Coherence (units of measurement)

{{Short description|Type of system of units of measurement}} [[File:James Clerk Maxwell.jpg|thumb|upright|[[James Clerk Maxwell]] played a major role in developing the concept of a coherent [[CGS system]] and in extending the [[metric system]] to include electrical units.]]

A '''coherent system of units''' is a [[system of units of measurement]] used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations directly relating the quantities.<ref name=definition> {{citation | author = Working Group 2 of the Joint Committee for Guides in Metrology (JCGM/WG 2). | publisher = [[International Bureau of Weights and Measures]] (BIPM) on behalf of the Joint Committee for Guides in Metrology | year = 2012 | url = https://www.bipm.org/documents/20126/2071204/JCGM_200_2012.pdf | title = International vocabulary of metrology – Basic and general concepts and associated terms (VIM) | edition = 3rd (2008 version with minor corrections) | at = 1.12 | accessdate = 2026-05-29 }}</ref><ref>{{citation |author=Thor, A. J. |year=1994 |title=New International Standards for Quantities and Units |journal=Metrologia |volume=30 |number=5|page=517|doi=10.1088/0026-1394/30/5/010 }}</ref> It is a system in which every quantity has a unique unit, or one that does not use [[conversion factor]]s.<ref> {{cite book |title = The International System of Units (SI) |url = https://archive.org/details/internationalsys3309tayl |at = p. 12 |first1 = Barry N. |last1 = Taylor |first2 = Ambler |last2 = Thompson |year = 2008 |publisher = U.S. Department of Commerce |location = Washington, D.C. }}</ref>

A '''coherent derived unit''' is a [[SI derived unit|derived unit]] that, for a given system of quantities and for a chosen set of [[Base unit (measurement)|base unit]]s, is a product of powers of base units, with the [[proportionality factor]] being one.<ref name=definition/>

If a system of quantities has equations that relate quantities and the associated system of units has corresponding base units, with only one unit for each base quantity, then it is coherent if and only if every derived unit of the system is coherent.

The concept of coherence was developed in the mid-nineteenth century by, amongst others, [[William Thomson, 1st Baron Kelvin|Kelvin]] and [[James Clerk Maxwell]] and promoted by the [[British Science Association]]. The concept was initially applied to the [[centimetre–gram–second]] (CGS) in 1873 and the [[foot–pound–second system]]s (FPS) of units in 1875. The [[International System of Units]] (SI) was designed in 1960 to incorporate the principle of coherence.

== Examples == In the SI, the derived unit {{val|ul=m/s}} is a coherent derived unit for [[speed]] or [[velocity]]<ref>{{SIbrochure9th|page=139}}</ref> but {{val|ul=km|up=hour}} is not a coherent derived unit. Speed or velocity is defined by the change in distance divided by a change in time. The derived unit {{val|u=m/s}} uses the base units of the SI system.<ref name=definition/> The derived unit {{val|u=km|up=hour}} requires numerical factors to relate to the SI base units: {{val|1000|u=m|up=km}} and {{val|3600|u=second|up=hour}}.

In the [[cgs]] system, {{val|u=m/s}} is not a coherent derived unit. The numerical factor of {{val|100|u=cm|up=s}} is needed to express {{val|u=m/s}} in the cgs system.

== History ==

=== Before the metric system === The earliest units of measure devised by humanity bore no relationship to each other.{{Cn|date=May 2021}} As both humanity's understanding of [[Philosophy|philosophical concepts]] and the organisation of [[society]] developed, so units of measurement were standardized—first particular units of measure had the same value across a [[community]], then different units of the same [[quantity]] (for example feet and inches) were given a fixed relationship. Apart from [[Ancient China]] where the units of capacity and of mass were linked to [[millet|red millet seed]], there is little evidence of the linking of different quantities until the [[Age of Enlightenment|Enlightenment]].<ref> {{cite book |title = The Basis of Measurement: Volume 1—Historical Aspects |at = Chapter 1: Some Ancient Units |last1 = McGreevy |first1 = Thomas |editor1-last = Cunningham |editor1-first = Peter |year = 1995 |publisher = Picton Publishing |location = Chippenham |isbn = 0-948251-82-4 }}</ref>

