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                         International System of Units

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   "SI" redirects here. For other uses, see Si (disambiguation).
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   For broader coverage of this topic, see Metric system.

   [IMG]
   Enlarge
   Approximation of the SI logo produced by the BIPM, showing the seven SI
   base units and the seven defining constants^[1]

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

                             SI defining constants
   Symbol Name                                   Exact value             
   Δν_Cs  hyperfine transition frequency of Cs   9192631770 Hz           
   c      speed of light                         299792458 m/s           
   h      Planck constant                        6.62607015×10^−34 J⋅s   
   e      elementary charge                      1.602176634×10^−19 C    
   k      Boltzmann constant                     1.380649×10^−23 J/K     
   N_A    Avogadro constant                      6.02214076×10^23 mol^−1 
   K_cd   luminous efficacy of 540 THz radiation 683 lm/W                

   The International System of Units, known by the international abbreviation
   SI^[a] in all languages^[2]^: 125 ^[3]^: iii ^[4] and sometimes
   pleonastically as the SI system,^[b] is the modern form^[2]^: 117 ^[7]^[8]
   of the metric system^[g] and the world's most widely used system of
   measurement.^[2]^: 123 ^[10]^[11] Established and maintained^[12] by the
   General Conference on Weights and Measures^[j] (CGPM^[k]), it is the only
   system of measurement with an official status^[m] in nearly every country
   in the world,^[n] employed in science, technology, industry, and everyday
   commerce.

   The SI comprises a coherent^[o] system of units of measurement starting
   with seven base units, which are the second (symbol s, the unit of time),
   metre (m, length), kilogram (kg, mass), ampere (A, electric current),
   kelvin (K, thermodynamic temperature), mole (mol, amount of substance),
   and candela (cd, luminous intensity). The system can accommodate coherent
   units for an unlimited number of additional quantities. These are called
   coherent derived units, which can always be represented as products of
   powers of the base units.^[p] Twenty-two coherent derived units have been
   provided with special names and symbols.^[q]

   The seven base units and the 22 coherent derived units with special names
   and symbols may be used in combination to express other coherent derived
   units.^[r] Since the sizes of coherent units will be convenient for only
   some applications and not for others, the SI provides twenty prefixes
   which, when added to the name and symbol of a coherent unit^[s] produce
   twenty additional (non-coherent) SI units for the same quantity; these
   non-coherent units are always decimal (i.e. power-of-ten) multiples and
   sub-multiples of the coherent unit.^[t]^[u] 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.

   Since 2019, the magnitudes of all SI units have been defined by declaring
   that seven defining constants have certain exact numerical values when
   expressed in terms of their SI units. These defining constants are the
   speed of light in vacuum c, the hyperfine transition frequency of caesium
   Δν_Cs, the Planck constant h, the elementary charge e, the Boltzmann
   constant k, the Avogadro constant N_A, and the luminous efficacy K_cd. The
   nature of the defining constants ranges from fundamental constants of
   nature such as c to the purely technical constant K_cd. Prior to 2019, h,
   e, k, and N_A were not defined a priori but were rather very precisely
   measured quantities. In 2019, their values were fixed by definition to
   their best estimates at the time, ensuring continuity with previous
   definitions of the base units.

   The current way of defining the SI is a result of a decades-long move
   towards increasingly abstract and idealised formulation in which the
   realisations of the units are separated conceptually from the definitions.
   A consequence is that as science and technologies develop, new and
   superior realisations may be introduced without the need to redefine the
   unit. One problem with artefacts is that they can be lost, damaged, or
   changed; another is that they introduce uncertainties that cannot be
   reduced by advancements in science and technology. The last artefact used
   by the SI was the International Prototype of the Kilogram, a cylinder of
   platinum-iridium.

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

Contents

     * 1 Introduction
          * 1.1 Controlling body
          * 1.2 Overview of the units
               * 1.2.1 SI base units
               * 1.2.2 SI derived units
               * 1.2.3 Why SI kept the distinction between base and derived
                 units
               * 1.2.4 SI metric prefixes and the decimal nature of the SI
               * 1.2.5 Coherent and non-coherent SI units
               * 1.2.6 Permitted non-SI units
               * 1.2.7 New units
          * 1.3 Defining magnitudes of units
          * 1.4 Specifying fundamental constants vs. other methods of
            definition
          * 1.5 History
     * 2 Controlling authority
     * 3 Units and prefixes
          * 3.1 Base units
          * 3.2 Derived units
          * 3.3 Prefixes
          * 3.4 Non-SI units accepted for use with SI
          * 3.5 Common notions of the metric units
     * 4 Lexicographic conventions
          * 4.1 Unit names
          * 4.2 Unit symbols and the values of quantities
               * 4.2.1 General rules
               * 4.2.2 Printing SI symbols
     * 5 International System of Quantities
     * 6 Realisation of units
     * 7 Evolution of the SI
          * 7.1 Changes to the SI
          * 7.2 2019 redefinitions
     * 8 History
          * 8.1 The improvisation of units
          * 8.2 Metre Convention
          * 8.3 The CGS and MKS systems
          * 8.4 The Practical system of units
          * 8.5 Birth of the SI
          * 8.6 Historical definitions
     * 9 Metric units that are not recognised by the SI
     * 10 See also
     * 11 Notes
     * 12 References
     * 13 Further reading
     * 14 External links
          * 14.1 Spelling and Usage

Introduction[edit]

   [IMG]
   Enlarge
   Countries using the metric (SI), imperial, and US customary systems as of
   2019.

   The International System of Units, or SI,^[2]^: 123  is a decimal^[v] and
   metric^[w] system of units established in 1960 and periodically updated
   since then. The SI has an official status in most countries,^[x] including
   the United States,^[y] Canada, and the United Kingdom, although these
   three countries are amongst a handful of nations that, to various degrees,
   also continue to use their customary systems. Nevertheless, with this
   nearly universal level of acceptance, the SI "has been used around the
   world as the preferred system of units, the basic language for science,
   technology, industry and trade."^[2]^: 123 

   The only other types of measurement system that still have widespread use
   across the world are the Imperial and US customary measurement
   systems^[z], and they are legally defined in terms of the SI.^[aa] There
   are other, less widespread systems of measurement that are occasionally
   used in particular regions of the world. In addition, there are many
   individual non-SI units that don't belong to any comprehensive system of
   units, but that are nevertheless still regularly used in particular fields
   and regions. Both of these categories of unit are also typically defined
   legally in terms of SI units.^[ab]

  Controlling body[edit]

   The SI was established and is maintained by the General Conference on
   Weights and Measures (CGPM^[k]).^[12] In practice, the CGPM follows the
   recommendations of the Consultative Committee for Units (CCU), which is
   the actual body conducting technical deliberations concerning new
   scientific and technological developments related to the definition of
   units and the SI. The CCU reports to the International Committee for
   Weights and Measures (CIPM^[ac]), which, in turn, reports to the CGPM. See
   below for more details.

   All the decisions and recommendations concerning units are collected in a
   brochure called The International System of Units (SI)^[ad], which is
   published by the International Bureau of Weights and Measures (BIPM^[ae])
   and periodically updated.

  Overview of the units[edit]

    SI base units[edit]

   The SI selects seven units to serve as base units, corresponding to seven
   base physical quantities.^[af]^[ag] They are the second, with the symbol
   s, which is the SI unit of the physical quantity of time; the metre,
   symbol m, the SI unit of length; kilogram (kg, the unit of mass); ampere
   (A, electric current); kelvin (K, thermodynamic temperature); mole (mol,
   amount of substance); and candela (cd, luminous intensity).^[2] All units
   in the SI can be expressed in terms of the base units, and the base units
   serve as a preferred set for expressing or analysing the relationships
   between units.

    SI derived units[edit]

   The system allows for an unlimited number of additional units, called
   derived units, which can always be represented as products of powers of
   the base units, possibly with a nontrivial numeric multiplier. When that
   multiplier is one, the unit is called a coherent derived unit.^[ah] The
   base and coherent derived units of the SI together form a coherent system
   of units (the set of coherent SI units).^[ai] Twenty-two coherent derived
   units have been provided with special names and symbols.^[q] The seven
   base units and the 22 derived units with special names and symbols may be
   used in combination to express other derived units,^[r] which are adopted
   to facilitate measurement of diverse quantities.

    Why SI kept the distinction between base and derived units[edit]

   Prior to its redefinition in 2019, the SI was defined through the seven
   base units from which the derived units were constructed as products of
   powers of the base units. After the redefinition, the SI is defined by
   fixing the numerical values of seven defining constants. This has the
   effect that the distinction between the base units and derived units is,
   in principle, not needed, since all units, base as well as derived, may be
   constructed directly from the defining constants. Nevertheless, the
   distinction is retained because 'it is useful and historically well
   established', and also because the ISO/IEC 80000 series of standards^[aj]
   specifies base and derived quantities that necessarily have the
   corresponding SI units.^[2]^: 129 

    SI metric prefixes and the decimal nature of the SI[edit]

   Like all metric systems, the SI uses metric prefixes to systematically
   construct, for the same physical quantity, a set of units that are decimal
   multiples of each other over a wide range.

   For example, while the coherent unit of length is the metre,^[ak] the SI
   provides a full range of smaller and larger units of length, any of which
   may be more convenient for any given application – for example, driving
   distances are normally given in kilometres (symbol km) rather than in
   metres. Here the metric prefix 'kilo-' (symbol 'k') stands for a factor of
   1000; thus, 1 km = 1000 m.^[al]

   The current version of the SI provides twenty metric prefixes that signify
   decimal powers ranging from 10^−24 to 10^24.^[2]^: 143–4  Most prefixes
   correspond to integer powers of 1000; the only ones that do not are those
   for 10, 1/10, 100, and 1/100.

   In general, given any coherent unit with a separate name and symbol,^[am]
   one forms a new unit by simply adding an appropriate metric prefix to the
   name of the coherent unit (and a corresponding prefix symbol to the
   coherent unit's symbol).^[an] Since the metric prefix signifies a
   particular power of ten, the new unit is always a power-of-ten multiple or
   sub-multiple of the coherent unit. Thus, the conversion between different
   SI units for one and the same physical quantity is always through a power
   of ten.^[ao] This is why the SI (and metric systems more generally) are
   called decimal systems of measurement units.^[18]^[ap]

   The grouping formed by a prefix symbol attached to a unit symbol (e.g.
   'km', 'cm') constitutes a new inseparable unit symbol. This new symbol can
   be raised to a positive or negative power and can be combined with other
   unit symbols to form compound unit symbols.^[2]^: 143  For example, g/cm^3
   is an SI unit of density, where cm^3 is to be interpreted as (cm)^3.

    Coherent and non-coherent SI units[edit]

   When prefixes are used with the coherent SI units, the resulting units are
   no longer coherent, because the prefix introduces a numerical factor other
   than one.^[2]^: 137  The one exception is the kilogram, the only coherent
   SI unit whose name and symbol, for historical reasons, include a
   prefix.^[an]

   The complete set of SI units consists of both the coherent set and the
   multiples and sub-multiples of coherent units formed by using the SI
   prefixes.^[2]^: 138  For example, the metre, kilometre, centimetre,
   nanometre, etc. are all SI units of length, though only the metre is a
   coherent SI unit. A similar statement holds for derived units: for
   example, kg/m^3, g/dm^3, g/cm^3, Pg/km^3, etc. are all SI units of
   density, but of these, only kg/m^3 is a coherent SI unit.

   Moreover, the metre is the only coherent SI unit of length. Every physical
   quantity has exactly one coherent SI unit, although this unit may be
   expressible in different forms by using some of the special names and
   symbols.^[2]^: 140  For example, the coherent SI unit of linear momentum
   may be written as either kg⋅m/s or as N⋅s, and both forms are in use (e.g.
   compare respectively here^[19]^:205 and here^[20]^:135).

   On the other hand, several different quantities may share same coherent SI
   unit. For example, the joule per kelvin (symbol J/K) is the coherent SI
   unit for two distinct quantities: heat capacity and entropy; another
   example is the ampere, which is the coherent SI unit for both electric
   current and magnetomotive force. This is why it is important not to use
   the unit alone to specify the quantity.^[aq]

   Furthermore, the same coherent SI unit may be a base unit in one context,
   but a coherent derived unit in another. For example, the ampere is a base
   unit when it is a unit of electric current, but a coherent derived unit
   when it is a unit of magnetomotive force.^[2]^: 140  As perhaps a more
   familiar example, consider rainfall, defined as volume of rain (measured
   in m^3) that fell per unit area (measured in m^2). Since m^3/m^2=m, it
   follows that the coherent derived SI unit of rainfall is the metre, even
   though the metre is also the base SI unit of length.^[ar]

    Permitted non-SI units[edit]

   There is a special group of units that are called "non-SI units that are
   accepted for use with the SI".^[2]^: 145  See Non-SI units mentioned in
   the SI for a full list. Most of these, in order to be converted to the
   corresponding SI unit, require conversion factors that are not powers of
   ten. Some common examples of such units are the customary units of time,
   namely the minute (conversion factor of 60 s/min, since 1 min = 60 s), the
   hour (3600 s), and the day (86400 s); the degree (for measuring plane
   angles, 1° = π/180 rad); and the electronvolt (a unit of energy, 1 eV =
   1.602176634×10^−19 J).

    New units[edit]

   The SI is intended to be an evolving system; units^[as] and prefixes are
   created and unit definitions are modified through international agreement
   as the technology of measurement progresses and the precision of
   measurements improves.

