View source for Quantity calculus

'''Quantity calculus''' is the formal method for describing the mathematical relations between ''abstract'' [[Physical quantity|physical quantities]].<ref name="deBoer">{{citation | title = On the History of Quantity Calculus and the International System | first = J. | last = de Boer | year = 1995 |journal = [[Metrologia]] | volume = 31 | issue = 6 | pages = 405–429 | doi = 10.1088/0026-1394/31/6/001|bibcode = 1995Metro..31..405D }}</ref>{{efn|Here the term ''calculus'' should be understood in its broader sense of "a system of computation", rather than in the sense of [[differential calculus]] and [[integral calculus]].}} 

Its roots can be traced to [[Joseph Fourier|Fourier's]] concept of [[dimensional analysis]] (1822).<ref>{{citation | last = Fourier | first = Joseph | authorlink = Joseph Fourier | title = Théorie analytique de la chaleur | year = 1822}}</ref> The basic [[axiom]] of quantity calculus is [[James Clerk Maxwell|Maxwell's]] description<ref>{{citation | last = Maxwell | first = J. C. | authorlink = James Clerk Maxwell | title = A Treatise on Electricity and Magnetism | url = https://archive.org/details/electricandmagne01maxwrich | location = Oxford | publisher = Oxford University Press | year = 1873| hdl = 2027/uc1.l0065867749 | hdl-access = free }}</ref> of a physical quantity as the [[product (mathematics)|product]] of a "numerical value" and a "reference quantity" (i.e. a "unit quantity" or a "[[unit of measurement]]"). De Boer summarized the multiplication, division, addition, association and commutation rules of quantity calculus and proposed that a full [[axiomatization]] has yet to be completed.<ref name="deBoer" />

Measurements are expressed as products of a numeric value with a unit symbol, e.g. "12.7&nbsp;m".  Unlike algebra, the unit symbol represents a measurable quantity such as a meter, not an [[Variable (mathematics)|algebraic variable]] i.e. the unit symbol does not satisfy the axioms of arithmetic.<ref>{{Cite journal|author=A. Majhi |title=A logico-linguistic inquiry into the foundations of physics: Part 1|journal= Axiomathes|volume=32|pp= 153-198|year=2022 |doi=10.1007/s10516-021-09593-0|url=https://arxiv.org/abs/2110.03514|arxiv=2110.03514}}</ref>

A careful distinction needs to be made between [[abstract quantity|abstract quantities]] and measurable quantities. The multiplication and division rules of quantity calculus are applied to [[SI base unit]]s (which are measurable quantities) to define [[SI derived unit]]s, including [[dimensionless quantity|dimensionless]] derived units, such as the [[radian]] (rad) and [[steradian]] (sr) which are useful for clarity, although they are both algebraically equal to 1. Thus there is some disagreement about whether it is meaningful to multiply or divide units. Emerson suggests that if the units of a quantity are algebraically simplified, they then are no longer units of that quantity.<ref name="Emerson">{{citation | title = On quantity calculus and units of measurement | first = W.H. | last = Emerson | year = 2008 |journal = [[Metrologia]] | volume = 45 | issue = 2 | pages = 134–138 | doi=10.1088/0026-1394/45/2/002|bibcode = 2008Metro..45..134E }}</ref> Johansson proposes that there are logical flaws in the application of quantity calculus, and that the so-called dimensionless quantities should be understood as "unitless quantities".<ref name="Johansson">{{citation | title = Metrological thinking needs the notions of ''parametric'' quantities, units and dimensions | first = I. | last = Johansson | year = 2010 |journal = [[Metrologia]] | volume = 47 | issue = 3 | pages = 219–230|bibcode = 2010Metro..47..219J |doi = 10.1088/0026-1394/47/3/012 }}</ref>

How to use quantity calculus for unit conversion and keeping track of units in algebraic manipulations is explained in the handbook ''[[Quantities, Units and Symbols in Physical Chemistry]]''.

==Notes==
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==References==
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==Further reading==
*[[International Organization for Standardization]]. ISO 80000-1:2009 ''Quantities and Units. Part 1 - General.''. ISO. Geneva
*{{SIBrochure|pages=131–35}}
*{{GreenBookRef|page=3}}

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[[Category:Physical quantities]]