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.^[5] 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,^[6] and has largely been revised in 2019–2020.^[7]

Controlling authority

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^{[lower-alpha 7]})^[8], the International Committee for Weights and Measures (CIPM^{[lower-alpha 8]}), and the International Bureau of Weights and Measures (BIPM^{[lower-alpha 9]}). All the decisions and recommendations concerning units are collected in a brochure called The International System of Units (SI),^[3] which is published in French and English by the BIPM and periodically updated. The writing and maintenance of the brochure is carried out by one of the committees of the 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.^[9] The brochure leaves some scope for local variations, particularly regarding unit names and terms in different languages. 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.^[10]

SI base units

The SI selects seven units to serve as base units, corresponding to seven base physical quantities.^{[lower-alpha 12][lower-alpha 13]} 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).^[3] 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 base units^{[10]: 6 [12][13]}

Unit name	Unit symbol	Dimension symbol	Quantity name	Typical symbols	Definition
second [n 1]	s	T	time	${\displaystyle t}$	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.
metre	m	L	length	${\displaystyle l}$, ${\displaystyle h}$, ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle r}$, etc.[n 2]	The distance travelled by light in vacuum in 1/299792458 seconds.
kilogram [n 3]	kg	M	mass	${\displaystyle m}$	The kilogram is defined by setting the Planck constant h exactly to 6.62607015×10−34 J⋅s (J = kg⋅m2⋅s−2), given the definitions of the metre and the second.[14]
ampere	A	I	electric current	${\displaystyle I,\;i}$	The flow of exactly 1/1.602176634×10−19 times the elementary charge e per second. Equalling approximately 6.2415090744×1018 elementary charges per second.
kelvin	K	Θ	thermodynamic temperature	${\displaystyle T}$	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⋅m2⋅s−2), given the definition of the kilogram, the metre, and the second.
mole	mol	N	amount of substance	${\displaystyle n}$	The amount of substance of exactly 6.02214076×1023 elementary entities.[n 4] This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit mol−1.
candela	cd	J	luminous intensity	${\displaystyle I_{v}}$	The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 5.4×1014 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.
Notes ↑ 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 (1⁄602) 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. ↑ Symbols for length vary greatly with context. Problems involving intuitive three-dimensional quantities often use ${\displaystyle l}$, ${\displaystyle w}$, and ${\displaystyle h}$ for length, width, 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. ${\displaystyle a}$, ${\displaystyle b}$) along said axis, or by the same symbol as the axis itself (e.g. ${\displaystyle x}$, ${\displaystyle y}$, or ${\displaystyle r}$ for horizontal, vertical, and radial axes, respectively). ↑ 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. ↑ 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

Arrangement of the principal measurements in physics based on the mathematical manipulation of length, time, and mass

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.^{[lower-alpha 14]} The base and coherent derived units of the SI together form a coherent system of units (the set of coherent SI units).^{[lower-alpha 15]} Twenty-two coherent derived units have been provided with special names and symbols.^{[lower-alpha 2]} The seven base units and the 22 derived units with special names and symbols may be used in combination to express other derived units,^{[lower-alpha 3]} which are adopted to facilitate measurement of diverse quantities.

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^{[lower-alpha 16]} specifies base and derived quantities that necessarily have the corresponding SI units.^[3]: 129

The derived units in the SI are formed by powers, products, or quotients of the base units and are potentially unlimited in number.^{[11]: 103 [10]: 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/m²).^[16]

Dimensionless units

The unit of a dimensionless quantity is one (symbol 1), but it is rarely shown.^[19] The radian and steradian are also dimensionless quantities, but use the symbols rad and sr respectively.^[20]

Prefixes

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, 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.

The current version of the SI provides twenty-four metric prefixes that signify decimal powers ranging from 10⁻³⁰ to 10³⁰, the most recent being adopted in 2022.^{[3]: 143–144 [21][22][23]} 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, 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).^{[lower-alpha 17]} 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. This is why the SI (and metric systems more generally) are called decimal systems of measurement units.^[24]

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. It can also be combined with other unit symbols to form compound unit symbols.^[3]: 143 For example, g/cm³ is an SI unit of density, where cm³ is to be interpreted as (cm)³.