  Defining magnitudes of units[edit]

   Since 2019, the magnitudes of all SI units have been defined in an
   abstract way, which is conceptually separated from any practical
   realisation of them.^[2]^: 126 ^[at] Namely, the SI units are defined by
   declaring that seven defining constants^[2]^: 125–9  have certain exact
   numerical values when expressed in terms of their SI units. Probably the
   most widely known of these constants is the speed of light in vacuum, c,
   which in the SI by definition has the exact value of c = 299792458 m/s.
   The other six constants are Δν_Cs, the hyperfine transition frequency of
   caesium; h, the Planck constant; e, the elementary charge; k, the
   Boltzmann constant; N_A, the Avogadro constant; and K_cd, the luminous
   efficacy of monochromatic radiation of frequency 540×10^12 Hz.^[au] The
   nature of the defining constants ranges from fundamental constants of
   nature such as c to the purely technical constant K_cd.^[2]^: 128–9  Prior
   to 2019, h, e, k, and N_A were not defined a priori but were rather very
   precisely measured quantities. In 2019, their values were fixed by
   definition to their best estimates at the time, ensuring continuity with
   previous definitions of the base units.

   As far as realisations, what are believed to be the current best practical
   realisations of units are described in the so-called 'mises en
   pratique',^[av] which are also published by the BIPM.^[23] The abstract
   nature of the definitions of units is what makes it possible to improve
   and change the mises en pratique as science and technology develop without
   having to change the actual definitions themselves.^[ay]

   In a sense, this way of defining the SI units is no more abstract than the
   way derived units are traditionally defined in terms of the base units.
   Consider a particular derived unit, for example, the joule, the unit of
   energy. Its definition in terms of the base units is kg⋅m^2/s^2. Even if
   the practical realisations of the metre, kilogram, and second are
   available, a practical realisation of the joule would require some sort of
   reference to the underlying physical definition of work or energy—some
   actual physical procedure for realising the energy in the amount of one
   joule such that it can be compared to other instances of energy (such as
   the energy content of gasoline put into a car or of electricity delivered
   to a household).

   The situation with the defining constants and all of the SI units is
   analogous. In fact, purely mathematically speaking, the SI units are
   defined as if we declared that it is the defining constant's units that
   are now the base units, with all other SI units being derived units. To
   make this clearer, first note that each defining constant can be taken as
   determining the magnitude of that defining constant's unit of
   measurement;^[2]^: 128  for example, the definition of c defines the unit
   m/s as 1 m/s =
   Link: mw-deduplicated-inline-style
   c/299792458 ('the speed of one metre per second is equal to one
   299792458th of the speed of light'). In this way, the defining constants
   directly define the following seven units:

     * the hertz (Hz), a unit of the physical quantity of frequency (note
       that problems can arise when dealing with frequency or the Planck
       constant because the units of angular measure (cycle or radian) are
       omitted in SI^[24]^[25]^[5]^[26]^[27]);
     * the metre per second (m/s), a unit of speed;
     * the joule-second (J⋅s), a unit of action;
     * the coulomb (C), a unit of electric charge;
     * the joule per kelvin (J/K), a unit of both entropy and heat capacity;
     * the inverse mole (mol^−1), a unit of a conversion constant between the
       amount of substance and the number of elementary entities (atoms,
       molecules, etc.);
     * and the lumen per watt (lm/W), a unit of luminous efficacy (conversion
       constant between the physical power carried by electromagnetic
       radiation and the intrinsic ability of that same radiation to produce
       visual perception of brightness in humans).

   Further, one can show, using dimensional analysis, that every coherent SI
   unit (whether base or derived) can be written as a unique product of
   powers of the units of the SI defining constants (in complete analogy to
   the fact that every coherent derived SI unit can be written as a unique
   product of powers of the base SI units). For example, the kilogram can be
   written as kg = (Hz)(J⋅s)/(m/s)^2.^[az] Thus, the kilogram is defined in
   terms of the three defining constants Δν_Cs, c, and h because, on the one
   hand, these three defining constants respectively define the units Hz,
   m/s, and J⋅s,^[ba] while, on the other hand, the kilogram can be written
   in terms of these three units, namely, kg = (Hz)(J⋅s)/(m/s)^2.^[bb] While
   the question of how to actually realise the kilogram in practice would, at
   this point, still be open, that is not really different from the fact that
   the question of how to actually realise the joule in practice is still in
   principle open even once one has achieved the practical realisations of
   the metre, kilogram, and second.

  Specifying fundamental constants vs. other methods of definition[edit]

   The current way of defining the SI is the result of a decades-long move
   towards increasingly abstract and idealised formulation in which the
   realisations of the units are separated conceptually from the
   definitions.^[2]^: 126 

   The great advantage of doing it this way is that as science and
   technologies develop, new and superior realisations may be introduced
   without the need to redefine the units.^[aw] Units can now be realised
   with 'an accuracy that is ultimately limited only by the quantum structure
   of nature and our technical abilities but not by the definitions
   themselves.^[ax] Any valid equation of physics relating the defining
   constants to a unit can be used to realise the unit, thus creating
   opportunities for innovation... with increasing accuracy as technology
   proceeds.'^[2]^: 122  In practice, the CIPM Consultative Committees
   provide so-called "mises en pratique" (practical techniques),^[23] which
   are the descriptions of what are currently believed to be best
   experimental realisations of the units.^[31]

   This system lacks the conceptual simplicity of using artefacts (referred
   to as prototypes) as realisations of units to define those units: with
   prototypes, the definition and the realisation are one and the same.^[bc]
   However, using artefacts has two major disadvantages that, as soon as it
   is technologically and scientifically feasible, result in abandoning them
   as means for defining units.^[bg] One major disadvantage is that artefacts
   can be lost, damaged,^[bi] or changed.^[bj] The other is that they largely
   cannot benefit from advancements in science and technology. The last
   artefact used by the SI was the International Prototype Kilogram (IPK), a
   particular cylinder of platinum-iridium; from 1889 to 2019, the kilogram
   was by definition equal to the mass of the IPK. Concerns regarding its
   stability on the one hand, and progress in precise measurements of the
   Planck constant and the Avogadro constant on the other, led to a revision
   of the definition of the base units, put into effect on 20 May 2019.^[38]
   This was the biggest change in the SI since it was first formally defined
   and established in 1960, and it resulted in the definitions described
   above.^[39]

   In the past, there were also various other approaches to the definitions
   of some of the SI units. One made use of a specific physical state of a
   specific substance (the triple point of water, which was used in the
   definition of the kelvin^[40]^: 113–4 ); others referred to idealised
   experimental prescriptions^[2]^: 125  (as in the case of the former SI
   definition of the ampere^[40]^: 113  and the former SI definition
   (originally enacted in 1979) of the candela^[40]^: 115 ).

   In the future, the set of defining constants used by the SI may be
   modified as more stable constants are found, or if it turns out that other
   constants can be more precisely measured.^[bk]

  History[edit]

   The original 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.

   Adopted in 1889, use of the MKS system of units succeeded the
   centimetre–gram–second system of units (CGS) in commerce and engineering.
   The metre and kilogram system served as the basis for the development of
   the International System of Units (abbreviated SI), which now serves as
   the international standard. Because of this, the standards of the CGS
   system were gradually replaced with metric standards incorporated from the
   MKS system.^[41]

   In 1901, Giovanni Giorgi proposed to the Associazione elettrotecnica
   italiana [it] (AEI) that this system, extended with a fourth unit to be
   taken from the units of electromagnetism, be used as an international
   system.^[42] This system was strongly promoted by electrical engineer
   George A. Campbell.^[43]

   The International System was published in 1960, based on the MKS units, as
   a result of an initiative that began in 1948.

Controlling authority[edit]

   The SI is regulated and continually developed by three international
   organisations that were established in 1875 under the terms of the Metre
   Convention. They are the General Conference on Weights and Measures
   (CGPM^[k]), the International Committee for Weights and Measures
   (CIPM^[ac]), and the International Bureau of Weights and Measures
   (BIPM^[ae]). The ultimate authority rests with the CGPM, which is a
   plenary body through which its Member States^[bl] act together on matters
   related to measurement science and measurement standards; it usually
   convenes every four years.^[13] The CGPM elects the CIPM, which is an
   18-person committee of eminent scientists. The CIPM operates based on the
   advice of a number of its Consultative Committees, which bring together
   the world's experts in their specified fields as advisers on scientific
   and technical matters.^[44]^[bm] One of these committees is the
   Consultative Committee for Units (CCU), which is responsible for matters
   related to the development of the International System of Units (SI),
   preparation of successive editions of the SI brochure, and advice to the
   CIPM on matters concerning units of measurement.^[45] It is the CCU which
   considers in detail all new scientific and technological developments
   related to the definition of units and the SI. In practice, when it comes
   to the definition of the SI, the CGPM simply formally approves the
   recommendations of the CIPM, which, in turn, follows the advice of the
   CCU.

   The CCU has the following as members:^[46]^[47] national laboratories of
   the Member States of the CGPM charged with establishing national
   standards;^[bn] relevant intergovernmental organisations and international
   bodies;^[bo] international commissions or committees;^[bp] scientific
   unions;^[bq] personal members;^[br] and, as an ex officio member of all
   Consultative Committees, the Director of the BIPM.

   All the decisions and recommendations concerning units are collected in a
   brochure called The International System of Units (SI)^[2]^[ad], which is
   published by the BIPM and periodically updated.

Units and prefixes[edit]

   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.^[40]^: 103–106  The units, excluding prefixed units,^[bs] 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/s^2 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.^[bt] Other useful derived quantities can be specified in terms of
   the SI base and derived units that have no named units in the SI, such as
   acceleration, which is defined in SI units as m/s^2.

  Base units[edit]

   Link: mw-deduplicated-inline-style
   Main article: SI base unit

   The SI base units are the building blocks of the system and all the other
   units are derived from them.

                        SI base units^[3]^: 6 ^[50]^[51]
Unit     Unit   Dimension Quantity      Typical        Definition                   
name     symbol symbol    name          symbols        
                                                       The duration of 9192631770   
                                                       periods of the radiation     
second                                                 corresponding to the         
^[n 1]   s      T         time          t              transition between the two   
                                                       hyperfine levels of the      
                                                       ground state of the          
                                                       caesium-133 atom.            
                                                       The distance travelled by    
                                        l, h, a, b, x, light in a vacuum in         
metre    m      L         length        y, r, etc.^[n  Link:                        
                                        2]             mw-deduplicated-inline-style 
                                                       1/299792458 seconds.         
                                                       The kilogram is defined by   
                                                       setting the Planck constant  
kilogram                                               h exactly to                 
^[n 3]   kg     M         mass          m              6.62607015×10^−34 J⋅s (J =   
                                                       kg⋅m^2⋅s^−2), given the      
                                                       definitions of the metre and 
                                                       the second.^[38]             
                                                       The flow of exactly          
                                                       Link:                        
                                                       mw-deduplicated-inline-style 
                                                       1/1.602176634×10^−19 times   
                          electric      {\displaystyle the elementary charge e per  
ampere   A      I         current       I,\;i}         second.                      
                                                                                    
                                                       Equalling approximately      
                                                       6.2415090744×10^18           
                                                       elementary charges per       
                                                       second.                      
                                                       The kelvin is defined by     
                                                       setting the fixed numerical  
                                                       value of the Boltzmann       
kelvin   K      Θ         thermodynamic T              constant k to                
                          temperature                  1.380649×10^−23 J⋅K^−1, (J = 
                                                       kg⋅m^2⋅s^−2), given the      
                                                       definition of the kilogram,  
                                                       the metre, and the second.   
                                                       The amount of substance of   
                                                       exactly 6.02214076×10^23     
                                                       elementary entities.^[n 4]   
mole     mol    N         amount of     n              This number is the fixed     
                          substance                    numerical value of the       
                                                       Avogadro constant, N_A, when 
                                                       expressed in the unit        
                                                       mol^−1.                      
                                                       The luminous intensity, in a 
                                                       given direction, of a source 
                                                       that emits monochromatic     
                                                       radiation of frequency       
candela  cd     J         luminous      I_v            5.4×10^14 hertz and that has 
                          intensity                    a radiant intensity in that  
                                                       direction of                 
                                                       Link:                        
                                                       mw-deduplicated-inline-style 
                                                       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 (1⁄60^2) 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. ^ Symbols for length vary greatly with context. Problems involving intuitive
    three-dimensional quantities often use l, w, and h for length, distance, and
    height, respectively. More generally, physicists tend to set up the coordinate
    system of a given problem so that one axis lies conveniently parallel to the
    length being measured. Length is then often denoted either by some constant
    (e.g. a, b) along said axis, or by the same symbol as the axis itself (e.g. x,
    y, or r for horizontal, vertical, and radial axes, respectively).
 3. ^ 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.
 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.

