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{{short description|Comparison of various scales}}
{{For|the conversion of units on wikipedia|Template:Convert}}
{{Refimprove|date=January 2011}}
'''Conversion of units''' is the conversion of the ] in which a ] is expressed, typically through a multiplicative '''conversion factor''' that changes the unit without changing the quantity. This is also often loosely taken to include replacement of a quantity with a corresponding quantity that describes the same physical property.


Unit conversion is often easier within a ] such as the ] than in others, due to the system's ] and its ]es that act as power-of-10 multipliers.
'''Conversion of units''' is the conversion between different ] for the same ], typically through multiplicative '''conversion factors'''.


==Techniques== == Overview ==
The definition and choice of units in which to express a quantity may depend on the specific situation and the intended purpose. This may be governed by regulation, ], ] or other published ]s. Engineering judgment may include such factors as:
* the ] of measurement and the associated ]
* the statistical ] or ] of the initial measurement
* the number of ] of the measurement
* the intended use of the measurement, including the ]s
* historical definitions of the units and their derivatives used in old measurements; e.g., ] vs. US ].


For some purposes, conversions from one system of units to another are needed to be exact, without increasing or decreasing the precision of the expressed quantity. An ''adaptive conversion'' may not produce an exactly equivalent expression. ] are sometimes allowed and used.
{{see also|Dimensional analysis}}


=== Process === == Factor–label method ==
{{further|Dimensional analysis}}


The '''factor–label method''', also known as the '''unit–factor method''' or the '''unity bracket method''',<ref name="BodóJones2013">{{cite book |author1=Béla Bodó |url=https://books.google.com/books?id=P46291mjqAsC&q=conversi%C3%B3n+walshaw+methode&pg=SA9-PA129 |title=Introduction to Soil Mechanics |author2=Colin Jones |date=26 June 2013 |publisher=John Wiley & Sons |isbn=978-1-118-55388-6 |pages=9–}}</ref> is a widely used technique for unit conversions that uses the rules of ].<ref>{{Cite book |last=Goldberg |first=David |title=Fundamentals of Chemistry |publisher=McGraw-Hill |year=2006 |isbn=978-0-07-322104-5 |edition=5th}}</ref><ref>{{Cite book |last=Ogden |first=James |title=The Handbook of Chemical Engineering |publisher=Research & Education Association |year=1999 |isbn=978-0-87891-982-6}}</ref><ref>{{Cite web |title=Dimensional Analysis or the Factor Label Method |url=http://www.kentchemistry.com/links/Measurements/dimensionalanalysis.htm |website=Mr Kent's Chemistry Page}}</ref>
The process of conversion depends on the specific situation and the intended purpose. This may be governed by regulation, ], ] or other published ]s. Engineering judgment may include such factors as:
* The ] of measurement and the associated ].
* The statistical ] or ] of the initial measurement.
* The number of ] of the measurement.
* The intended use of the measurement including the ]s.
* Historical definitions of the units and their derivatives used in old measurements; e.g., ] vs. US ].


The factor–label method is the sequential application of conversion factors expressed as fractions and arranged so that any dimensional unit appearing in both the numerator and denominator of any of the fractions can be cancelled out until only the desired set of dimensional units is obtained. For example, 10 ] can be converted to ] by using a sequence of conversion factors as shown below:
Some conversions from one system of units to another need to be exact, without increasing or decreasing the precision of the first measurement. This is sometimes called ''soft conversion''. It does not involve changing the physical configuration of the item being measured.