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.^{[11]: 122 [25]: 14} When prefixes are used to form multiples and submultiples of SI base and derived units, the resulting units are no longer coherent.^[11]: 7

The BIPM specifies 24 prefixes for the International System of Units (SI):

Unit names

According to the SI Brochure,^[3]: 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 capitalised only at the beginning of a sentence and in headings and publication titles. As a nontrivial application of this rule, the SI Brochure notes^[3]: 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 capitalised because it is a proper name.^[3]: 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, and International English uses deca-, metre, and litre. The name of the unit whose symbol is t and which is defined according to 1 t = 10³ kg is 'metric ton' in US English and 'tonne' in International English.^[10]: iii

Unit symbols and the values of quantities

Symbols of SI units are intended to be unique and universal, independent of the context language.^{[11]: 130–35} The SI Brochure has specific rules for writing them.^{[11]: 130–35} The guideline produced by the National Institute of Standards and Technology (NIST)^[28] clarifies language-specific details for American English that were left unclear by the SI Brochure, but is otherwise identical to the SI Brochure.^[29]

General rules

General rules^{[lower-alpha 19]} 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 (or nonbreaking space) (representing a multiplication sign) and a unit symbol; e.g., 2.21 kg, 7.3×10² m², 22 K. This rule explicitly includes the percent sign (%)^[11]: 134 and the symbol for degrees Celsius (°C).^[11]: 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., km² 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 m/s. In cases where a solidus is followed by a centre dot (or space), or more than one solidus is present, parentheses must be used to avoid ambiguity; e.g., kg/(m⋅s²), kg⋅m⁻¹⋅s⁻², and (kg/m)/s² are acceptable, but kg/m/s² and kg/m⋅s² are ambiguous and unacceptable.

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".^[25]
• 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.^[30]
• Thin spaces may be used as a thousands separator (1000000) in order to facilitate reading, but neither dots nor commas should be inserted between groups of three (1,000,000 or 1.000.000).^[11]: 133 When there are only four digits, a space is ordinarily not used to isolate a single digit. NIST guideline has the same recommendations.^[31]: 37
• 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.

Definition vs. realisation of units

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.^{[3]: 126 [lower-alpha 21]} Namely, the SI units are defined by declaring that seven defining constants^{[3]: 125–129} 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 540×10¹² Hz.^{[lower-alpha 22]} The nature of the defining constants ranges from fundamental constants of nature such as c to the purely technical constant K_cd.^{[3]: 128–129} 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 mises en pratique,^{[lower-alpha 23]} which are also published by the BIPM.^[41] 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.^{[lower-alpha 26]}

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⋅m²/s². 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 motor spirit 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;^[3]: 128 for example, the definition of c defines the unit m/s as 1 m/s = 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:

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.^{[lower-alpha 27]} 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,^{[lower-alpha 28]} while, on the other hand, the kilogram can be written in terms of these three units, namely, kg = (Hz)(J⋅s)/(m/s)².^{[lower-alpha 29]} 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

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.^[3]: 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.^{[lower-alpha 24]} 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.^{[lower-alpha 25]} 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.^[3]: 122 In practice, the CIPM Consultative Committees provide so-called "mises en pratique" (practical techniques),^[41] which are the descriptions of what are currently believed to be best experimental realisations of the units.^[45]

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.^{[lower-alpha 30]} 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. One major disadvantage is that artefacts can be lost, damaged, or changed.^{[lower-alpha 31]} 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 alloy; 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.^[14] 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.^[48]

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^{[11]: 113–114}); others referred to idealised experimental prescriptions^[3]: 125 (as in the case of the former SI definition of the ampere^[11]: 113 and the former SI definition (originally enacted in 1979) of the candela^[11]: 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.

