SI derived unit of area; and the kilogram per cubic metre (kg/m3 or kg⋅m−3), the SI derived unit of density.

The names of SI derived units, when written in full, are always in lowercase. However, the symbols for units named after persons are written with an uppercase initial letter. For example, the symbol for hertz is "Hz", while the symbol for metre is "m". 

## Special names

The International System of Units assigns special names to 22 derived units, which includes two dimensionless derived units, the radian (rad) and the steradian (sr).

Named units derived from SI base units 
Name Symbol Quantity Equivalents SI base unit
Equivalents
hertz Hz frequency 1/ss1
newton N force, weight kg⋅m/s2kg⋅m⋅s2
pascal Pa pressure, stress N/m2kg⋅m1⋅s2
joule J energy, work, heat m⋅N, C⋅V, W⋅skg⋅m2⋅s2
watt W power, radiant flux J/s, V⋅Akg⋅m2⋅s3
coulomb C electric charge or quantity of electricity s⋅A, F⋅Vs⋅A
volt V voltage, electrical potential difference, electromotive force W/A, J/Ckg⋅m2⋅s3⋅A1
farad F electrical capacitance C/V, s/Ωkg1⋅m2⋅s4⋅A2
ohm Ω electrical resistance, impedance, reactance 1/S, V/Akg⋅m2⋅s3⋅A2
siemens S electrical conductance 1/Ω, A/Vkg1⋅m2⋅s3⋅A2
weber Wb magnetic flux J/A, T⋅m2,V⋅skg⋅m2⋅s2⋅A1
tesla T magnetic induction, magnetic flux density V⋅s/m2, Wb/m2, N/(A⋅m)kg⋅s2⋅A1
henry H electrical inductance V⋅s/A, Ω⋅s, Wb/Akg⋅m2⋅s2⋅A2
degree Celsius °C temperature relative to 273.15 K KK
lumen lm luminous flux cd⋅srcd
lux lx illuminance lm/m2cd⋅m2
becquerel Bq radioactivity (decays per unit time)1/ss1
gray Gy absorbed dose (of ionizing radiation)J/kgm2⋅s2
sievert Sv equivalent dose (of ionizing radiation)J/kgm2⋅s2
katal kat catalytic activity mol/ss1⋅mol.