<math display="block"> \frac{\mathrm{10~\cancel{mi}}}{\mathrm{1~\cancel{h}}} \times \frac{\mathrm{1609.344~m}}{\mathrm{1~\cancel{mi}}} \times \frac{\mathrm{1~\cancel{h}}}{\mathrm{3600~s}} = \mathrm{4.4704~\frac{m}{s}}. </math>


Each conversion factor is chosen based on the relationship between one of the original units and one of the desired units (or some intermediary unit), before being rearranged to create a factor that cancels out the original unit. For example, as "mile" is the numerator in the original fraction and {{tmath|1= \mathrm{1~mi} = \mathrm{1609.344~m} }}, "mile" will need to be the denominator in the conversion factor. Dividing both sides of the equation by 1 mile yields {{tmath|1= \frac{\mathrm{1~mi} }{\mathrm{1~mi} } = \frac{\mathrm{1609.344~m} }{\mathrm{1~mi} } }}, which when simplified results in the dimensionless {{tmath|1= 1 = \frac{\mathrm{1609.344~m} }{\mathrm{1~mi} } }}. Because of the identity property of multiplication, multiplying any quantity (physical or not) by the dimensionless 1 does not change that quantity.<ref>{{cite web| title = Identity property of multiplication |url = http://www.basic-mathematics.com/identity-property-of-multiplication.html |access-date = 2015-09-09 }}</ref> Once this and the conversion factor for seconds per hour have been multiplied by the original fraction to cancel out the units ''mile'' and ''hour'', 10 miles per hour converts to 4.4704 metres per second.
By contrast, a ''hard conversion'' or an ''adaptive conversion'' may not be exactly equivalent. It changes the measurement to convenient and workable numbers and units in the new system. It sometimes involves a slightly different configuration, or size substitution, of the item. ] are sometimes allowed and used.


As a more complex example, the ] of ] (]) in the ] from an industrial ] can be converted to a ] expressed in grams per hour (g/h) of NO<sub>''x''</sub> by using the following information as shown below:
=== Multiplication factors ===
; NO<sub>''x''</sub> concentration := 10 ] by volume = 10&nbsp;ppmv = 10 volumes/10<sup>6</sup> volumes
Conversion between units in the ] can be discerned by their ] (for example, 1 kilogram = 1000&nbsp;grams, 1 milligram = 0.001&nbsp;grams) and are thus not listed in this article. Exceptions are made if the unit is commonly known by another name (for example, 1 micron = 10<sup>−6</sup> metre).
; NO<sub>''x''</sub> molar mass := 46&nbsp;kg/kmol = 46&nbsp;g/mol
; Flow rate of flue gas := 20 cubic metres per minute = 20&nbsp;m<sup>3</sup>/min
: The flue gas exits the furnace at 0&nbsp;°C temperature and 101.325&nbsp;kPa absolute pressure.
: The ] of a gas at 0&nbsp;°C temperature and 101.325&nbsp;kPa is 22.414&nbsp;m<sup>3</sup>/].


: <math chem="">
=== Table ordering ===
\frac{1000\ \ce{g\ NO}_x}{1 \cancel{\ce{kg\ NO}_x}} \times
Within each table, the units are listed alphabetically, and the ] units (base or derived) are highlighted.
\frac{46\ \cancel{\ce{kg\ NO}_x}}{1\ \cancel{\ce{kmol\ NO}_x}} \times
\frac{1\ \cancel{\ce{kmol\ NO}_x}}{22.414\ \cancel{\ce{m}^3\ \ce{NO}_x}} \times
\frac{10\ \cancel{\ce{m}^3\ \ce{NO}_x}}{10^6\ \cancel{\ce{m}^3\ \ce{gas}}} \times
\frac{20\ \cancel{\ce{m}^3\ \ce{gas}}}{1\ \cancel{\ce{minute}}} \times
\frac{60\ \cancel{\ce{minute}}}{1\ \ce{hour}} =
24.63\ \frac{\ce{g\ NO}_x}{\ce{hour}}
</math>


After cancelling any dimensional units that appear both in the numerators and the denominators of the fractions in the above equation, the NO<sub>''x''</sub> concentration of 10&nbsp;ppm<sub>v</sub> converts to mass flow rate of 24.63&nbsp;grams per hour.