The improvisation of units

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.^[53] The group, which included preeminent French men of science,^[54]: 89 used the same principles for relating length, volume, and mass that had been proposed by the English clergyman John Wilkins in 1668^[55][56] 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.^[57][58]

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.^[59]: 82

Thomson

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.^{[60][61][62][63][64]}

Later, during the process of adoption of the metric system, the names gramme and kilogramme, derived from Greek and Latin, replaced the former provincial French 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.^[65][66] 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.^[49]

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.^[60] 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.^{[lower-alpha 32][67]}

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 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

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.^{[lower-alpha 33][54]: 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.^{[61][11]: 96}

A set of 30 prototypes of the metre and 40 prototypes of the kilogram,^{[lower-alpha 34]} in each case made of a 90% platinum-10% iridium alloy, were manufactured by British metallurgy specialty firm 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.^[68]

The treaty also established a number of international organisations to oversee the keeping of international standards of measurement.^{[69][lower-alpha 35]}

The CGS and MKS systems

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.^[63]

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".^[70]

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.^[71] 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.^[64] 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.^[72] 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,^{[73][lower-alpha 36]} the darcy for permeability^[74] 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 US customary system and British Imperial system.

The Practical system of units

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".^[75] 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.^[76] 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.^{[11]: 104, 130}

Birth of the SI

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.^{[11]: 110 [77]}

The International Bureau of Weights and Measures (BIPM) has described SI as "the modern form of metric system".^[11]: 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.

Historical definitions

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/s². This is true whether the object is floating in space or in a gravity field e.g. at the Earth's surface. Weight is the force exerted on a body by a gravitational field, and hence its weight depends on the strength of the gravitational field. Weight of a 1 kg mass at the Earth's surface is m × g; mass times the acceleration due to gravity, which is 9.81 newtons at the Earth's surface and is about 3.5 newtons at the surface of Mars. Since the acceleration due to gravity is local and varies by location and altitude on the Earth, weight is unsuitable for precision measurements of a property of a body, and this makes a unit of weight unsuitable as a base unit.

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.^{[11]: 156, 158-159, 165 [78]}

The range of defined prefixes pico- (10⁻¹²) to tera- (10¹²) was extended to quecto- (10⁻³⁰) to quetta- (10³⁰).^{[11]: 152, 158, 164}

The 1960 definition of the standard metre in terms of wavelengths of a specific emission of the krypton-86 atom was replaced in 1983 with the distance that light travels in vacuum in exactly 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.^[79]

2019 redefinition

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. Here, ${\displaystyle a\rightarrow b}$ means that ${\displaystyle a}$ is used to define ${\displaystyle b}$.

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.^[81] 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:^[82]

• 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.^[83] The change was adopted by the European Union through Directive (EU) 2019/1258.^[84]

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:^[11]

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 m³.

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 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.

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).

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

Metric units that are not recognised by the SI

Although the term metric system is often used as an informal alternative name for the International System of Units,^[85] 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.^{[lower-alpha 38][lower-alpha 41]}

Examples include the centimetre–gram–second (CGS) system, the dominant metric system in the physical sciences and electrical engineering from the 1860s until at least the 1960s, and 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.).^{[lower-alpha 42][lower-alpha 44]} 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).^{[lower-alpha 45]}

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,^[91]:17 lambert, apostilb, skot, brill, troland, talbot, candlepower, candle), thermodynamics (calorie), and spectroscopy (reciprocal centimetre).

Some other SI-unrecognised metric units that do not 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).^{[92]: 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 deci pascal, etc. (a related group are the correspondences^{[lower-alpha 42]} such as 1 abampere ≘ 1 deca ampere, 1 abhenry ≘ 1 nano henry, etc.^{[lower-alpha 46]}). 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, The angstrom (1 Å = 10⁻¹⁰ m), still used in various fields, etc. (and correspondences^{[lower-alpha 42]} 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 US), the jansky (in radio astronomy), the gauss (in industry) and the CGS-Gaussian units^{[lower-alpha 45]} more generally (in some subfields of physics), the metric horsepower (for engine power, in most of the non-English speaking world), the kilogram-force (for rocket engine thrust, in China and sometimes in Europe), etc. Others are now rarely used, such as the sthene and the rutherford.

Unacceptable uses

Sometimes, SI unit name variations are introduced, mixing information about the corresponding physical quantity or the conditions of its measurement; however, this practice is unacceptable with the SI.^{[lower-alpha 47]} Instances include: "watt-peak" and "watt RMS"; "geopotential metre" and "vertical metre"; "standard cubic metre"; "atomic second", "ephemeris second", and "sidereal second".