  Derived units[edit]

   Link: mw-deduplicated-inline-style
   Main article: SI derived unit

   The derived units in the SI are formed by powers, products, or quotients
   of the base units and are potentially unlimited in
   number.^[40]^: 103 ^[3]^: 14, 16  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/m^2).^[52]

           SI derived units with special names and symbols^[3]^: 15 
   Name            Symbol Quantity             In SI base units   In other SI 
                                                                  units       
   radian^[N 1]    rad    plane angle          m/m                1           
   steradian^[N 1] sr     solid angle          m^2/m^2            1           
   hertz           Hz     frequency            s^−1               
   newton          N      force, weight        kg⋅m⋅s^−2          
   pascal          Pa     pressure, stress     kg⋅m^−1⋅s^−2       N/m^2       
   joule           J      energy, work, heat   kg⋅m^2⋅s^−2        N⋅m =       
                                                                  Pa⋅m^3      
   watt            W      power, radiant flux  kg⋅m^2⋅s^−3        J/s         
   coulomb         C      electric charge      s⋅A                
                          electrical potential                                
   volt            V      difference           kg⋅m^2⋅s^−3⋅A^−1   W/A = J/C
                          (voltage), emf       
   farad           F      capacitance          kg^−1⋅m^−2⋅s^4⋅A^2 C/V = C^2/J 
   ohm             Ω      resistance,          kg⋅m^2⋅s^−3⋅A^−2   V/A =       
                          impedance, reactance                    J⋅s/C^2     
   siemens         S      electrical           kg^−1⋅m^−2⋅s^3⋅A^2 Ω^−1        
                          conductance          
   weber           Wb     magnetic flux        kg⋅m^2⋅s^−2⋅A^−1   V⋅s         
   tesla           T      magnetic flux        kg⋅s^−2⋅A^−1       Wb/m^2      
                          density              
   henry           H      inductance           kg⋅m^2⋅s^−2⋅A^−2   Wb/A        
   degree Celsius  °C     temperature relative K                  
                          to 273.15 K          
   lumen           lm     luminous flux        cd⋅sr              cd⋅sr       
   lux             lx     illuminance          cd⋅sr⋅m^−2         lm/m^2      
                          activity referred to                    
   becquerel       Bq     a radionuclide       s^−1
                          (decays per unit     
                          time)                
   gray            Gy     absorbed dose (of    m^2⋅s^−2           J/kg        
                          ionising radiation)  
   sievert         Sv     equivalent dose (of  m^2⋅s^−2           J/kg        
                          ionising radiation)  
   katal           kat    catalytic activity   mol⋅s^−1           
   Notes           
    1. ^ ^a ^b The radian and steradian are defined as dimensionless derived
       units.      

   [IMG]
   Enlarge
   Arrangement of the principal measurements in physics based on the
   mathematical manipulation of length, time, and mass.

      Examples of coherent derived units in terms of base units^[3]^: 17 
   Name                      Symbol  Derived quantity        Typical symbol 
   square metre              m^2     area                    A              
   cubic metre               m^3     volume                  V              
   metre per second          m/s     speed, velocity         v              
   metre per second squared  m/s^2   acceleration            a              
   reciprocal metre          m^−1    wavenumber              σ, ṽ           
                                     vergence (optics)       V, 1/f         
   kilogram per cubic metre  kg/m^3  density                 ρ              
   kilogram per square metre kg/m^2  surface density         ρ_A            
   cubic metre per kilogram  m^3/kg  specific volume         v              
   ampere per square metre   A/m^2   current density         j              
   ampere per metre          A/m     magnetic field strength H              
   mole per cubic metre      mol/m^3 concentration           c              
   kilogram per cubic metre  kg/m^3  mass concentration      ρ, γ           
   candela per square metre  cd/m^2  luminance               L_v            

   Examples of derived units that include units with special names^[3]^: 18 
   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     m^2⋅kg⋅s^−2             
   newton per metre    N/m        surface tension     kg⋅s^−2                 
   radian per second   rad/s      angular velocity,   s^−1                    
                                  angular frequency   
   radian per second   rad/s^2    angular             s^−2                    
   squared                        acceleration        
   watt per square     W/m^2      heat flux density,  kg⋅s^−3                 
   metre                          irradiance          
   joule per kelvin    J/K        entropy, heat       m^2⋅kg⋅s^−2⋅K^−1        
                                  capacity            
   joule per                      specific heat                               
   kilogram-kelvin     J/(kg⋅K)   capacity, specific  m^2⋅s^−2⋅K^−1
                                  entropy             
   joule per kilogram  J/kg       specific energy     m^2⋅s^−2                
   watt per            W/(m⋅K)    thermal             m⋅kg⋅s^−3⋅K^−1          
   metre-kelvin                   conductivity        
   joule per cubic     J/m^3      energy density      m^−1⋅kg⋅s^−2            
   metre               
   volt per metre      V/m        electric field      m⋅kg⋅s^−3⋅A^−1          
                                  strength            
   coulomb per cubic   C/m^3      electric charge     m^−3⋅s⋅A                
   metre                          density             
                                  surface charge                              
   coulomb per square             density, electric   
   metre               C/m^2      flux density,       m^−2⋅s⋅A
                                  electric            
                                  displacement        
   farad per metre     F/m        permittivity        m^−3⋅kg^−1⋅s^4⋅A^2      
   henry per metre     H/m        permeability        m⋅kg⋅s^−2⋅A^−2          
   joule per mole      J/mol      molar energy        m^2⋅kg⋅s^−2⋅mol^−1      
   joule per           J/(mol⋅K)  molar entropy,      m^2⋅kg⋅s^−2⋅K^−1⋅mol^−1 
   mole-kelvin                    molar heat capacity 
   coulomb per         C/kg       exposure (x- and    kg^−1⋅s⋅A               
   kilogram                       γ-rays)             
   gray per second     Gy/s       absorbed dose rate  m^2⋅s^−3                
   watt per steradian  W/sr       radiant intensity   m^2⋅kg⋅s^−3             
   watt per square     W/(m^2⋅sr) radiance            kg⋅s^−3                 
   metre-steradian     
   katal per cubic     kat/m^3    catalytic activity  m^−3⋅s^−1⋅mol           
   metre                          concentration       

  Prefixes[edit]

   Link: mw-deduplicated-inline-style
   Main article: Metric prefix

   Prefixes are added to unit names to produce multiples and submultiples 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.^[40]^: 122 ^[53]^: 14  When prefixes are used to form
   multiples and submultiples of SI base and derived units, the resulting
   units are no longer coherent.^[40]^: 7 

   The BIPM specifies 20 prefixes for the International System of Units
   (SI):^[bu]

                                  SI prefixes
     * v
     * t
     * e
Prefix       Base                              English word                Adoption^[nb Etymology
Name  Symbol 10     Decimal                    Short scale   Long scale    1]           Language Source    
                                                                                                 word      
yotta Y      10^24  1000000000000000000000000  septillion    quadrillion   1991         Latin    eight^[nb 
                                                                                                 2]        
zetta Z      10^21  1000000000000000000000     sextillion    trilliard     1991         Latin    seven^[nb 
                                                                                                 2]        
exa   E      10^18  1000000000000000000        quintillion   trillion      1975         Greek    six       
peta  P      10^15  1000000000000000           quadrillion   billiard      1975         Greek    five^[nb  
                                                                                                 2]        
                                                                                                 four,^[nb 
tera  T      10^12  1000000000000              trillion      billion       1960         Greek    2]        
                                                                                                 monster   
giga  G      10^9   1000000000                 billion       milliard      1960         Greek    giant     
mega  M      10^6   1000000                    million                     1873         Greek    great     
kilo  k      10^3   1000                       thousand                    1795         Greek    thousand  
hecto h      10^2   100                        hundred                     1795         Greek    hundred   
deca  da     10^1   10                         ten                         1795         Greek    ten       
             10^0   1                          one                         –            
deci  d      10^−1  0.1                        tenth                       1795         Latin    ten       
centi c      10^−2  0.01                       hundredth                   1795         Latin    hundred   
milli m      10^−3  0.001                      thousandth                  1795         Latin    thousand  
micro μ      10^−6  0.000001                   millionth                   1873         Greek    small     
nano  n      10^−9  0.000000001                billionth     milliardth    1960         Greek    dwarf     
                                                                                                 peak, a   
pico  p      10^−12 0.000000000001             trillionth    billionth     1960         Spanish  little    
                                                                                                 bit       
                                                                                                 fifteen,  
femto f      10^−15 0.000000000000001          quadrillionth billiardth    1964         Danish   Fermi^[nb 
                                                                                                 3]        
atto  a      10^−18 0.000000000000000001       quintillionth trillionth    1964         Danish   eighteen  
zepto z      10^−21 0.000000000000000000001    sextillionth  trilliardth   1991         Latin    seven^[nb 
                                                                                                 2]        
yocto y      10^−24 0.000000000000000000000001 septillionth  quadrillionth 1991         Latin    eight^[nb 
                                                                                                 2]        
 1. ^ Prefixes adopted before 1960 already existed before SI. The introduction of the CGS system was in
    1873.
 2. ^ ^a ^b ^c ^d ^e ^f Part of the beginning of the prefix was modified from the word it was derived
    from, ex: "peta" (prefix) vs "penta" (source word).
 3. ^ The fermi was introduced earlier with the symbol "fm", which prompted the reinterpretation of the
    "f" as a prefix to "m", with femto being derived from the Danish word femten due to its similarity.

  Non-SI units accepted for use with SI[edit]

   Link: mw-deduplicated-inline-style
   Main article: 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:^[40]

   [IMG]
   Enlarge
   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 m^3.

   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
   Link: mw-deduplicated-inline-style
   1/2π 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 SI units
   Quantity     Name          Symbol Value in SI units                        
                minute        min    1 min = 60 s                             
   time         hour          h      1 h = 60 min = 3600 s                    
                day           d      1 d = 24 h = 86400 s                     
   length       astronomical  au     1 au = 149597870700 m                    
                unit          
                                     1° =                                     
                degree        °      Link: mw-deduplicated-inline-style       
                                     π/180 rad                                
                                     1′ =                                     
                                     Link: mw-deduplicated-inline-style       
   plane and    minute        ′      1/60° =                                  
   phase angle                       Link: mw-deduplicated-inline-style       
                                     π/10800 rad                              
                                     1″ =                                     
                                     Link: mw-deduplicated-inline-style       
                second        ″      1/60′ =                                  
                                     Link: mw-deduplicated-inline-style       
                                     π/648000 rad                             
   area         hectare       ha     1 ha = 1 hm^2 = 10^4 m^2                 
   volume       litre         l, L   1 l = 1 L = 1 dm^3 = 10^3 cm^3 = 10^−3   
                                     m^3                                      
                tonne (metric t      1 t = 1 Mg = 10^3 kg                     
   mass         ton)          
                dalton        Da     1 Da = 1.660539040(20)×10^−27 kg         
   energy       electronvolt  eV     1 eV = 1.602176634×10^−19 J              
   logarithmic  neper         Np     In using these units it is important     
   ratio                             that the nature of the quantity be       
   quantities   bel           B      specified and that any reference value   
                decibel       dB     used be specified.                       

   These units are used in combination with SI units in common units such as
   the kilowatt-hour (1 kW⋅h = 3.6 MJ).

  Common notions of the metric units[edit]

   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.^[bv]

     * A second is
       Link: mw-deduplicated-inline-style
       1/60 of a minute, which is
       Link: mw-deduplicated-inline-style
       1/60 of an hour, which is
       Link: mw-deduplicated-inline-style
       1/24 of a day, so a second is
       Link: mw-deduplicated-inline-style
       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.^[bw]
     * The length of the equator is close to 40000000 m (more precisely
       40075014.2 m).^[55] In fact, the dimensions of our planet were used by
       the French Academy in the original definition of the metre.^[56]
     * The metre is close to the length of a pendulum that has a period of 2
       seconds;^[bx] most dining tabletops are about 0.75 metres high;^[57] a
       very tall human (basketball forward) is about 2 metres tall.^[58]
     * 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 weighs
       7.5 g;^[59] a Sacagawea US 1-dollar coin weighs 8.1 g;^[60] a UK
       50-pence coin weighs 8.0 g.^[61]
     * 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 candelas.^[by]
     * 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.
     * Since all gases have the same volume per mole at a given temperature
       and pressure far from their points of liquefaction and solidification
       (see Perfect gas), and air is about
       Link: mw-deduplicated-inline-style
       1/5 oxygen (molecular mass 32) and
       Link: mw-deduplicated-inline-style
       4/5 nitrogen (molecular mass 28), the density of any near-perfect gas
       relative to air can be obtained to a good approximation by dividing
       its molecular mass by 29 (because
       Link: mw-deduplicated-inline-style
       4/5 × 28 +
       Link: mw-deduplicated-inline-style
       1/5 × 32 = 28.8 ≈ 29). For example, carbon monoxide (molecular mass
       28) has almost the same density as air.
     * A temperature difference of one kelvin is the same as one degree
       Celsius:
       Link: mw-deduplicated-inline-style
       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 rated at 120 V (US mains voltage)
       consumes 0.5 A at this voltage. A 60 W bulb rated at 230 V (European
       mains voltage) consumes 0.26 A at this voltage.^[bz]

Lexicographic conventions[edit]

  Unit names[edit]

   According to the SI Brochure,^[2]^: 148  unit names should be treated as
   common nouns of the context language. This means that they should be
   typeset in the same character set as other common nouns (e.g. Latin
   alphabet in English, Cyrillic script in Russian, etc.), following the
   usual grammatical and orthographical rules of the context language. For
   example, in English and French, even when the unit is named after a person
   and its symbol begins with a capital letter, the unit name in running text
   should start with a lowercase letter (e.g., newton, hertz, pascal) and is
   capitalized only at the beginning of a sentence and in headings and
   publication titles. As a nontrivial application of this rule, the SI
   Brochure notes^[2]^: 148  that the name of the unit with the symbol °C is
   correctly spelled as 'degree Celsius': the first letter of the name of the
   unit, 'd', is in lowercase, while the modifier 'Celsius' is capitalized
   because it is a proper name.^[ca]^[2]^: 148 

   The English spelling and even names for certain SI units and metric
   prefixes depend on the variety of English used. US English uses the
   spelling deka-, meter, and liter, whilst International English uses deca-,
   metre, and litre. Additionally, the name of the unit whose symbol is t and
   which is defined according to 1 t = 10^3 kg is 'metric ton' US English but
   'tonne' in International English.^[3]^: iii 