== Tables of conversion factors ==
This article gives lists of conversion factors for each of a number of physical quantities, which are listed in the index. For each physical quantity, a number of different units (some only of historical interest) are shown and expressed in terms of the corresponding SI unit.


=== Checking equations that involve dimensions ===
{| class="wikitable"
The factor–label method can also be used on any mathematical equation to check whether or not the dimensional units on the left hand side of the equation are the same as the dimensional units on the right hand side of the equation. Having the same units on both sides of an equation does not ensure that the equation is correct, but having different units on the two sides (when expressed in terms of base units) of an equation implies that the equation is wrong.
|+ Legend
! Symbol
! Definition
|-
! ≡
| exactly equal to
|-
! ≈
| approximately equal to
|-
! <var>{{overline|digits}}</var>
| indicates that <var>digits</var> repeat infinitely (e.g. {{gaps|8.294|{{overline|369}}}} corresponds to {{gaps|8.294|369|369|369|369|…}})
|-
! (H)
| of chiefly historical interest
|}


For example, check the ] equation of {{nowrap|1=''PV'' = ''nRT''}}, when:
===Length===
* the pressure ''P'' is in pascals (Pa)
{| class="wikitable"
* the volume ''V'' is in cubic metres (m<sup>3</sup>)
|+ ]
* the amount of substance ''n'' is in moles (mol)
!Name of unit
* the ] ''R'' is 8.3145&nbsp;Pa⋅m<sup>3</sup>/(mol⋅K)
!Symbol
* the temperature ''T'' is in kelvins (K)
!Definition
!Relation to SI units
|-
| ] || Å
| ≡ {{val|1|e=-10|u=m}}
| ≡ 0.1&nbsp;nm
|-
| ] || AU
| ≡ {{val|149597870700}}&nbsp;m ≈ Distance from Earth to Sun
| ≡ {{val|149597870700}}&nbsp;m <ref>{{cite web|author=jobs |url=http://www.nature.com/news/the-astronomical-unit-gets-fixed-1.11416 |title=The astronomical unit gets fixed : Nature News & Comment |doi=10.1038/nature.2012.11416 |publisher=Nature.com |date=2012-09-14 |accessdate=2013-08-31}}</ref>
|-
| ] (H) || &nbsp;
| = {{frac|3}} ] (see note above about rounding)
| ≈ 8.4{{overline|6}}{{e|-3}} m
|-
| bohr, ] || a<sub>0</sub>
| = ] of hydrogen
| ≈ {{val|5.2917721092|e=-11|u=m}} &nbsp;±&nbsp;{{val|1.7|e=-20|u=m}}<ref>"(2010). ]. Retrieved October 17, 2014.</ref>
|-
| cable length (imperial) || &nbsp;
| ≡ 608 ]
| ≈ 185.3184 m
|-
| ] (International) || &nbsp;
| ≡ {{frac|10}} ]
| ≡ 185.2 m
|-
| cable length (US) || &nbsp;
| ≡ 720 ]
| = 219.456 m
|-
| ] (]; Surveyor's) || ch
| ≡ 66 ](US) ≡ 4 ] <ref name=nist></ref>
| ≈ {{val|20.11684|u=m}}
|-
| ] (H) || &nbsp;
| ≡ Distance from fingers to elbow ≈ 18 in
| ≈ 0.5 m
|-
| ] (H) || ell
| ≡ 45 in <ref name=CRC71>Lide, D. (Ed.). (1990). ''Handbook of Chemistry and Physics'' (71st ed). Boca Raton, FL: CRC Press. Section 1.</ref> (In England usually)