==== Relating quantities of the same kind ====  The history of the measurement of length dates back to the early civilization of the [[Middle East]] (10000 BC – 8000 BC). Archaeologists have been able to reconstruct the units of measure in use in [[Ancient Mesopotamian units of measurement|Mesopotamia]], [[History of measurement systems in India|India]], [[Biblical and Talmudic units of measurement|the Jewish culture]] and many others. Archaeological and other evidence shows that in many civilizations, the ratios between different units for the same quantity of measure were adjusted so that they were integer numbers. In many early cultures such as [[Ancient Egyptian units of measurement|Ancient Egypt]], multiples with prime factors aside from 2, 3 and 5 were sometimes used—the Egyptian royal cubit being 28 fingers or 7 [[Hand (unit)|hands]].<ref name="MC"> {{cite book |last=Clagett |first=Marshall |title=Ancient Egyptian science, a Source Book. Volume Three: Ancient Egyptian Mathematics. |year=1999 |publisher=[[American Philosophical Society]] |location=Philadelphia |isbn=978-0-87169-232-0 |url=https://archive.org/details/bub_gb_8c10QYoGa4UC |accessdate = 2013-05-02 |page = [https://archive.org/details/bub_gb_8c10QYoGa4UC/page/n15 7] }}</ref> In 2150 BC, the [[Akkadian Empire|Akkadian]] emperor [[Naram-Sin of Akkad|Naram-Sin]] rationalized the Babylonian system of measure, adjusting the ratios of many units of measure to multiples of which the only prime factors were 2, 3 and 5; for example there were 6 ''she'' ([[barley]]corns) in a ''shu-si'' ([[Finger (unit)|finger]]) and 30 shu-si in a ''kush'' ([[cubit]]).<ref> {{cite web |url = http://it.stlawu.edu/~dmelvill/mesomath/obmetrology.html |title = Old Babylonian Weights and Measures |first1 = Duncan J. |last1 = Melville |accessdate = 2013-05-02 |year = 2001 |publisher = [[St. Lawrence University]] |archive-date = 2008-05-13 |archive-url = https://web.archive.org/web/20080513131154/http://it.stlawu.edu/~dmelvill/mesomath/obmetrology.html |url-status = dead }}</ref> [[File:Nippur_cubit.JPG|thumb|550px|center|[[Measuring rod]] on exhibition in the Archeological Museum of [[Istanbul]] (Turkey) dating to the (3rd millennium BC) excavated at [[Nippur]], [[Mesopotamia]]. The rod shows the various units of measure in use.]]

==== Relating quantities of different kinds ==== Non-[[Unit commensurability#Commensurability|commensurable]] quantities have different [[Dimensional analysis|physical dimension]]s, which means that adding or subtracting them is not meaningful. For instance, adding the [[mass]] of an object to its [[volume]] has no physical meaning. However, new quantities (and, as such, units) can be [[Derived units|derived]] via multiplication and [[exponentiation]] of other units. As an example, the [[SI unit]] for force is the [[Newton (unit)|newton]], which is defined as kg⋅m⋅s−2. Since a coherent derived unit is one which is defined by means of multiplication and exponentiation of other units but not multiplied by any scaling factor other than 1, the [[Pascal (unit)|pascal]] is a coherent unit of [[pressure]] (defined as kg⋅m−1⋅s−2), but the [[Bar (unit)|bar]] (defined as {{val|100000|u=kg⋅m−1⋅s−2}}) is not.

Note that coherence of a given unit depends on the definition of the base units. Should the standard unit of length change such that it is shorter by a factor of {{val|100000}}, then the bar would be a coherent derived unit. However, a coherent unit remains coherent (and a non-coherent unit remains non-coherent) if the base units are redefined in terms of other units with the numerical factor always being unity.