  Unit symbols and the values of quantities [edit]

   Symbols of SI units are intended to be unique and universal, independent
   of the context language.^[40]^: 130–135  The SI Brochure has specific
   rules for writing them.^[40]^: 130–135  The guideline produced by the
   National Institute of Standards and Technology (NIST)^[63] clarifies
   language-specific details for American English that were left unclear by
   the SI Brochure, but is otherwise identical to the SI Brochure.^[64]

    General rules[edit]

   General rules^[cb] for writing SI units and quantities apply to text that
   is either handwritten or produced using an automated process:

     * The value of a quantity is written as a number followed by a space
       (representing a multiplication sign) and a unit symbol; e.g., 2.21 kg,
       7.3×10^2 m^2, 22 K. This rule explicitly includes the percent sign
       (%)^[40]^: 134  and the symbol for degrees Celsius (°C).^[40]^: 133 
       Exceptions are the symbols for plane angular degrees, minutes, and
       seconds (°, ′, and ″, respectively), which are placed immediately
       after the number with no intervening space.
     * Symbols are mathematical entities, not abbreviations, and as such do
       not have an appended period/full stop (.), unless the rules of grammar
       demand one for another reason, such as denoting the end of a sentence.
     * A prefix is part of the unit, and its symbol is prepended to a unit
       symbol without a separator (e.g., k in km, M in MPa, G in GHz, μ in
       μg). Compound prefixes are not allowed. A prefixed unit is atomic in
       expressions (e.g., km^2 is equivalent to (km)^2).
     * Unit symbols are written using roman (upright) type, regardless of the
       type used in the surrounding text.
     * Symbols for derived units formed by multiplication are joined with a
       centre dot (⋅) or a non-breaking space; e.g., N⋅m or N m.
     * Symbols for derived units formed by division are joined with a solidus
       (/), or given as a negative exponent. E.g., the "metre per second" can
       be written m/s, m s^−1, m⋅s^−1, or
       Link: mw-deduplicated-inline-style
       m/s. A solidus followed without parentheses by a centre dot (or space)
       is ambiguous and must be avoided; e.g., kg/(m⋅s^2) and kg⋅m^−1⋅s^−2
       are acceptable, but kg/m/s^2 is ambiguous and unacceptable.
   [IMG]
   Enlarge
   In the expression of acceleration due to gravity, a space separates the
   value and the units, both the 'm' and the 's' are lowercase because
   neither the metre nor the second are named after people, and
   exponentiation is represented with a superscript '2'.
     * The first letter of symbols for units derived from the name of a
       person is written in upper case; otherwise, they are written in lower
       case. E.g., the unit of pressure is named after Blaise Pascal, so its
       symbol is written "Pa", but the symbol for mole is written "mol".
       Thus, "T" is the symbol for tesla, a measure of magnetic field
       strength, and "t" the symbol for tonne, a measure of mass. Since 1979,
       the litre may exceptionally be written using either an uppercase "L"
       or a lowercase "l", a decision prompted by the similarity of the
       lowercase letter "l" to the numeral "1", especially with certain
       typefaces or English-style handwriting. The American NIST recommends
       that within the United States "L" be used rather than "l".
     * Symbols do not have a plural form, e.g., 25 kg, not 25 kgs.
     * Uppercase and lowercase prefixes are not interchangeable. E.g., the
       quantities 1 mW and 1 MW represent two different quantities (milliwatt
       and megawatt).
     * The symbol for the decimal marker is either a point or comma on the
       line. In practice, the decimal point is used in most English-speaking
       countries and most of Asia, and the comma in most of Latin America and
       in continental European countries.^[65]
     * Spaces should be used as a thousands separator (1000000) in contrast
       to commas or periods (1,000,000 or 1.000.000) to reduce confusion
       resulting from the variation between these forms in different
       countries.
     * Any line-break inside a number, inside a compound unit, or between
       number and unit should be avoided. Where this is not possible, line
       breaks should coincide with thousands separators.
     * Because the value of "billion" and "trillion" varies between
       languages, the dimensionless terms "ppb" (parts per billion) and "ppt"
       (parts per trillion) should be avoided. The SI Brochure does not
       suggest alternatives.

    Printing SI symbols[edit]

   The rules covering printing of quantities and units are part of ISO
   80000-1:2009.^[66]

   Further rules^[cb] are specified in respect of production of text using
   printing presses, word processors, typewriters, and the like.

International System of Quantities[edit]

                                                     SI Brochure

   [IMG]
   Enlarge
   Cover of brochure The International System of Units

   The CGPM publishes a brochure that defines and presents the SI.^[40] Its
   official version is in French, in line with the Metre
   Convention.^[40]^: 102  It leaves some scope for local variations,
   particularly regarding unit names and terms in different
   languages.^[cc]^[3]

   The writing and maintenance of the CGPM brochure is carried out by one of
   the committees of the International Committee for Weights and Measures
   (CIPM). The definitions of the terms "quantity", "unit", "dimension" etc.
   that are used in the SI Brochure are those given in the International
   vocabulary of metrology.^[67]

   Link: mw-deduplicated-inline-style
   Main article: International System of Quantities

   The quantities and equations that provide the context in which the SI
   units are defined are now referred to as the International System of
   Quantities (ISQ). The ISQ is based on the quantities underlying each of
   the seven base units of the SI. Other quantities, such as area, pressure,
   and electrical resistance, are derived from these base quantities by clear
   non-contradictory equations. The ISQ defines the quantities that are
   measured with the SI units.^[68] The ISQ is formalised, in part, in the
   international standard ISO/IEC 80000, which was completed in 2009 with the
   publication of ISO 80000-1,^[69] and has largely been revised in 2019–2020
   with the remainder being under review.

Realisation of units[edit]

   Link: mw-deduplicated-inline-style
   Main article: Realisation (metrology)
   [IMG]
   Enlarge
   Silicon sphere for the Avogadro project used for measuring the Avogadro
   constant to a relative standard uncertainty of 2×10^−8 or less, held by
   Achim Leistner^[70]

   Metrologists carefully distinguish between the definition of a unit and
   its realisation. The definition of each base unit of the SI is drawn up so
   that it is unique and provides a sound theoretical basis on which the most
   accurate and reproducible measurements can be made. The realisation of the
   definition of a unit is the procedure by which the definition may be used
   to establish the value and associated uncertainty of a quantity of the
   same kind as the unit. A description of the mise en pratique^[cd] of the
   base units is given in an electronic appendix to the SI
   Brochure.^[71]^[40]^: 168–169 

   The published mise en pratique is not the only way in which a base unit
   can be determined: the SI Brochure states that "any method consistent with
   the laws of physics could be used to realise any SI unit."^[40]^: 111 
   Various consultative committees of the CIPM decided in 2016 that more than
   one mise en pratique would be developed for determining the value of each
   unit.^[72] These methods include the following:

     * At least three separate experiments be carried out yielding values
       having a relative standard uncertainty in the determination of the
       kilogram of no more than 5×10^−8 and at least one of these values
       should be better than 2×10^−8. Both the Kibble balance and the
       Avogadro project should be included in the experiments and any
       differences between these be reconciled.^[73]^[74]
     * The definition of the kelvin measured with a relative uncertainty of
       the Boltzmann constant derived from two fundamentally different
       methods such as acoustic gas thermometry and dielectric constant gas
       thermometry be better than one part in 10^−6 and that these values be
       corroborated by other measurements.^[75]

Evolution of the SI[edit]

  Changes to the SI[edit]

   The International Bureau of Weights and Measures (BIPM) has described SI
   as "the modern form of metric system".^[40]^: 95  Changing technology has
   led to an evolution of the definitions and standards that has followed two
   principal strands – changes to SI itself, and clarification of how to use
   units of measure that are not part of SI but are still nevertheless used
   on a worldwide basis.

   Since 1960 the CGPM has made a number of changes to the SI to meet the
   needs of specific fields, notably chemistry and radiometry. These are
   mostly additions to the list of named derived units, and include the mole
   (symbol mol) for an amount of substance, the pascal (symbol Pa) for
   pressure, the siemens (symbol S) for electrical conductance, the becquerel
   (symbol Bq) for "activity referred to a radionuclide", the gray (symbol
   Gy) for ionising radiation, the sievert (symbol Sv) as the unit of dose
   equivalent radiation, and the katal (symbol kat) for catalytic
   activity.^[40]^: 156 ^[76]^[40]^: 156 ^[40]^: 158 ^[40]^: 159 ^[40]^: 165 

   The range of defined prefixes pico- (10^−12) to tera- (10^12) was extended
   to 10^−24 to 10^24.^[40]^: 152 ^[40]^: 158 ^[40]^: 164 

   The 1960 definition of the standard metre in terms of wavelengths of a
   specific emission of the krypton-86 atom was replaced with the distance
   that light travels in vacuum in exactly
   Link: mw-deduplicated-inline-style
   1/299792458 second, so that the speed of light is now an exactly specified
   constant of nature.

   A few changes to notation conventions have also been made to alleviate
   lexicographic ambiguities. An analysis under the aegis of CSIRO, published
   in 2009 by the Royal Society, has pointed out the opportunities to finish
   the realisation of that goal, to the point of universal zero-ambiguity
   machine readability.^[77]

  2019 redefinitions[edit]

   [IMG]
   Enlarge
   Reverse dependencies of the SI base units on seven physical constants,
   which are assigned exact numerical values in the 2019 redefinition. Unlike
   in the previous definitions, the base units are all derived exclusively
   from constants of nature. Arrows are shown in the opposite direction
   compared to typical dependency graphs, i.e. a\rightarrow b in this chart
   means b depends on a: a is used to define b.
   Link: mw-deduplicated-inline-style
   Main article: 2019 redefinition of the SI base units

   After the metre was redefined in 1960, the International Prototype of the
   Kilogram (IPK) was the only physical artefact upon which base units
   (directly the kilogram and indirectly the ampere, mole and candela)
   depended for their definition, making these units subject to periodic
   comparisons of national standard kilograms with the IPK.^[78] During the
   2nd and 3rd Periodic Verification of National Prototypes of the Kilogram,
   a significant divergence had occurred between the mass of the IPK and all
   of its official copies stored around the world: the copies had all
   noticeably increased in mass with respect to the IPK. During extraordinary
   verifications carried out in 2014 preparatory to redefinition of metric
   standards, continuing divergence was not confirmed. Nonetheless, the
   residual and irreducible instability of a physical IPK undermined the
   reliability of the entire metric system to precision measurement from
   small (atomic) to large (astrophysical) scales.

   A proposal was made that:^[79]

     * In addition to the speed of light, four constants of nature – the
       Planck constant, an elementary charge, the Boltzmann constant, and the
       Avogadro constant – be defined to have exact values
     * The International Prototype of the Kilogram be retired
     * The current definitions of the kilogram, ampere, kelvin, and mole be
       revised
     * The wording of base unit definitions should change emphasis from
       explicit unit to explicit constant definitions.

   The new definitions were adopted at the 26th CGPM on 16 November 2018, and
   came into effect on 20 May 2019.^[80] The change was adopted by the
   European Union through Directive (EU) 2019/1258.^[81]

History[edit]

   [IMG]
   Enlarge
   Stone marking the Austro-Hungarian/Italian border at Pontebba displaying
   myriametres, a unit of 10 km used in Central Europe in the 19th century
   (but since deprecated)^[82]
   Link: mw-deduplicated-inline-style
   Main article: History of the metric system

  The improvisation of units[edit]

   The units and unit magnitudes of the metric system which became the SI
   were improvised piecemeal from everyday physical quantities starting in
   the mid-18th century. Only later were they moulded into an orthogonal
   coherent decimal system of measurement.

   The degree centigrade as a unit of temperature resulted from the scale
   devised by Swedish astronomer Anders Celsius in 1742. His scale
   counter-intuitively designated 100 as the freezing point of water and 0 as
   the boiling point. Independently, in 1743, the French physicist
   Jean-Pierre Christin described a scale with 0 as the freezing point of
   water and 100 the boiling point. The scale became known as the
   centi-grade, or 100 gradations of temperature, scale.

   The metric system was developed from 1791 onwards by a committee of the
   French Academy of Sciences, commissioned to create a unified and rational
   system of measures.^[83] The group, which included preeminent French men
   of science,^[84]^: 89  used the same principles for relating length,
   volume, and mass that had been proposed by the English clergyman John
   Wilkins in 1668^[85]^[86] and the concept of using the Earth's meridian as
   the basis of the definition of length, originally proposed in 1670 by the
   French abbot Mouton.^[87]^[88]

   [IMG]
   Enlarge
   Carl Friedrich Gauss

   In March 1791, the Assembly adopted the committee's proposed principles
   for the new decimal system of measure including the metre defined to be
   1/10,000,000 of the length of the quadrant of Earth's meridian passing
   through Paris, and authorised a survey to precisely establish the length
   of the meridian. In July 1792, the committee proposed the names metre,
   are, litre and grave for the units of length, area, capacity, and mass,
   respectively. The committee also proposed that multiples and submultiples
   of these units were to be denoted by decimal-based prefixes such as centi
   for a hundredth and kilo for a thousand.^[89]^: 82 

   William Thomson, (Lord Kelvin)
   Thomson
   James Clerk Maxwell
   Maxwell
   William Thomson (Lord Kelvin) and James Clerk Maxwell played a prominent
   role in the development of the principle of coherence and in the naming of
   many units of measure.^[90]^[91]^[92]^[93]^[94]

   Later, during the process of adoption of the metric system, the Latin
   gramme and kilogramme, replaced the former provincial terms gravet (1/1000
   grave) and grave. In June 1799, based on the results of the meridian
   survey, the standard mètre des Archives and kilogramme des Archives were
   deposited in the French National Archives. Subsequently, that year, the
   metric system was adopted by law in France.^[95] ^[96] The French system
   was short-lived due to its unpopularity. Napoleon ridiculed it, and in
   1812, introduced a replacement system, the mesures usuelles or "customary
   measures" which restored many of the old units, but redefined in terms of
   the metric system.