| = 1.143 m
|-
| ] || fm
| ≡ 6&nbsp;ft <ref name=CRC71/>
| = 1.8288 m
|-
| ] || fm
| ≡ {{val|1|e=-15|u=m}}<ref name=CRC71/>
| ≡ {{val|1|e=-15|u=m}}
|-
| finger || &nbsp;
| ≡ {{frac|7|8}} in
| = {{val|0.022225|u=m}}
|-
| finger (cloth) || &nbsp;
| ≡ {{frac|4|1|2}} in
| = 0.1143 m
|-
| ] (Benoît) (H) || ft (Ben)
|
| ≈ {{val|0.304799735|u=m}}
|-
| foot (Cape) (H) || &nbsp;
|Legally defined as 1.033 English feet in 1859
| ≈ {{val|0.314858|u=m}}
|-
| foot (Clarke's) (H) || ft (Cla)
|
| ≈ {{val|0.3047972654|u=m}}
|-
| foot (Indian) (H) || ft Ind
|
| ≈ {{val|0.304799514|u=m}}
|-
| foot (International) || ft
| ≡ 0.3048 m ≡ {{frac|3}} yd ≡ 12&nbsp;inches
| ≡ 0.3048 m
|-
| foot (Sear's) (H) || ft (Sear)
|
| ≈ {{val|0.30479947|u=m}}
|-
| foot (US Survey) || ft (US)
| ≡ {{frac|1200|3937}} m <ref name="nbs">National Bureau of Standards. (June 30, 1959). ''Refinement of values for the yard and the pound''. Federal Register, viewed September 20, 2006 at .</ref>
| ≈ {{val|0.304800610|u=m}}
|-
| ]; charriere || F
| ≡ {{frac|3}} mm
| = 0.{{overline|3}} {{e|-3}} m
|-
| ] || fur
| ≡ 10 chains = 660&nbsp;ft = 220 yd <ref name=CRC71/>
| = 201.168 m
|-
| ] || &nbsp;
| ≡ 4 in <ref name=CRC71/>
| ≡ 0.1016 m
|-
| ] (International) || in
| ≡ 2.54&nbsp;cm ≡ {{frac|36}} yd ≡ {{frac|12}}&nbsp;ft
| ≡ .0254 m
|-
| ] (land) || lea
| ≈ 1 hour walk, Currently defined in US as 3 Statute miles,<ref name=nist/> but historically varied from 2 to 9&nbsp;km
| ≈ {{val|4828|u=m}}
|-
| ] || &nbsp;
| ≡ 24 light-hours
| ≡ {{val|2.59020683712|e=13|u=m}}
|-
| ] || &nbsp;
| ≡ 60 light-minutes
| ≡ {{val|1.0792528488|e=12|u=m}}
|-
| ] || &nbsp;
| ≡ 60 light-seconds
| ≡ {{val|1.798754748|e=10|u=m}}
|-
| ] || &nbsp;
| ≡ Distance light travels in one second in vacuum
| ≡ {{val|299792458|u=m}}
|-
| ] || ly
| ≡ Distance light travels in vacuum in 365.25 days <ref></ref>
| = {{val|9.4607304725808|e=15|u=m}}
|-
| ] || ln
| ≡ {{frac|12}} in <ref>
Klein, Herbert Arthur.
(1988). ''The Science of Measurement: a Historical Survey''. Mineola, NY: Dover Publications ].</ref>
| = {{gaps|0.002|11{{overline|6}}}} m
|-
| ] (Gunter's; Surveyor's) || lnk
| ≡ {{frac|100}} ch <ref name=CRC71/> ≡ 0.66&nbsp;ft ≡ 7.92&nbsp;in
| = {{val|0.201168|u=m}}
|-
| link (Ramsden's; Engineer's) || lnk
| ≡ 1&nbsp;ft <ref name=CRC71/>
| = 0.3048 m
|- style="background:#dfd;"
| ] (])<br /><small>(meter)</small> || m
| ≡ Distance light travels in {{frac|{{val|299792458}}}} of a second in vacuum.<ref name=sibaseunits>{{citation | title=The International System of Units, Section 2.1 | publisher=] | url=http://www.bipm.org/en/si/si_brochure/chapter2/2-1/ | edition=8 | year=2006 | accessdate=August 26, 2009}}</ref><br/> ≈ {{frac|{{val|10000000}}}} of the distance from equator to pole.