## Examples of derived quantities and units

Kinematic SI derived units
NameSymbolQuantityExpression in terms
of SI base units
metre per second m/s speed, velocity m⋅s1
metre per second squared m/s2 acceleration m⋅s2
metre per second cubedm/s3 jerk, jolt m⋅s3
metre per second to the fourthm/s4 snap, jounce m⋅s4
hertz per secondHz/s frequency drift s2
cubic metre per second m3/s volumetric flow m3⋅s1
Mechanical SI derived units
NameSymbolQuantityExpression in terms
of SI base units
square metre m2 area m2
cubic metre m3 volume m3
newton-second N⋅s momentum, impulse m⋅kg⋅s1
newton metre secondN⋅m⋅s angular momentum m2⋅kg⋅s1
newton-metre N⋅m = J/rad torque, moment of force m2⋅kg⋅s2
newton per secondN/s yank m⋅kg⋅s3
reciprocal metre m1 wavenumber, optical power, curvature, spatial frequency m1
kilogram per square metrekg/m2 area density m2⋅kg
kilogram per cubic metre kg/m3 density, mass densitym3⋅kg
cubic metre per kilogramm3/kg specific volume m3⋅kg1
joule-second J⋅s action m2⋅kg⋅s1
joule per kilogramJ/kg specific energy m2⋅s2
joule per cubic metreJ/m3 energy density m1⋅kg⋅s2
newton per metreN/m = J/m2 surface tension, stiffness kg⋅s2
watt per square metreW/m2heat flux density, irradiance kg⋅s3
square metre per secondm2/s kinematic viscosity, thermal diffusivity, diffusion coefficientm2⋅s1
pascal-second Pa⋅s = N⋅s/m2dynamic viscosity m1⋅kg⋅s1
kilogram per metrekg/m linear mass density m1⋅kg
kilogram per secondkg/s mass flow rate kg⋅s1
watt per metreW/m spectral power m⋅kg⋅s3
gray per secondGy/s absorbed dose ratem2⋅s3
metre per cubic metrem/m3 fuel efficiency m2
watt per cubic metreW/m3 spectral irradiance, power density m1⋅kg⋅s3
joule per square metre secondJ/(m2⋅s) energy flux density kg⋅s3
reciprocal pascalPa1 compressibility m⋅kg1⋅s2
joule per square metreJ/m2 radiant exposure kg⋅s2
kilogram square metrekg⋅m2 moment of inertia m2⋅kg
newton metre second per kilogramN⋅m⋅s/kg specific angular momentum m2⋅s1
watt per steradian metreW/(sr⋅m) spectral intensity m⋅kg⋅s3
Molar SI derived units
NameSymbolQuantityExpression in terms
of SI base units
mole per cubic metremol/m3 molarity, amount of substance concentrationm3⋅mol
cubic metre per mole m3/mol molar volume m3⋅mol1
joule per kelvin moleJ/(K⋅mol) molar heat capacity, molar entropym2⋅kg⋅s2⋅K1⋅mol1
joule per mole J/molmolar energym2⋅kg⋅s2⋅mol1
siemens square metre per moleS⋅m2/mol molar conductivity kg−1⋅s3⋅A2⋅mol1
mole per kilogrammol/kg molality kg1⋅mol
kilogram per molekg/mol molar mass kg⋅mol1
cubic metre per mole secondm3/(mol⋅s) catalytic efficiency m3⋅s1⋅mol1
Electromagnetic SI derived units
NameSymbolQuantityExpression in terms
of SI base units
coulomb per square metreC/m2 electric displacement field, polarization density m2⋅s⋅A
coulomb per cubic metreC/m3electric charge density m3⋅s⋅A
ampere per square metreA/m2electric current density m2⋅A
siemens per metre S/m electrical conductivity m3⋅kg1⋅s3⋅A2
henry per metreH/m magnetic permeability m⋅kg⋅s2⋅A2
volt per metreV/m electric field strengthm⋅kg⋅s3⋅A1
ampere per metreA/m magnetization, magnetic field strengthm1⋅A
coulomb per kilogramC/kg exposure (X and gamma rays)kg1⋅s⋅A
ohm metre Ω⋅m resistivity m3⋅kg⋅s3⋅A2
coulomb per metreC/m linear charge density m1⋅s⋅A
joule per teslaJ/T magnetic dipole moment m2⋅A
square metre per volt secondm2/(V⋅s) electron mobility kg1⋅s2⋅A
reciprocal henryH1 magnetic reluctance m2⋅kg1⋅s2⋅A2
weber per metreWb/m magnetic vector potential m⋅kg⋅s2⋅A1
weber metreWb⋅m magnetic moment m3⋅kg⋅s2⋅A1
tesla metreT⋅m magnetic rigidity m⋅kg⋅s2⋅A1
metre per henrym/H magnetic susceptibility m1⋅kg1⋅s2⋅A2
Photometric SI derived units
NameSymbolQuantityExpression in terms
of SI base units
lumen second lm⋅s luminous energy s⋅cd
lux second lx⋅s luminous exposure m2⋅s⋅cd
candela per square metre cd/m2 luminance m2⋅cd
lumen per wattlm/W luminous efficacy m2⋅kg1⋅s3⋅cd
Thermodynamic SI derived units
NameSymbolQuantityExpression in terms
of SI base units
joule per kelvinJ/K heat capacity, entropy m2⋅kg⋅s2⋅K1
joule per kilogram kelvinJ/(K⋅kg) specific heat capacity, specific entropym2⋅s2⋅K1
watt per metre kelvinW/(m⋅K) thermal conductivity m⋅kg⋅s3⋅K1
kelvin per wattK/W thermal resistance m2⋅kg1⋅s3⋅K
reciprocal kelvinK1 thermal expansion coefficient K1
kelvin per metreK/m temperature gradient m1⋅K

## Other units used with SI

Some other units such as the hour, litre, tonne, bar, and electronvolt are not SI units, but are widely used in conjunction with SI units.

## Supplementary units

Until 1995, the SI classified the radian and the steradian as supplementary units, but this designation was abandoned and the units were grouped as derived units. 

## Related Research Articles The candela is the unit of luminous intensity in the International System of Units (SI). It measures luminous power per unit solid angle emitted by a light source in a particular direction. Luminous intensity is analogous to radiant intensity, but instead of simply adding up the contributions of every wavelength of light in the source's spectrum, the contribution of each wavelength is weighted by the standard luminosity function. A common wax candle emits light with a luminous intensity of roughly one candela. If emission in some directions is blocked by an opaque barrier, the emission would still be approximately one candela in the directions that are not obscured. The kilogram is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially. It means 'one thousand grams'. The litre or liter is a metric unit of volume. It is equal to 1 cubic decimetre (dm3), 1000 cubic centimetres (cm3) or 0.001 cubic metre (m3). A cubic decimetre occupies a volume of 10 cm × 10 cm × 10 cm and is thus equal to one-thousandth of a cubic metre. Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to a basic reference quantity of the same kind. The scope and application of measurement are dependent on the context and discipline. In natural sciences and engineering, measurements do not apply to nominal properties of objects or events, which is consistent with the guidelines of the International vocabulary of metrology published by the International Bureau of Weights and Measures. However, in other fields such as statistics as well as the social and behavioural sciences, measurements can have multiple levels, which would include nominal, ordinal, interval and ratio scales.