   During the first half of the 19th century there was little consistency in
   the choice of preferred multiples of the base units: typically the
   myriametre (10000 metres) was in widespread use in both France and parts
   of Germany, while the kilogram (1000 grams) rather than the myriagram was
   used for mass.^[82]

   In 1832, the German mathematician Carl Friedrich Gauss, assisted by
   Wilhelm Weber, implicitly defined the second as a base unit when he quoted
   the Earth's magnetic field in terms of millimetres, grams, and
   seconds.^[90] Prior to this, the strength of the Earth's magnetic field
   had only been described in relative terms. The technique used by Gauss was
   to equate the torque induced on a suspended magnet of known mass by the
   Earth's magnetic field with the torque induced on an equivalent system
   under gravity. The resultant calculations enabled him to assign dimensions
   based on mass, length and time to the magnetic field.^[ce]^[97]

   A candlepower as a unit of illuminance was originally defined by an 1860
   English law as the light produced by a pure spermaceti candle weighing
   Link: mw-deduplicated-inline-style
   1⁄6 pound (76 grams) and burning at a specified rate. Spermaceti, a waxy
   substance found in the heads of sperm whales, was once used to make
   high-quality candles. At this time the French standard of light was based
   upon the illumination from a Carcel oil lamp. The unit was defined as that
   illumination emanating from a lamp burning pure rapeseed oil at a defined
   rate. It was accepted that ten standard candles were about equal to one
   Carcel lamp.

  Metre Convention[edit]

   Link: mw-deduplicated-inline-style
   Main article: Metre Convention

   A French-inspired initiative for international cooperation in metrology
   led to the signing in 1875 of the Metre Convention, also called Treaty of
   the Metre, by 17 nations.^[cf]^[84]^: 353–354  Initially the convention
   only covered standards for the metre and the kilogram. In 1921, the Metre
   Convention was extended to include all physical units, including the
   ampere and others thereby enabling the CGPM to address inconsistencies in
   the way that the metric system had been used.^[91]^[40]^: 96 

   A set of 30 prototypes of the metre and 40 prototypes of the
   kilogram,^[cg] in each case made of a 90% platinum-10% iridium alloy, were
   manufactured by British metallurgy specialty firm^[who?] and accepted by
   the CGPM in 1889. One of each was selected at random to become the
   International prototype metre and International prototype kilogram that
   replaced the mètre des Archives and kilogramme des Archives respectively.
   Each member state was entitled to one of each of the remaining prototypes
   to serve as the national prototype for that country.^[98]

   The treaty also established a number of international organisations to
   oversee the keeping of international standards of measurement.^[99]^[ch]

  The CGS and MKS systems[edit]

   Link: mw-deduplicated-inline-style
   See also: CGS system of units and MKS system of units
   [IMG]
   Enlarge
   Closeup of the National Prototype Metre, serial number 27, allocated to
   the United States

   In the 1860s, James Clerk Maxwell, William Thomson (later Lord Kelvin) and
   others working under the auspices of the British Association for the
   Advancement of Science, built on Gauss's work and formalised the concept
   of a coherent system of units with base units and derived units christened
   the centimetre–gram–second system of units in 1874. The principle of
   coherence was successfully used to define a number of units of measure
   based on the CGS, including the erg for energy, the dyne for force, the
   barye for pressure, the poise for dynamic viscosity and the stokes for
   kinematic viscosity.^[93]

   In 1879, the CIPM published recommendations for writing the symbols for
   length, area, volume and mass, but it was outside its domain to publish
   recommendations for other quantities. Beginning in about 1900, physicists
   who had been using the symbol "μ" (mu) for "micrometre" or "micron", "λ"
   (lambda) for "microlitre", and "γ" (gamma) for "microgram" started to use
   the symbols "μm", "μL" and "μg".^[100]

   At the close of the 19th century three different systems of units of
   measure existed for electrical measurements: a CGS-based system for
   electrostatic units, also known as the Gaussian or ESU system, a CGS-based
   system for electromechanical units (EMU) and an International system based
   on units defined by the Metre Convention.^[101] for electrical
   distribution systems. Attempts to resolve the electrical units in terms of
   length, mass, and time using dimensional analysis was beset with
   difficulties—the dimensions depended on whether one used the ESU or EMU
   systems.^[94] This anomaly was resolved in 1901 when Giovanni Giorgi
   published a paper in which he advocated using a fourth base unit alongside
   the existing three base units. The fourth unit could be chosen to be
   electric current, voltage, or electrical resistance.^[102] Electric
   current with named unit 'ampere' was chosen as the base unit, and the
   other electrical quantities derived from it according to the laws of
   physics. This became the foundation of the MKS system of units.

   In the late 19th and early 20th centuries, a number of non-coherent units
   of measure based on the gram/kilogram, centimetre/metre, and second, such
   as the Pferdestärke (metric horsepower) for power,^[103]^[ci] the darcy
   for permeability^[104] and "millimetres of mercury" for barometric and
   blood pressure were developed or propagated, some of which incorporated
   standard gravity in their definitions.

   At the end of the Second World War, a number of different systems of
   measurement were in use throughout the world. Some of these systems were
   metric system variations; others were based on customary systems of
   measure, like the U.S customary system and Imperial system of the UK and
   British Empire.

  The Practical system of units[edit]

   In 1948, the 9th CGPM commissioned a study to assess the measurement needs
   of the scientific, technical, and educational communities and "to make
   recommendations for a single practical system of units of measurement,
   suitable for adoption by all countries adhering to the Metre
   Convention".^[105] This working document was Practical system of units of
   measurement. Based on this study, the 10th CGPM in 1954 defined an
   international system derived from six base units including units of
   temperature and optical radiation in addition to those for the MKS system
   mass, length, and time units and Giorgi's current unit. Six base units
   were recommended: the metre, kilogram, second, ampere, degree Kelvin, and
   candela.

   The 9th CGPM also approved the first formal recommendation for the writing
   of symbols in the metric system when the basis of the rules as they are
   now known was laid down.^[106] These rules were subsequently extended and
   now cover unit symbols and names, prefix symbols and names, how quantity
   symbols should be written and used, and how the values of quantities
   should be expressed.^[40]^: 104, 130 

  Birth of the SI[edit]

   In 1960, the 11th CGPM synthesised the results of the 12-year study into a
   set of 16 resolutions. The system was named the International System of
   Units, abbreviated SI from the French name, Le Système International
   d'Unités.^[40]^: 110 ^[107]

  Historical definitions[edit]

   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.

   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/s^2. 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.

                        SI base units^[3]^: 6 ^[50]^[51]
   Unit     Definition^[n 1]                                                  
   name     
              * Prior: (1675)                                                 
                Link: mw-deduplicated-inline-style                            
                1/86400 of a day of 24 hours of 60 minutes of 60 seconds.^TLB 
              * Interim (1956):                                               
   second       Link: mw-deduplicated-inline-style                            
                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. 
              * Prior (1793):                                                 
                Link: mw-deduplicated-inline-style                            
                1/10000000 of the meridian through Paris between the North    
                Pole and the Equator.^FG                                      
              * Interim (1889): The Prototype of the metre chosen by the      
                CIPM, at the temperature of melting ice, represents the       
   metre        metric unit of length.                                        
              * Interim (1960): 1650763.73 wavelengths in vacuum of the       
                radiation corresponding to the transition between the 2p^10   
                and 5d^5 quantum levels of the krypton-86 atom.               
              * Current (1983): The distance travelled by light in vacuum in  
                Link: mw-deduplicated-inline-style                            
                1/299792458 second.                                           
              * Prior (1793): The grave was defined as being the mass (then   
                called weight) of one litre of pure water at its freezing     
                point.^FG                                                     
              * Interim (1889): The mass of a small squat cylinder of ≈47     
                cubic centimetres of platinum-iridium alloy kept in the       
                International Burueau of Weights and Measures (BIPM),         
   kilogram     Pavillon de Breteuil, France.^[cj] Also, in practice, any of  
                numerous official replicas of it.                             
              * Current (2019): The kilogram is defined by setting the Planck 
                constant h exactly to 6.62607015×10^−34 J⋅s (J =              
                kg⋅m^2⋅s^−2), given the definitions of the metre and the      
                second.^[38] Then the formula would be kg =                   
                Link: mw-deduplicated-inline-style                            
                h/6.62607015×10^−34⋅m^2⋅s^−1                                  
              * 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.^[108] ^IEC                                            
              * Interim (1946): The constant current which, if maintained in  
   ampere       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.                       
              * Current (2019): The flow of                                   
                Link: mw-deduplicated-inline-style                            
                1/1.602176634×10^−19 times the elementary charge e per        
                second.                                                       
              * Prior (1743): The centigrade scale is obtained by assigning   
                0 °C to the freezing point of water and 100 °C to the boiling 
                point of water.                                               
              * Interim (1954): The triple point of water (0.01 °C) defined   
                to be exactly 273.16 K.^[n 2]                                 
              * Previous (1967):                                              
   kelvin       Link: mw-deduplicated-inline-style                            
                1/273.16 of the thermodynamic temperature of the triple point 
                of water.                                                     
              * Current (2019): 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⋅m^2⋅s^−2), given the          
                definition of the kilogram, the metre, and the second.        
              * Prior (1900): A stoichiometric quantity which is the          
                equivalent mass in grams of Avogadro's number of molecules of 
                a substance.^ICAW                                             
              * Interim (1967): The amount of substance of a system which     
                contains as many elementary entities as there are atoms in    
   mole         0.012 kilogram of carbon-12.                                  
              * Current (2019): The amount of substance of exactly            
                6.02214076×10^23 elementary entities. This number is the      
                fixed numerical value of the Avogadro constant, N_A, when     
                expressed in the unit mol^−1 and is called the Avogadro       
                number.                                                       
              * Prior (1946): The value of the new candle (early name for the 
                candela) 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   
                5.4×10^14 hertz and that has a radiant intensity in that      
   candela      direction of                                                  
                Link: mw-deduplicated-inline-style                            
                1/683 watt per steradian.                                     
                                                                              
               Note: both old and new definitions are approximately the       
               luminous intensity of a spermaceti candle burning modestly     
               bright, in the late 19th century called a "candlepower" or a   
               "candle".                                                      
   Notes    
            
   Link: mw-deduplicated-inline-style
    1. ^ Interim definitions are given here only when there has been a
       significant difference in the definition.
    2. ^ 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.
            
   The Prior definitions of the various base units in the above table were
   made by the following authors and authorities:
            
         * TLB = Tito Livio Burattini, Misura universale, Vilnius, 1675
         * FG = French Government
         * IEC = International Electrotechnical Commission
         * ICAW = International Committee on Atomic Weights
            
   All other definitions result from resolutions by either CGPM or the CIPM
   and are catalogued in the SI Brochure.

Metric units that are not recognised by the SI[edit]

   Link: mw-deduplicated-inline-style
   Main article: Metric units

   Although the term metric system is often used as an informal alternative
   name for the International System of Units,^[109] other metric systems
   exist, some of which were in widespread use in the past or are even still
   used in particular areas. There are also individual metric units such as
   the sverdrup and the darcy that exist outside of any system of units. Most
   of the units of the other metric systems are not recognised by the
   SI.^[ck]^[cn]

   Here are some examples. The centimetre–gram–second (CGS) system was the
   dominant metric system in the physical sciences and electrical engineering
   from the 1860s until at least the 1960s, and is still in use in some
   fields. It includes such SI-unrecognised units as the gal, dyne, erg,
   barye, etc. in its mechanical sector, as well as the poise and stokes in
   fluid dynamics. When it comes to the units for quantities in electricity
   and magnetism, there are several versions of the CGS system. Two of these
   are obsolete: the CGS electrostatic ('CGS-ESU', with the SI-unrecognised
   units of statcoulomb, statvolt, statampere, etc.) and the CGS
   electromagnetic system ('CGS-EMU', with abampere, abcoulomb, oersted,
   maxwell, abhenry, gilbert, etc.).^[co]^[cq] A 'blend' of these two systems
   is still popular and is known as the Gaussian system (which includes the
   gauss as a special name for the CGS-EMU unit maxwell per square
   centimetre).^[cr]

   In engineering (other than electrical engineering), there was formerly a
   long tradition of using the gravitational metric system, whose
   SI-unrecognised units include the kilogram-force (kilopond), technical
   atmosphere, metric horsepower, etc. The metre–tonne–second (mts) system,
   used in the Soviet Union from 1933 to 1955, had such SI-unrecognised units
   as the sthène, pièze, etc. Other groups of SI-unrecognised metric units
   are the various legacy and CGS units related to ionising radiation
   (rutherford, curie, roentgen, rad, rem, etc.), radiometry (langley,
   jansky), photometry (phot, nox, stilb, nit, metre-candle,^[115]^:17
   lambert, apostilb, skot, brill, troland, talbot, candlepower, candle),
   thermodynamics (calorie), and spectroscopy (reciprocal centimetre).