| ≡ 1 m
|-
| mickey || &nbsp;
| ≡ {{frac|200}} in
| = {{val|1.27|e=-4|u=m}}
|-
| ]|| µ
|
| ≡ {{val|1|e=-6|u=m}}
|-
| mil; ] || mil
| ≡ {{val|1|e=-3|u=in}}
| ≡ {{val|2.54|e=-5|u=m}}
|-
| ] (Sweden and Norway) || mil
| ≡ 10&nbsp;km
| = {{val|10000|u=m}}
|-
| ] (H) || <!--Please provide a reference for any symbols-->
| ≡ {{val|6082|u=ft}}
| = {{val|1853.7936|u=m}}
|-
| ] (international) || mi
| ≡ 80 chains ≡ {{val|5280|u=ft}} ≡ {{val|1760|u=yd}}
| ≡ {{val|1609.344|u=m}}
|-
| ] (tactical or data) ||
| ≡ {{val|6000|u=ft}}
| ≡ {{val|1828.8|u=m}}
|-
| mile (telegraph) (H) || mi
| ≡ {{val|6087|u=ft}}
| = {{val|1855.3176|u=m}}
|-
| mile (US Survey) || mi
| ≡ {{val|5280}} US Survey feet ≡ ({{val|5280}} × {{frac|{{val|1200}}|{{val|3937}}}}) m
| ≈ {{val|1609.347219|u=m}}
|-
| nail (cloth) || &nbsp;
| ≡ {{frac|2|1|4}} in <ref name=CRC71/>
| = {{val|0.05715|u=m}}
|-
| nanometre || nm
| ≡ {{val|1|e=-9|u=m}}
| ≡ {{val|1|e=-9|u=m}}
|-
| nautical league || NL; nl
| ≡ 3 nmi <ref name=CRC71/>
| = {{val|5556|u=m}}
|-
| nautical mile (Admiralty) || NM (Adm); nmi (Adm)
| = {{val|6080|u=ft}}
| = {{val|1853.184|u=m}}
|-
| ] (international) || NM; nmi
| ≡ {{val|1852|u=m}}<ref name=Table8> 8th ed. (2006), ], Section 4.1 Table 8.</ref>
| ≡ {{val|1852|u=m}}
|-
| nautical mile (US pre 1954) ||
| ≡ 1853.248 m
| ≡ 1853.248 m
|-
| pace || &nbsp;
| ≡ 2.5&nbsp;ft <ref name=CRC71/>
| = 0.762 m
|-
| ] || &nbsp;
| ≡ 3 in <ref name=CRC71/>
| = 0.0762 m
|-
| ] || pc
| Distance of star with '''''par'''''allax shift of one arc '''''sec'''''ond from a base of one astronomical unit
| ≈ {{val|3.08567782|e=16}} ± {{val|6|e=6}} m <ref name=Seidelmann>P. Kenneth Seidelmann, Ed. (1992). ''Explanatory Supplement to the Astronomical Almanac.'' Sausalito, CA: University Science Books. p. 716 and s.v. parsec in Glossary.</ref>
|-
| ] || &nbsp;
| ≡ 12 points
| Dependent on point measures.
|-
| ] (American, English) <ref name=whitelaw>Whitelaw, Ian. (2007). . New York: Macmillan ]. p. 152.