A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a value, which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For example, the physical quantity of mass can be quantified as '32.3 kg ', where '32.3' is the numerical value and 'kg' is the Unit. The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI), and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit. The radian is defined in the SI as being a dimensionless unit with 1 rad = 1. Its symbol is accordingly often omitted, especially in mathematical writing.

The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and based on the metre as the unit of length and either the kilogram as the unit of mass or the kilogram-force as the unit of force.</ref> and the world's most widely used system of measurement. Established and maintained by the General Conference on Weights and Measures (CGPM), it is the only system of measurement with an official status in nearly every country in the world, employed in science, technology, industry, and everyday commerce.

A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand: one kilogram is equal to one thousand grams. The prefix milli-, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousandth of a metre. The steradian or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radians, projected onto a circle, gives a length on the circumference, a solid angle in steradians, projected onto a sphere, gives an area on the surface. The name is derived from the Greek στερεός stereos 'solid' + radian. The caesium standard is a primary frequency standard in which the photon absorption by transitions between the two hyperfine ground states of caesium-133 atoms is used to control the output frequency. The first caesium clock was built by Louis Essen in 1955 at the National Physical Laboratory in the UK. and promoted worldwide by Gernot M. R. Winkler of the United States Naval Observatory. The metric system is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the International System of Units (SI) in the mid-20th century, under the oversight of an international standards body. Adopting the metric system is known as metrication.

A dimensionless quantity is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one, which is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Dimensionless quantities are distinct from quantities that have associated dimensions, such as time. Dimensionless units are dimensionless values that serve as units of measurement for expressing other quantities, such as radians (rad) or steradians (sr) for plane angles and solid angles, respectively. For example, optical extent is defined as having units of metres multiplied by steradians. In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement. Commonly used are parts-per-million, parts-per-billion, parts-per-trillion and parts-per-quadrillion. This notation is not part of the International System of Units (SI) system and its meaning is ambiguous.

A geometrized unit system, geometric unit system or geometrodynamic unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, c, and the gravitational constant, G, are set equal to unity.

The International System of Quantities (ISQ) consists of the quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This system underlies the International System of Units (SI) but does not itself determine the units of measurement used for the quantities. It is inherently incomplete because the number of quantities is potentially infinite.

The Unified Code for Units of Measure (UCUM) is a system of codes for unambiguously representing measurement units. Its primary purpose is machine-to-machine communication rather than communication between humans. A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multiple of the unit of measurement.

Quantity calculus is the formal method for describing the mathematical relations between abstract physical quantities. Its roots can be traced to Fourier's concept of dimensional analysis (1822). The basic axiom of quantity calculus is Maxwell's description of a physical quantity as the product of a "numerical value" and a "reference quantity". De Boer summarized the multiplication, division, addition, association and commutation rules of quantity calculus and proposed that a full axiomatization has yet to be completed. Such axiomatization needs to begin from a definition of quantity in terms of physical dimension(see dimensional analysis) which is indeed a more fundamental concept than of unit or unit quantity or unit of measurement. In 2019, four of the seven SI base units specified in the International System of Quantities were redefined in terms of natural physical constants, rather than human artifacts such as the standard kilogram. Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are now defined by setting exact numerical values, when expressed in SI units, for the Planck constant, the elementary electric charge, the Boltzmann constant, and the Avogadro constant, respectively. The second, metre, and candela had previously been redefined using physical constants. The four new definitions aimed to improve the SI without changing the value of any units, ensuring continuity with existing measurements. In November 2018, the 26th General Conference on Weights and Measures (CGPM) unanimously approved these changes, which the International Committee for Weights and Measures (CIPM) had proposed earlier that year after determining that previously agreed conditions for the change had been met. These conditions were satisfied by a series of experiments that measured the constants to high accuracy relative to the old SI definitions, and were the culmination of decades of research. A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations directly relating the quantities.

1. Suplee, Curt (2 July 2009). "Special Publication 811".
2. International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN   92-822-2213-6, archived (PDF) from the original on 4 June 2021, retrieved 16 December 2021
3. "Resolution 8 of the CGPM at its 20th Meeting (1995)". Bureau International des Poids et Mesures . Retrieved 23 September 2014.