   The angstrom is still used in various fields. Some other SI-unrecognised
   metric units that don't fit into any of the already mentioned categories
   include the are, bar, barn, fermi, gradian (gon, grad, or grade), metric
   carat, micron, millimetre of mercury, torr, millimetre (or centimetre, or
   metre) of water, millimicron, mho, stere, x unit, γ (unit of mass), γ
   (unit of magnetic flux density), and λ (unit of volume).^[116]^: 20–21  In
   some cases, the SI-unrecognised metric units have equivalent SI units
   formed by combining a metric prefix with a coherent SI unit. For example,
   1 γ (unit of magnetic flux density) = 1 nT, 1 Gal = 1 cm⋅s^−2, 1 barye =
   1 decipascal, etc. (a related group are the correspondences^[co] such as
   1 abampere ≘ 1 decaampere, 1 abhenry ≘ 1 nanohenry, etc.^[cs]). Sometimes,
   it is not even a matter of a metric prefix: the SI-nonrecognised unit may
   be exactly the same as an SI coherent unit, except for the fact that the
   SI does not recognise the special name and symbol. For example, the nit is
   just an SI-unrecognised name for the SI unit candela per square metre and
   the talbot is an SI-unrecognised name for the SI unit lumen second.
   Frequently, a non-SI metric unit is related to an SI unit through a
   power-of-ten factor, but not one that has a metric prefix, e.g., 1 dyn =
   10^−5 newton, 1 Å = 10^−10 m, etc. (and correspondences^[co] like 1 gauss
   ≘ 10^−4 tesla). Finally, there are metric units whose conversion factors
   to SI units are not powers of ten, e.g., 1 calorie = 4.184 joules and
   1 kilogram-force = 9.806650 newtons. Some SI-unrecognised metric units are
   still frequently used, e.g., the calorie (in nutrition), the rem (in the
   U.S.), the jansky (in radio astronomy), the gauss (in industry) and the
   CGS-Gaussian units^[cr] more generally (in some subfields of physics), the
   metric horsepower (for engine power, in Europe), the kilogram-force (for
   rocket engine thrust, in China and sometimes in Europe), etc. Others are
   now rarely used, such as the sthène and the rutherford.

See also[edit]

     * Non-SI units mentioned in the SI
     * Conversion of units – Comparison of various scales
     * Outline of the metric system – Overview of and topical guide to the
       metric system
     * List of international common standards

   Organisations

     * International Bureau of Weights and Measures – Intergovernmental
       measurement science and measurement standards setting organisation
     * Institute for Reference Materials and Measurements (EU)
     * National Institute of Standards and Technology – Measurement standards
       laboratory in the United States (US)

   Standards and conventions

     * Conventional electrical unit
     * Coordinated Universal Time (UTC) – Primary time standard
     * Unified Code for Units of Measure

Notes[edit]

   Link: mw-deduplicated-inline-style
    1. ^ 'SI' is an initialism of Système international, which is an
       abbreviated form of its full French name Système international
       d’unités,^[2]^: 165  which literally means 'International System of
       Units'. By Resolution 12 of the 11th CGPM (1960), the international
       abbreviation of the name of the system is: SI.^[2]^: 165 
    2. ^ When we say 'SI system', we are basically saying the word 'system'
       twice: 'International System system' (note that 'SI' stands for the
       French name Système international, which literally means
       'International System'). This is a type of linguistic redundancy
       called pleonasm. Some examples of such pleonastic usage include the
       list of 'alternate titles' in the Encyclopedia Britannica article on
       the SI,^[4] the last paragraph in an editorial in the journal
       Nature,^[5] and the footnote 1 to Table 5 in the style manual of the
       International Astronomical Union.^[6]
    3. ^ In a decimal system, different units for a given kind of physical
       quantity are related by factors of 10, so that, within such a system,
       unit conversions involve the simple process of moving the decimal
       point to the right or to the left.^[9] So instead of relations like 1
       mile = 1760 yards, as we have in imperial and US customary measurement
       systems (which are not decimal), in the SI (which is decimal) we
       instead have 1 kilometre = 1000 metres. Here the kilometre is
       comparable in size to the mile (1 km ≈ 0.6 mi) and the metre to the
       yard (1 m ≈ 1.1 yd).
    4. ^ Or one of its decimal multiples or submultiples, like the
       centimetre.
    5. ^ Or one of its decimal multiples or submultiples, like the gram.
    6. ^ Or one of its decimal multiples or submultiples, like the
       gram-force.
    7. ^ A metric system of units is any system of weights and measures that
       is decimal^[c] and based on the metre^[d] as the unit of length and
       either the kilogram^[e] as the unit of mass or the kilogram-force^[f]
       as the unit of force.
    8. ^ As of 19 January 2021.
    9. ^ ^a ^b The latter group includes economic unions such as the
       Caribbean Community (CARICOM).
   10. ^ This is an international organization with^[h] 63 member states and
       39 Associate States and Economies of the General Conference.^[i]^[13]
       It was established in 1875 under the terms of the Metre
       Convention.^[12]^[14]
   11. ^ ^a ^b ^c From French: Conférence générale des poids et mesures.
   12. ^ ^a ^b It shall be lawful throughout the United States of America to
       employ the weights and measures of the metric system; and no contract
       or dealing, or pleading in any court, shall be deemed invalid or
       liable to objection because the weights or measures expressed or
       referred to therein are weights or measures of the metric system.
       (15 U.S.C. § 204)
   13. ^ Here 'official status' means that the SI is recognized in some way
       by the laws and regulations of the country. In many countries, this
       means that using the SI units is mandatory for most commercial and
       administrative purposes (e.g. in the European Union). On the other
       hand, when it comes to the US, 'official status' means that federal
       law specifically allows, but doesn't require, the SI units to be
       used.^[l] In fact, federal law even states that it is the declared
       policy of the United States to designate the metric system of
       measurement as the preferred system of weights and measures for United
       States trade and commerce (15 U.S.C. § 205b).
       See metrication for more information.
   14. ^ This includes the United States, Canada, and the United Kingdom,
       despite the fact these three countries also continue to use their
       customary systems to various degrees.
   15. ^ Although the precise definition of coherence is complicated, the
       basic idea is that mathematical relations between the units for
       quantities should mirror the mathematical relations between the
       corresponding quantities themselves. For example, the coherent unit of
       volume is equal to the volume of a cube whose sides are one unit of
       length; the coherent unit of pressure is equal to the pressure exerted
       by a unit-magnitude force over a surface of unit area; etc. As an
       example of lack of coherence, consider how, in the US customary
       system, the units of fluid volume are related to the units of length.
       The principal units of length are inches, feet, yards, and miles;
       meanwhile, the principal units of fluid volume are based on the (US)
       gallon, which, at 231 cubic inches, is not a cubic inch, or a cubic
       foot, or a cubic yard, or a cubic mile (note that 231=3×7×11).
   16. ^ For example, the SI unit of velocity is the metre per second,
       m⋅s^−1; of acceleration is the metre per second squared, m⋅s^−2; etc.
       These can also be written as m/s and m/s^2, respectively.
   17. ^ ^a ^b For example the newton (N), the unit of force, equivalent to
       kg⋅m⋅s^−2; the joule (J), the unit of energy, equivalent to
       kg⋅m^2⋅s^−2, etc. The most recently named derived unit, the katal, was
       defined in 1999.
   18. ^ ^a ^b For example, the recommended unit for the electric field
       strength is the volt per metre, V/m, where the volt is the derived
       unit for electric potential difference. The volt per metre is equal to
       kg⋅m⋅s^−3⋅A^−1 when expressed in terms of base units.
   19. ^ This must be one of 29 coherent units with a separate name and
       symbol, i.e. either one of the seven base units or one of the 22
       coherent derived units with special names and symbols.
   20. ^ For example, the coherent SI unit of length is the metre, about the
       height of kitchen counter (just over 3 ft). But for driving distances,
       one would normally use kilometres, where one kilometre is 1000 metres;
       here the metric prefix 'kilo-' (symbol 'k') stands for a factor of
       1000. On the other hand, for tailoring measurements, one would usually
       use centimetres, where one centimetre is 1/100 of a metre; here the
       metric prefix 'centi-' (symbol 'c') stands for a factor of 1/100.
   21. ^ Non-coherent, customary systems have another tendency,
       well-illustrated by the U.S. customary system. In that system, some
       liquid commodities are measured neither in the coherent units of
       volume (e.g. cubic inches) nor in gallons, but in barrels.
       Furthermore, the size of the barrel depends on the commodity: it means
       31 US gallons for beer,^[15] but 42 gallons for petroleum.^[16] So
       different units for one and the same quantity (e.g. volume) are used
       depending on what is being measured, and these different units may not
       be related to each other in any obvious way—even if they have the same
       name.
   22. ^ Meaning that different units for a given quantity, such as length,
       are related by factors of 10. Therefore, calculations involve the
       simple process of moving the decimal point to the right or to the
       left.^[9]

       For example, the coherent SI unit of length is the metre, which is
       about the height of the kitchen counter. But if one wishes to talk
       about driving distances using the SI units, one will normally use
       kilometres, where one kilometre is 1000 metres. On the other hand,
       tailoring measurements would usually be expressed in centimetres,
       where one centimetre is 1/100 of a metre. This makes communication
       much simpler when relating quantities, compared with nonmetric units.
       For example, converting miles and feet to inches requires fractional
       arithmetic.
   23. ^ Although the terms the metric system and the SI are often used as
       synonyms, there are in fact many mutually incompatible metric systems.
       Moreover, there exist metric units that are not recognised by any
       larger metric system. See § Metric units that are not recognised by
       the SI, below.
   24. ^ As of May 2020, only for the following countries is it uncertain
       whether the SI has any official status: Myanmar, Liberia, the
       Federated States of Micronesia, the Marshall Islands, Palau, and
       Samoa.
   25. ^ In the US, the history of legislation begins with the Metric Act of
       1866, which legally protected use of the metric system in commerce.
       The first section is still part of US law (15 U.S.C. § 204).^[l] In
       1875, the US became one of the original signatories of the Metre
       Convention. In 1893, the Mendenhall Order stated that the Office of
       Weights and Measures ... will in the future regard the International
       Prototype Metre and Kilogramme as fundamental standards, and the
       customary units — the yard and the pound — will be derived therefrom
       in accordance with the Act of July 28, 1866. In 1954, the US adopted
       the International Nautical Mile, which is defined as exactly 1852 m,
       in lieu of the U.S. Nautical Mile, defined as 6080.20 ft = 1853.248 m.
       In 1959, the U.S. National Bureau of Standards officially adapted the
       International yard and pound, which are defined exactly in terms of
       the metre and the kilogram. In 1968, the Metric Study Act (Pub. L.
       90-472, August 9, 1968, 82 Stat. 693) authorised a three-year study of
       systems of measurement in the U.S., with particular emphasis on the
       feasibility of adopting the SI. The Metric Conversion Act of 1975
       followed, later amended by the Omnibus Trade and Competitiveness Act
       of 1988, the Savings in Construction Act of 1996, and the Department
       of Energy High-End Computing Revitalization Act of 2004. As a result
       of all these acts, the US current law (15 U.S.C. § 205b) states that

         It is therefore the declared policy of the United States-

         (1) to designate the metric system of measurement as the preferred
         system of weights and measures for United States trade and commerce;

         (2) to require that each Federal agency, by a date certain and to
         the extent economically feasible by the end of the fiscal year 1992,
         use the metric system of measurement in its procurements, grants,
         and other business-related activities, except to the extent that
         such use is impractical or is likely to cause significant
         inefficiencies or loss of markets to United States firms, such as
         when foreign competitors are producing competing products in
         non-metric units;

         (3) to seek out ways to increase understanding of the metric system
         of measurement through educational information and guidance and in
         Government publications; and

         (4) to permit the continued use of traditional systems of weights
         and measures in non-business activities.