=== Metric system === The concept of coherence was only introduced into the metric system in the third quarter of the nineteenth century; in its original form the metric system was non-coherent – in particular the [[litre]] was 0.001 m3 and the [[Are (unit)|are]] (from which we get the [[hectare]]) was 100 m2. A precursor to the concept of coherence was however present in that the units of mass and length were related to each other through the physical properties of water, the gram having been designed as being the mass of one cubic centimetre of water at its freezing point.<ref name=France1795> {{cite web |url=http://aviatechno.free.fr/unites/nouveausys.php |title=La loi du 18 Germinal an 3 ''la mesure [républicaine] de superficie pour les terrains, égale à un carré de dix mètres de côté'' |language = French |trans-title=The law of 18 Germinal year 3 "The republican measures of land area equal to a square with sides of ten metres" |publisher = Le CIV (Centre d'Instruction de Vilgénis) – Forum des Anciens |accessdate = 2010-03-02 }}</ref>

The [[CGS system]] had two units of energy, the [[erg]] that was related to [[mechanics]] and the [[calorie]] that was related to [[thermal energy]], so only one of them (the erg, equivalent to the g⋅cm2/s2) could bear a coherent relationship to the base units. By contrast, coherence was a design aim of the SI, resulting in only one unit of energy being defined – the [[joule]].<ref>SI brochure, §1.2 Two classes of SI Units, p. 92</ref>

== List of coherent units == This list catalogues coherent relationships in various systems of units.

=== SI === {{main|SI coherent derived unit}}

The following is a list of quantities, each with its corresponding coherent SI unit: : [[frequency]] ([[hertz]]) = [[Multiplicative inverse|reciprocal]] of time ([[inverse second]]) : [[force (physics)|force]] ([[newton (unit)|newton]]) = mass (kilogram) × acceleration (m/s2) : [[pressure]] ([[Pascal (unit)|pascal]]) = force (newton) ÷ [[area]] (m2) : [[energy]] ([[joule]]) = force (newton) × distance (metre) : [[Power (physics)|power]] ([[watt]]) = energy (joule) ÷ time (second) : [[potential difference]] ([[volt]]) = power (watt) ÷ electric current (ampere) : [[electric charge]] ([[coulomb]]) = electric current (ampere) × time (second) : [[equivalent dose|equivalent radiation dose]] ([[sievert]]) = energy (joule) ÷ mass (kilogram) : [[Absorbed dose|absorbed radiation dose]] ([[Gray (unit)|gray]]) = energy (joule) ÷ mass (kilogram) : [[Radioactive decay|radioactive activity]] ([[becquerel]]) = reciprocal of time (s−1) : [[capacitance]] ([[farad]]) = electric charge (coulomb) ÷ potential difference (volt) : [[Electrical resistance and conductance|electrical resistance]] ([[ohm]]) = potential difference (volt) ÷ electric current (ampere) : [[Electrical resistance and conductance|electrical conductance]] ([[siemens (unit)|siemens]]) = electric current (ampere) ÷ potential difference (volt) : [[magnetic flux]] ([[Weber (unit)|weber]]) = potential difference ([[volt]]) × time (second) : [[Magnetic field|magnetic flux density]] ([[Tesla (unit)|tesla]]) = magnetic flux (weber) ÷ area (square metre)

=== CGS === The following is a list of coherent [[centimetre–gram–second]] (CGS) system of units: : [[acceleration]] ([[gal (unit)|gals]]) = distance (centimetre) ÷ time (s2) : force ([[dyne]]) = mass (gram) × acceleration (cm/s2) : energy ([[erg]]) = force (dyne) × distance (centimetre) : pressure ([[barye]]) = force (dyne) ÷ [[area]] (cm2) : dynamic [[viscosity]] ([[Poise (unit)|poise]]) = mass (gram) ÷ (distance (centimetre) × time (second)) : kinematic [[viscosity]] ([[stokes (unit)|stokes]]) = area (cm2) ÷ time (second)

=== FPS === The following is a list of coherent [[foot–pound–second]] (FPS) system of units: : force ([[poundal|pdl]]) = mass ([[Pound (mass)|lb]]) × acceleration ([[Foot (unit)|ft]]/s2)

== See also == * [[Systems of measurement]] * [[Geometrized unit system]] * [[Planck units]] * [[Atomic units]] * [[Metre–kilogram–second system]] (MKS) * [[Metre–tonne–second system]] (MTS) * [[QES|Quadrant–eleventh-gram–second system]] (QES)

== References == {{reflist}}

{{DEFAULTSORT:Coherent units of measurement}} [[Category:Systems of units]] [[Category:Dimensional analysis]]