</ref><ref name=DeVinne>De Vinne, Theodore Low (1900). 2nd ed. New York: The Century Co. p. 142&ndash;150.</ref> || pt
| ≡ {{frac|72.272}} ]
| ≈ {{val|0.000351450|u=m}}
|-
| point (Didot; European) <ref name=DeVinne/><ref>Pasko, Wesley Washington (1894). . (1894). New York: Howard Lockwood. p. 521.</ref> || pt
| ≡ {{frac|12}} × {{frac|72}} of ];<br /><br />After 1878:<br />≡ 5/133&nbsp;cm
| ≈ {{val|0.00037597|u=m}};<br /><br />After 1878:<br />≈ {{nowrap|0.000 375 939 85}} m
|-
| point (]) <ref name=whitelaw/>|| pt
| ≡ {{frac|72}} ]
| = {{gaps|0.000|352{{overline|7}}}} m
|-
| point (]) <ref name=whitelaw/>|| pt
| ≡ {{frac|72.27}} ]
| = 0.00{{overline|{{gaps|0|351|4598}}}} m
|-
| quarter || &nbsp;
| ≡ {{frac|4}} yd
| = 0.2286 m
|-
| ]; pole; perch (H) || rd
| ≡ {{frac|16|1|2}} ft
| = 5.0292 m
|-
| ] (H) || rope
| ≡ 20&nbsp;ft <ref name=CRC71/>
| = 6.096 m
|-
| ] (Japan) || &nbsp;
| &nbsp;
| = 0.3030 m
|-
| span (H) || &nbsp;
| ≡ 9 in <ref name=CRC71/>
| = 0.2286 m
|-
| ] <ref name=howmany>{{citation | last=Rowlett | first=Russ | url=http://www.unc.edu/~rowlett/units/index.html | title=How Many? A Dictionary of Units of Measurement | year=2005}}</ref> ||
|
| ≡ {{val|1|e=12|u=m}}
|-
| stick (H) || &nbsp;
| ≡ 2 in
| = 0.0508 m
|-
| stigma; bicron (]) || pm
|
| ≡ {{val|1|e=-12|u=m}}
|-
| ] || twp
| ≡ {{frac|1440}} in
| = 1.763{{overline|8}}{{e|−5}} m
|-
| ]; siegbahn || xu
|
| ≈ {{val|1.0021|e=-13}} m <ref name=CRC71/>
|-
| ] (International) || yd
| ≡ 0.9144 m <ref name="nbs"/> ≡ 3&nbsp;ft ≡ 36 in
| ≡ 0.9144 m
|}


<math display="block">\mathrm{Pa{\cdot}m^3} = \frac{\cancel{\mathrm{mol}}}{1} \times
===Area===
\frac{\mathrm{Pa{\cdot}m^3}}{\cancel{\mathrm{mol}}\ \cancel{\mathrm{K}}} \times \frac{\cancel{\mathrm{K}}}{1}
{| class="wikitable"
</math>
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] (international) || ac
| ≡ 1&nbsp;ch × 10&nbsp;ch = {{nowrap|4840}} sq yd
| ≡ {{nowrap|4 046.856 4224}} m<sup>2</sup>
|-
| ] (U. S. survey) || ac
| ≡ 10 sq ch = {{nowrap|4840}} sq yd, also 43560 sq ft.