   26. ^ The United States system should be referred to as the US customary
       system and not the imperial system because there are differences
       between the US customary system and the United Kingdom's imperial
       system.
   27. ^ And have been defined in terms of the SI's metric predecessors since
       at least the 1890s.
   28. ^ See e.g. here for the various definitions of the catty, a
       traditional Chinese unit of mass, in various places across East and
       Southeast Asia. Similarly, see this article on the traditional
       Japanese units of measurement, as well as this one on the traditional
       Indian units of measurement.
   29. ^ ^a ^b from French: Comité international des poids et mesures
   30. ^ ^a ^b The SI Brochure for short. As of May 2020, the latest edition
       is the ninth, published in 2019. It is Ref.^[2] of this article.
   31. ^ ^a ^b from French: Bureau international des poids et mesures
   32. ^ The latter are formalised in the International System of Quantities
       (ISQ).^[2]^: 129 
   33. ^ The choice of which and even how many quantities to use as base
       quantities is not fundamental or even unique – it is a matter of
       convention.^[2]^: 126  For example, four base quantities could have
       been chosen as velocity, angular momentum, electric charge and energy.
   34. ^ Here are some examples of coherent derived SI units: the unit of
       velocity, which is the metre per second, with the symbol m/s; the unit
       of acceleration, which is the metre per second squared, with the
       symbol m/s^2; etc.
   35. ^ A useful property of a coherent system is that when the numerical
       values of physical quantities are expressed in terms of the units of
       the system, then the equations between the numerical values have
       exactly the same form, including numerical factors, as the
       corresponding equations between the physical quantities;^[17]^: 6  An
       example may be useful to clarify this. Suppose we are given an
       equation relating some physical quantities, e.g. T =
       Link: mw-deduplicated-inline-style
       1/2{m}{v}^2, expressing the kinetic energy T in terms of the mass m
       and the velocity v. Choose a system of units, and let {T}, {m}, and
       {v} be the numerical values of T, m, and v when expressed in that
       system of units. If the system is coherent, then the numerical values
       will obey the same equation (including numerical factors) as the
       physical quantities, i.e. we will have that T =
       Link: mw-deduplicated-inline-style
       1/2{m}{v}^2. Therefore, SI units can be converted without numerical
       factors: 1 J = 1 N·m = 1 C·V = 1 W·s.
       On the other hand, if the chosen system of units is not coherent, this
       property may fail. For example, the following is not a coherent
       system: one where energy is measured in calories, while mass and
       velocity are measured in their SI units. After all, in that case,
       Link: mw-deduplicated-inline-style
       1/2{m}{v}^2 will give a numerical value whose meaning is the kinetic
       energy when expressed in joules, and that numerical value is
       different, by a factor of 4.184, from the numerical value when the
       kinetic energy is expressed in calories. Thus, in that system, the
       equation satisfied by the numerical values is instead {T} =
       Link: mw-deduplicated-inline-style
       1/4.184
       Link: mw-deduplicated-inline-style
       1/2{m}{v}^2.
   36. ^ Which define the International System of Quantities (ISQ).
   37. ^ It is correct to say that an SI base unit (like the metre) is a
       coherent unit for its corresponding physical quantity. Recall that the
       set of coherent SI units consists of the base units and the coherent
       derived units. This usage is consistent with the definition of a
       coherent unit as one that is equal to 'a product of powers of the base
       units with a prefactor of 1'. After all, each base unit is obviously
       so representable—it is equal to itself to the power of 1 and with a
       prefactor of 1.
   38. ^ One kilometre is about 0.62 miles, a length equal to about two and a
       half laps around a typical athletic track. Walking at a moderate pace
       for one hour, an adult human will cover about five kilometres (about
       three miles). The distance from London, UK, to Paris, France is about
       350 km; from London to New York, 5600 km.
   39. ^ In other words, given any base unit or any coherent derived unit
       with a special name and symbol.
   40. ^ ^a ^b For historical reasons, names and symbols for decimal
       multiples and sub-multiples of the unit of mass are formed as if it is
       the gram which is the base unit, i.e. by attaching prefix names and
       symbols, respectively, to the unit name "gram" and the unit symbol
       "g". For example, 10^−6 kg is written as milligram, mg, not as
       microkilogram, μkg.^[2]^: 144 
   41. ^ This last statement in fact applies to all SI units, not only those
       with special names and symbols. Consider the example of the SI units
       of torque. Because the SI does not have a unit with a special name and
       symbol for torque, its coherent SI unit is the newton-metre, N⋅m. The
       following are some examples of non-coherent SI units of torque: N⋅mm,
       kN⋅μm, mN⋅cm, etc. Note that these non-coherent units are obtained
       from the original coherent unit by replacing some (or all) of the
       units with special names and symbols that are present in the original
       coherent unit by their decimal multiples or submultiples. But then
       these different powers of ten combine into one overall power of ten.
       For example, kN⋅μm = (10^3 N)⋅(10^−6 m) = 10^3–6 N⋅m = 10^−3 N⋅m.
   42. ^ Note, however, that there is a special group of units that are
       called non-SI units accepted for use with SI, most of which are not
       decimal multiples of the corresponding SI units; see below.
   43. ^ As the SI Brochure states,^[2]^: 140  this applies not only to
       technical texts, but also, for example, to measuring instruments (i.e.
       the instrument read-out needs to indicate both the unit and the
       quantity measured).
   44. ^ Customarily, however, rainfall is measured in non-coherent SI units
       such as millimetres in height collected on each square metre during a
       certain period, equivalent to litres per square metre.
   45. ^ Even base units; the mole was added as a base SI unit only in
       1971.^[2]^: 156 
   46. ^ See the next section for why this type of definition is considered
       advantageous.
   47. ^ Their exactly defined values are as follows:^[2]^: 128 
       {\displaystyle \Delta \nu _{\text{Cs}}} = 9192631770 Hz
       c = 299792458 m/s
       h = 6.62607015×10^−34 J⋅s
       e = 1.602176634×10^−19 C
       k = 1.380649×10^−23 J/K
       {\displaystyle N_{\text{A}}} = 6.02214076×10^23 mol^−1
       {\displaystyle K_{\text{cd}}} = 683 lm/W.
   48. ^ A mise en pratique is French for 'putting into practice;
       implementation'.^[21]^[22]
   49. ^ ^a ^b The sole exception is the definition of the second, which is
       still given not in terms of fixed values of fundamental constants but
       in terms of a particular property of a particular naturally occurring
       object, the caesium atom. And indeed, it has been clear for some time
       that relatively soon, by using atoms other than caesium, it will be
       possible to have definitions of the second that are more precise than
       the current one. Taking advantage of these more precise methods will
       necessitate the change in the definition of the second, probably
       sometime around the year 2030.^[28]^: 196 ^[29]
   50. ^ ^a ^b Again, except for the second, as explained in the previous
       note.
       The second may eventually get fixed by defining an exact value for yet
       another fundamental constant (whose derived unit includes the second),
       for example the Rydberg constant. For this to happen, the uncertainty
       in the measurement of that constant must become so small as to be
       dominated by the uncertainty in the measurement of whatever clock
       transition frequency is being used to define the second at that point.
       Once that happens, the definitions will be reversed: the value of the
       constant will be fixed by definition to an exact value, namely its
       most recent best measured value, while the clock transition frequency
       will become a quantity whose value is no longer fixed by definition
       but which has to be measured. Unfortunately, it is unlikely that this
       will happen in the foreseeable future, because presently there are no
       promising strategies for measuring any additional fundamental
       constants with the necessary precision.^[30]^: 4112–3 
   51. ^ The one exception being the definition of the second; see Notes
       ^[aw] and ^[ax] in the following section.
   52. ^ To see this, recall that Hz = s^−1 and J = kg⋅m^2⋅s^−2. Thus,
       (Hz) (J⋅s) / (m/s)^2
       = (s^−1) [(kg⋅m^2⋅s^−2)⋅s] (m⋅s^−1)^−2
       = s^(−1−2+1+2)⋅m^(2–2)⋅kg
       = kg,
       since all the powers of metres and seconds cancel out. It can further
       be shown that (Hz) (J⋅s) / (m/s)^2 is the only combination of powers
       of the units of the defining constants (that is, the only combination
       of powers of Hz, m/s, J⋅s, C, J/K, mol^−1, and lm/W) that results in
       the kilogram.
   53. ^ Namely,
       1 Hz =
       Link: mw-deduplicated-inline-style
       Δν_Cs/9192631770
       1 m/s =
       Link: mw-deduplicated-inline-style
       c/299792458 , and
       1 J⋅s =
       Link: mw-deduplicated-inline-style
       h/6.62607015×10^−34.
   54. ^ The SI Brochure prefers to write the relationship between the
       kilogram and the defining constants directly, without going through
       the intermediary step of defining 1 Hz, 1 m/s, and 1 J⋅s, like
       this:^[2]^: 131  1 kg =
       Link: mw-deduplicated-inline-style
       (299792458)^2/(6.62607015×10^−34)(9192631770)
       Link: mw-deduplicated-inline-style
       h Δν_Cs/c^2.
   55. ^ For example, from 1889 until 1960, the metre was defined as the
       length of the International Prototype Metre, a particular bar made of
       platinum-iridium alloy that was (and still is) kept at the
       International Bureau of Weights and Measures, located in the Pavillon
       de Breteuil in Saint-Cloud, France, near Paris. The final
       artefact-based definition of the metre, which stood from 1927 to the
       redefinition of the metre in 1960, read as follows:^[2]^: 159 
       Link: mw-deduplicated-inline-style

         The unit of length is the metre, defined by the distance, at 0°,
         between the axes of the two central lines marked on the bar of
         platinum-iridium kept at the Bureau International des Poids et
         Mesures and declared Prototype of the metre by the 1st Conférence
         Générale des Poids et Mesures, this bar being subject to standard
         atmospheric pressure and supported on two cylinders of at least one
         centimetre diameter, symmetrically placed in the same horizontal
         plane at a distance of 571 mm from each other.

       The '0°' refers to the temperature of 0 °C. The support requirements
       represent the Airy points of the prototype—the points, separated by
       Link: mw-deduplicated-inline-style
       4/7 of the total length of the bar, at which the bending or droop of
       the bar is minimised.^[32]
   56. ^ The latter was called the 'quadrant', the length of a meridian from
       the equator to the North Pole. The originally chosen meridian was the
       Paris meridian.
   57. ^ At the time 'weight' and 'mass' were not always carefully
       distinguished.
   58. ^ This volume is 1 cm^3 = 1 mL, which is 1×10^−6 m^3. Thus, the
       original definition of mass used not the coherent unit of volume
       (which would be the m^3) but a decimal submultiple of it.
   59. ^ Indeed, the original idea of the metric system was to define all
       units using only natural and universally available measurable
       quantities. For example, the original definition of the unit of
       length, the metre, was a definite fraction (one ten-millionth) of the
       length of a quarter of the Earth's meridian.^[bd] Once the metre was
       defined, one could define the unit of volume as the volume of a cube
       whose sides are one unit of length. And once the unit of volume was
       determined, the unit of mass could be defined as the mass of a unit of
       volume of some convenient substance at standard conditions. In fact,
       the original definition of the gram was 'the absolute weight^[be] of a
       volume of pure water equal to the cube of the hundredth part of a
       metre,^[bf] and at the temperature of melting ice.'

       However, it soon became apparent that these particular 'natural'
       realisations of the units of length and mass simply could not, at that
       time, be as precise (and as convenient to access) as the needs of
       science, technology, and commerce demanded. Therefore, prototypes were
       adopted instead. Care was taken to manufacture the prototypes so that
       they would be as close as possible, given the available science and
       technology of the day, to the idealised 'natural' realisations. But
       once the prototypes were completed, the units of length and mass
       became equal by definition to these prototypes (see Mètre des Archives
       and Kilogramme des Archives).

       Nevertheless, throughout the history of the SI, one keeps seeing
       expressions of hope that one day, one would be able to dispense with
       the prototypes and define all units in terms of standards found in
       nature. The first such standard was the second. It was never defined
       using a prototype, being originally defined as 1/86400 of the length
       of a day (since there are 60 s/min × 60 min/hr × 24 hr/day = 86400
       s/day). As we mentioned, the vision of defining all units in terms of
       universally available natural standards was at last fulfilled in 2019,
       when the sole remaining prototype used by the SI, the one for the
       kilogram, was finally retired.
   60. ^ The following references are useful for identifying the authors of
       the preceding reference: Ref.,,^[34] Ref.,^[35] and Ref.^[36]
   61. ^ ^a ^b As happened with British standards for length and mass in
       1834, when they were lost or damaged beyond the point of useability in
       a great fire known as the burning of Parliament. A commission of
       eminent scientists was assembled to recommend the steps to be taken
       for the restoration of the standards, and in its report, it described
       the destruction caused by the fire as follows:^[33]^[bh]
       Link: mw-deduplicated-inline-style

         We shall in the first place describe the state of the Standards
         recovered from the ruins of the House of Commons, as ascertained in
         our inspection of them made on 1st June, 1838, at the Journal
         Office, where they are preserved under the care of Mr. James Gudge,
         Principal Clerk of the Journal Office. The following list, taken by
         ourselves from inspection, was compared with a list produced by Mr.
         Gudge, and stated by him to have been made by Mr. Charles Rowland,
         one of the Clerks of the Journal Office, immediately after the fire,
         and was found to agree with it. Mr. Gudge stated that no other
         Standards of Length or Weight were in his custody.

         No. 1. A brass bar marked "Standard [G. II. crown emblem] Yard,
         1758", which on examination was found to have its right hand stud
         perfect, with the point and line visible, but with its left hand
         stud completely melted out, a hole only remaining. The bar was
         somewhat bent, and discoloured in every part.

         No. 2. A brass bar with a projecting cock at each end, forming a bed
         for the trial of yard-measures; discoloured.

         No. 3. A brass bar marked "Standard [G. II. crown emblem] Yard,
         1760", from which the left hand stud was completely melted out, and
         which in other respects was in the same condition as No. 1.

         No. 4. A yard-bed similar to No. 2; discoloured.

         No. 5. A weight of the form [drawing of a weight] marked [2 lb. T.
         1758], apparently of brass or copper; much discoloured.

         No. 6. A weight marked in the same manner for 4 lbs., in the same
         state.

         No. 7. A weight similar to No. 6, with a hollow space at its base,
         which appeared at first sight to have been originally filled with
         some soft metal that had been now melted out, but which on a rough
         trial was found to have nearly the same weight as No. 6.

         No. 8. A similar weight of 8 lbs., similarly marked (with the
         alteration of 8 lbs. for 4 lbs.), and in the same state.

         No. 9. Another exactly like No. 8.

         Nos. 10 and 11. Two weights of 16 lbs., similarly marked.

         Nos. 12 and 13. Two weights of 32 lbs., similarly marked.

         No. 14. A weight with a triangular ring-handle, marked "S.F. 1759 17
         lbs. 8 dwts. Troy", apparently intended to represent the stone of 14
         lbs. avoirdupois, allowing 7008 troy grains to each avoirdupois
         pound.

         It appears from this list that the bar adopted in the Act 5th Geo.
         IV., cap. 74, sect. 1, for the legal standard of one yard, (No. 3 of
         the preceding list), is so far injured, that it is impossible to
         ascertain from it, with the most moderate accuracy, the statutable
         length of one yard. The legal standard of one troy pound is missing.
         We have therefore to report that it is absolutely necessary that
         steps be taken for the formation and legalising of new Standards of
         Length and Weight.