| ≈ {{nowrap|4 046.873}} m<sup>2</sup> <ref>Thompson, A. and Taylor, B.N. (2008). ''Guide for the Use of the International System of Units (SI)''. ] Special Publication 811. p. 57.</ref>
|-
| ] || a
| ≡ 100 m<sup>2</sup>
| = 100 m<sup>2</sup>
|-
| ] || b
| ≡ 10<sup>−28</sup> m<sup>2</sup>
| = 10<sup>−28</sup> m<sup>2</sup>
|-
| barony || &nbsp;
| ≡ {{nowrap|4000}} ac
| ≈ {{nowrap|1.618 742{{e|7}}}} m<sup>2</sup>
|-
| board || bd
| ≡ 1 in × 1&nbsp;ft
| = {{nowrap|7.741 92{{e|−3}}}} m<sup>2</sup>
|-
| boiler horsepower equivalent direct radiation || bhp EDR
| ≡ (1&nbsp;ft<sup>2</sup>) (1&nbsp;bhp) / (240 BTU<sub>IT</sub>/h)
| ≈ {{nowrap|12.958 174}} m<sup>2</sup>
|-
| circular ] || circ in
| ≡ π/4 sq in
| ≈ {{nowrap|5.067 075{{e|−4}}}} m<sup>2</sup>
|-
| circular mil; circular thou || circ mil
| ≡ π/4 mil<sup>2</sup>
| ≈ {{nowrap|5.067 075{{e|−10}}}} m<sup>2</sup>
|-
| cord || &nbsp;
| ≡ 192 bd
| = {{nowrap|1.486 448 64}} m<sup>2</sup>
|-
| ] (PR Survey) || cda
| ≡ 1 cda x 1 cda = {{nowrap|0.971222}} acre
| ≡ {{nowrap|3930.395625}} m<sup>2</sup>
|-
| ] || &nbsp;
| ≡ {{nowrap|1 000}} m<sup>2</sup>
| = {{nowrap|1 000}} m<sup>2</sup>
|-
| ] || &nbsp;
| ≡ 121 sq yd
| ≈ 101.17 m<sup>2</sup>
|-
| ] || ha
| ≡ {{nowrap|10 000}} m<sup>2</sup>
| ≡ {{nowrap|10 000}} m<sup>2</sup>
|-
| ] || &nbsp;
| ≈ 120 ac (variable)<!-- Definition is "amount of land required to support one peasant family" -->
| ≈ {{nowrap|5{{e|5}}}} m<sup>2</sup>
|-
| rood || ro
| ≡ ¼ ac
| = {{nowrap|1 011.714 1056}} m<sup>2</sup>
|-
| section ||
| ≡ 1&nbsp;mi × 1&nbsp;mi
| = {{nowrap|2.589 988 110 336{{e|6}}}} m<sup>2</sup>
|-
| ] || &nbsp;
| ≡ 10<sup>−52</sup> m<sup>2</sup>
| = 10<sup>−52</sup> m<sup>2</sup>
|-
| square (roofing) ||
| ≡ 10&nbsp;ft × 10&nbsp;ft
| = {{nowrap|9.290 304}} m<sup>2</sup>
|-
| square chain (international) || sq ch
| ≡ 66&nbsp;ft × 66&nbsp;ft = 1/10 ac
| ≡ {{nowrap|404.685 642 24}} m<sup>2</sup>
|-
| square chain (US Survey) || sq ch
| ≡ 66&nbsp;ft(US) × 66&nbsp;ft(US) = 1/10 ac
| ≈ {{nowrap|404.687 3}} m<sup>2</sup>
|-
| ] || sq ft
| ≡ 1&nbsp;ft × 1&nbsp;ft
| ≡ {{nowrap|9.290 304{{e|−2}}}} m<sup>2</sup>
|-
| square ] (US Survey) || sq ft
| ≡ 1&nbsp;ft (US) × 1&nbsp;ft (US)
| ≈ {{nowrap|9.290 341 161 327 49{{e|-2}}}} m<sup>2</sup>
|-
| ] || sq in
| ≡ 1 in × 1 in
| ≡ 6.4516{{e|−4}} m<sup>2</sup>
|-
| ] || km<sup>2</sup>
| ≡ 1&nbsp;km × 1&nbsp;km
| = 10<sup>6</sup> m<sup>2</sup>
|-
| square link (Gunter's)(International) || sq lnk
| ≡ 1 lnk × 1 lnk ≡ 0.66&nbsp;ft × 0.66&nbsp;ft
| = {{nowrap|4.046 856 4224{{e|−2}}}} m<sup>2</sup>
|-
| square link (Gunter's)(US Survey) || sq lnk
| ≡ 1 lnk × 1 lnk ≡ 0.66&nbsp;ft(US) × 0.66&nbsp;ft(US)
| ≈ {{nowrap|4.046 872{{e|−2}}}} m<sup>2</sup>
|-
| square link (Ramsden's) || sq lnk
| ≡ 1 lnk × 1 lnk ≡ 1&nbsp;ft × 1&nbsp;ft