   62. ^ Indeed, one of the motivations for the 2019 redefinition of the SI
       was the instability of the artefact that served as the definition of
       the kilogram.

       Before that, one of the reasons the United States started defining the
       yard in terms of the metre in 1893 was that^[37]^: 381 
       Link: mw-deduplicated-inline-style

         [t]he bronze yard No. 11, which was an exact copy of the British
         imperial yard both in form and material, had shown changes when
         compared with the imperial yard in 1876 and 1888 which could not
         reasonably be said to be entirely due to changes in No. 11.
         Suspicion as to the constancy of the length of the British standard
         was therefore aroused.

       In the above, the bronze yard No. 11 is one of two copies of the new
       British standard yard that were sent to the US in 1856, after Britain
       completed the manufacture of new imperial standards to replace those
       lost in the fire of 1834 (see ^[bi]). As standards of length, the new
       yards, especially bronze No. 11, were far superior to the standard the
       US had been using up to that point, the so-called Troughton scale.
       They were therefore accepted by the Office of Weights and Measures (a
       predecessor of NIST) as the standards of the United States. They were
       twice taken to England and recompared with the imperial yard, in 1876
       and in 1888, and, as mentioned above, measurable discrepancies were
       found.^[37]^: 381 

       In 1890, as a signatory of the Metre Convention, the US received two
       copies of the International Prototype Metre, the construction of which
       represented the most advanced ideas of standards of the time.
       Therefore it seemed that US measures would have greater stability and
       higher accuracy by accepting the international metre as fundamental
       standard, which was formalised in 1893 by the Mendenhall
       Order.^[37]^: 379–81 

   63. ^ As mentioned above, it is all but certain that the defining constant
       {\displaystyle \Delta \nu _{\text{Cs}}} will have to be replaced
       relatively soon, as it is becoming increasingly clear that atoms other
       than caesium can provide more precise time standards. However, it is
       not excluded that some of the other defining constants would
       eventually have to be replaced as well. For example, the elementary
       charge e corresponds to a coupling strength of the electromagnetic
       force via the fine-structure constant \alpha. Some theories predict
       that \alpha can vary over time. The presently known experimental
       limits of the maximum possible variation of \alpha are so low that
       'any effect on foreseeable practical measurements can be
       excluded',^[2]^: 128  even if one of these theories turns out to be
       correct. Nevertheless, if the fine-structure constant turns out to
       slightly vary over time, science and technology may in the future
       advance to a point where such changes become measurable. At that
       point, one might consider replacing, for the purposes of defining the
       SI, the elementary charge with some other quantity, the choice of
       which will be informed by what we learn about the time variation of
       \alpha.
   64. ^ The official term is "States Parties to the Metre Convention"; the
       term "Member States" is its synonym and used for easy reference.^[13]
       As of 13 January 2020,.^[13] there are 63 Member States and 39
       Associate States and Economies of the General Conference.^[i]
   65. ^ Among the tasks of these Consultative Committees are the detailed
       consideration of advances in physics that directly influence
       metrology, the preparation of Recommendations for discussion at the
       CIPM, the identification, planning and execution of key comparisons of
       national measurement standards, and the provision of advice to the
       CIPM on the scientific work in the laboratories of the BIPM.^[44]
   66. ^ As of April 2020, these include those from Spain (CEM), Russia
       (FATRiM), Switzerland (METAS), Italy (INRiM), South Korea (KRISS),
       France (LNE), China (NIM), US (NIST), Japan (AIST/NIMJ), UK (NPL),
       Canada (NRC), and Germany (PTB).
   67. ^ As of April 2020, these include International Electrotechnical
       Commission (IEC), International Organization for Standardization
       (ISO), and International Organization of Legal Metrology (OIML).
   68. ^ As of April 2020, these include International Commission on
       Illumination (CIE), CODATA Task Group on Fundamental Constants,
       International Commission on Radiation Units and Measurements (ICRU),
       and International Federation of Clinical Chemistry and Laboratory
       Medicine (IFCC).
   69. ^ As of April 2020, these include International Astronomical Union
       (IAU), International Union of Pure and Applied Chemistry (IUPAC), and
       International Union of Pure and Applied Physics (IUPAP).
   70. ^ These are individuals with a long-term involvement in matters
       related to units, having actively contributed to publications on
       units, and having a global view and understanding of science as well
       as knowledge on the development and functioning of the International
       System of Units.^[48] As of April 2020, these include^[47]^[49] Prof.
       Marc Himbert and Dr. Terry Quinn.
   71. ^ For historical reasons, the kilogram rather than the gram is treated
       as the coherent unit, making an exception to this characterisation.
   72. ^ Ohm's law: 1 Ω = 1 V/A from the relationship E = I × R, where E is
       electromotive force or voltage (unit: volt), I is current (unit:
       ampere), and R is resistance (unit: ohm).
   73. ^ The 22nd CGPM meeting Draft Resolution includes a proposal to add
       new unit prefixes ronna (R) for powers of 27, ronto (r) for −27,
       quetta (Q) for 30, and quecto (q) for −30.^[54]
   74. ^ While the second is readily determined from the Earth's rotation
       period, the metre, originally defined in terms of the Earth's size and
       shape, is less amenable; however, the fact that the Earth's
       circumference is very close to 40000 km may be a useful mnemonic.
   75. ^ This is evident from the formula s = v_0 t +
       Link: mw-deduplicated-inline-style
       1/2 a t^2 with v_0 = 0 and a = 9.81 m/s^2.
   76. ^ This is evident from the formula T = 2π √L / g.
   77. ^ A 60 watt light bulb has about 800 lumens^[62] which is radiated
       equally in all directions (i.e. 4π steradians), thus is equal to I_v =
       Link: mw-deduplicated-inline-style
       800 lm/4π sr ≈ 64 cd.
   78. ^ This is evident from the formula P = I V.
   79. ^ The unit is named after Anders Celsius.
   80. ^ ^a ^b Except where specifically noted, these rules are common to
       both the SI Brochure and the NIST brochure.
   81. ^ For example, the United States' National Institute of Standards and
       Technology (NIST) has produced a version of the CGPM document (NIST SP
       330) which clarifies usage for English-language publications that use
       American English
   82. ^ This term is a translation of the official [French] text of the SI
       Brochure.
   83. ^ The strength of the Earth's magnetic field was designated 1 G
       (gauss) at the surface (= 1 cm^−1/2⋅g^1/2⋅s^−1).
   84. ^ Argentina, Austria-Hungary, Belgium, Brazil, Denmark, France, German
       Empire, Italy, Peru, Portugal, Russia, Spain, Sweden and Norway,
       Switzerland, Ottoman Empire, United States, and Venezuela.
   85. ^ The text "Des comparaisons périodiques des étalons nationaux avec
       les prototypes internationaux" (English: the periodic comparisons of
       national standards with the international prototypes) in article 6.3
       of the Metre Convention distinguishes between the words "standard"
       (OED: "The legal magnitude of a unit of measure or weight") and
       "prototype" (OED: "an original on which something is modelled").
   86. ^ These included:
          * General Conference on Weights and Measures (Conférence générale
            des poids et mesures or CGPM)
          * International Committee for Weights and Measures (Comité
            international des poids et mesures or CIPM)
          * International Bureau of Weights and Measures (Bureau
            international des poids et mesures or BIPM) – an international
            metrology centre at Sèvres in France that has custody of the
            International prototype kilogram, provides metrology services for
            the CGPM and CIPM.
   87. ^ Pferd is German for "horse" and Stärke is German for "strength" or
       "power". The Pferdestärke is the power needed to raise 75 kg against
       gravity at the rate of one metre per second. (1 PS = 0.985 HP).
   88. ^ It is known as the International Prototype of the Kilogram.
   89. ^ Meaning, they are neither part of the SI nor one of the non-SI units
       accepted for use with that system.
   90. ^ Almost invariably either the meter or the centimeter.
   91. ^ All major systems of units in which force rather than mass is a base
       unit are of a type known as gravitational system (also known as
       technical or engineering system). In the most prominent metric example
       of such a system, the unit of force is taken to be the kilogram-force
       (kp), which is the weight of the standard kilogram under standard
       gravity, g = 9.80665 m/s^2. The unit of mass is then a derived unit,
       defined as the mass that is accelerated at a rate of 1 m/s^2 when
       acted upon by a net force of 1 kp; often called the hyl, it therefore
       has a value of 1 hyl = 9.80665 kg, so that it is not a decimal
       multiple of the gram.
   92. ^ Having said that, some units are recognised by all metric systems.
       The second is a base unit in all of them. The metre is recognised in
       all of them, either as the base unit of length or as a decimal
       multiple or submultiple of the base unit of length. On the other hand,
       not every metric system recognises the gram as a unit (either the base
       unit or a decimal multiple of the base unit). In particular, in
       gravitational metric systems, the unit of force (gram-force or
       kilogram-force) replaces the unit of mass as a base unit. The unit of
       mass is then a derived unit, defined as the mass that, when acted upon
       by a net unit force, is accelerated at the unit rate (i.e. at a rate
       of 1 base unit of length^[cl] per second squared).^[cm]
   93. ^ ^a ^b ^c Interconversion between different systems of units is
       usually straightforward; however, the units for electricity and
       magnetism are an exception, and a surprising amount of care is
       required. The problem is that, in general, the physical quantities
       that go by the same name and play the same role in the CGS-ESU,
       CGS-EMU, and SI—e.g. 'electric charge', 'electric field strength',
       etc.—do not merely have different units in the three systems;
       technically speaking, they are actually different physical
       quantities.^[110]^: 422 ^[110]^: 423  Consider 'electric charge',
       which in each of the three systems can be identified as the quantity
       two instances of which enter in the numerator of Coulomb's law (as
       that law is written in each system). This identification produces
       three different physical quantities: the 'CGS-ESU charge', the
       'CGS-EMU charge', and the 'SI charge'.^[111]^: 35 ^[110]^: 423  They
       even have different dimensions when expressed in terms of the base
       dimensions: mass^1/2 × length^3/2 × time^−1 for the CGS-ESU charge,
       mass^1/2 × length^1/2 for the CGS-EMU charge, and current × time for
       the SI charge (where, in the SI, the dimension of current is
       independent of those of mass, length, and time). On the other hand,
       these three quantities are clearly quantifying the same underlying
       physical phenomenon. Thus, we say not that 'one abcoulomb equals ten
       coulomb', but rather that 'one abcoulomb corresponds to ten
       coulomb',^[110]^: 423  written as 1 abC ≘ 10 C.^[111]^: 35  By that we
       mean, 'if the CGS-EMU electric charge is measured to have the
       magnitude of 1 abC, then the SI electric charge will have the
       magnitude of 10 C'.^[111]^: 35 ^[112]^: 57–58 
   94. ^ Neither EMU nor ESU units had sizes that were convenient for
       practical work of electrical engineers. It was therefore decided to
       establish a 'practical' system of units, where each unit is an
       appropriate decimal multiple or submultiple of the corresponding EMU
       unit, so that the resulting units have convenient sizes and form a
       coherent system. These practical units were given names derived from
       the names of eminent scientists, and many of these units—both the
       names and the magnitudes—were later incorporated into the SI: the
       volt, the ampere, the ohm, etc.
   95. ^ For several decades, the ESU and EMU units did not have special
       names; one would just say, for example, the ESU unit of resistance. In
       1903, A. E. Kennelly suggested that the names of the EMU units be
       obtained by prefixing the name of the corresponding ‘practical unit'
       by ‘ab-’ (short for ‘absolute’, giving the ‘abohm’, ‘abvolt’, the
       ‘abampere’, etc.), and that the names of the ESU units be analogously
       obtained by using the prefix ‘abstat-’, which was later shortened to
       ‘stat-’ (giving the ‘statohm’, ‘statvolt’, ‘statampere’,
       etc.).^[113]^: 534–5  In a sense, this brings us back full circle:
       historically, the magnitudes of the 'practical units' were first
       defined in terms of the magnitudes of the corresponding units of the
       EMU system;^[cp] then, Kennelly proposed that the names of units in
       the ESU and EMU systems be derived from the corresponding names of
       practical units. Kennelly's naming system was widely used in the U.S.,
       but, apparently, not in Europe.^[114]
   96. ^ ^a ^b The CGS-Gaussian units are a blend of the CGS-ESU and CGS-EMU,
       taking units related to magnetism from the latter and all the rest
       from the former. In addition, the system introduces the gauss as a
       special name for the CGS-EMU unit maxwell per square centimetre.
   97. ^ Authors often abuse notation slightly and write these with an
       'equals' sign ('=') rather than a 'corresponds to' sign ('≘').

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Further reading[edit]

     * International Union of Pure and Applied Chemistry (1993). Quantities,
       Units and Symbols in Physical Chemistry, 2nd edition, Oxford:
       Blackwell Science.
       Link: mw-deduplicated-inline-style
       ISBN 0-632-03583-8. Electronic version.
     * Unit Systems in Electromagnetism
     * MW Keller et al. Metrology Triangle Using a Watt Balance, a Calculable
       Capacitor, and a Single-Electron Tunneling Device
     * "The Current SI Seen From the Perspective of the Proposed New SI".
       Barry N. Taylor. Journal of Research of the National Institute of
       Standards and Technology, Vol. 116, No. 6, Pgs. 797–807, Nov–Dec 2011.
     * B. N. Taylor, Ambler Thompson, International System of Units (SI),
       National Institute of Standards and Technology 2008 edition,
       Link: mw-deduplicated-inline-style
       ISBN 1437915582.

External links[edit]

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   Official

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     * NIST On-line official publications on the SI
          * NIST Special Publication 330, 2019 Edition: The International
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