| = {{nowrap|0.09290304}} m<sup>2</sup>
|- style="background:#dfd;"
| ] (SI unit) || m<sup>2</sup>
| ≡ 1 ] × 1 m
| = 1 m<sup>2</sup>
|-
| square mil; square thou || sq mil
| ≡ 1 mil × 1 mil
| = 6.4516{{e|−10}} m<sup>2</sup>
|-
| square ] || sq mi
| ≡ 1&nbsp;mi × 1&nbsp;mi
| = {{nowrap|2.589 988 110 336{{e|6}}}} m<sup>2</sup>
|-
| square ] (US Survey) || sq mi
| ≡ 1&nbsp;mi (US) × 1&nbsp;mi (US)
| ≈ {{nowrap|2.589 998 47{{e|6}}}} m<sup>2</sup>
|-
| square rod/pole/perch || sq rd
| ≡ 1 rd × 1 rd
| = {{nowrap|25.292 852 64}} m<sup>2</sup>
|-
| ] (International) || sq yd
| ≡ 1 yd × 1 yd
| ≡ {{nowrap|0.836 127 36}} m<sup>2</sup>
|-
| ] || &nbsp;
| ≡ {{nowrap|1 000}} m<sup>2</sup>
| = {{nowrap|1 000}} m<sup>2</sup>
|-
| ] || &nbsp;
| ≡ 36 sq mi (US)
| ≈ {{nowrap|9.323 994{{e|7}}}} m<sup>2</sup>
|-
| ] || &nbsp;
| ≈ 30 ac
| ≈ {{nowrap|1.2{{e|5}}}} m<sup>2</sup>
|}


As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's right hand side are cancelled out, both sides of the equation have the same dimensional units. Dimensional analysis can be used as a tool to construct equations that relate non-associated physico-chemical properties. The equations may reveal undiscovered or overlooked properties of matter, in the form of left-over dimensions – dimensional adjusters – that can then be assigned physical significance. It is important to point out that such 'mathematical manipulation' is neither without prior precedent, nor without considerable scientific significance. Indeed, the ], a fundamental physical constant, was 'discovered' as a purely mathematical abstraction or representation that built on the ] for preventing the ]. It was assigned and ascended to its quantum physical significance either in tandem or post mathematical dimensional adjustment – not earlier.
===Volume===
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] || ac ft
| ≡ 1 ac x 1&nbsp;ft = {{nowrap|43 560}} ft<sup>3</sup>
| = {{nowrap|1 233.481 837 547 52}} m<sup>3</sup>
|-
| acre-inch || &nbsp;
| ≡ 1 ac × 1 in
| = {{nowrap|102.790 153 128 96}} m<sup>3</sup>
|-
| ] (imperial) || bl (imp)
| ≡ 36 gal (imp)
| = {{nowrap|0.163 659 24}} m<sup>3</sup>
|-
| barrel (petroleum) || bl; bbl
| ≡ 42 gal (US)
| = {{nowrap|0.158 987 294 928}} m<sup>3</sup>
|-
| barrel (US dry) || bl (US)
| ≡ 105 qt (US) = 105/32 bu (US lvl)
| = {{nowrap|0.115 628 198 985 075}} m<sup>3</sup>
|-
| barrel (US fluid) || fl bl (US)
| ≡ 31½ gal (US)
| = {{nowrap|0.119 240 471 196}} m<sup>3</sup>
|-
| ] || fbm
| ≡ 144 cu in
| ≡ {{nowrap|2.359 737 216{{e|−3}}}} m<sup>3</sup>
|-
| bucket (imperial) || bkt
| ≡ 4 gal (imp)
| = {{nowrap|0.018 184 36}} m<sup>3</sup>
|-
| ] (imperial) || bu (imp)
| ≡ 8 gal (imp)
| = {{nowrap|0.036 368 72}} m<sup>3</sup>
|-
| bushel (US dry heaped) || bu (US)
| ≡ 1 ¼ bu (US lvl)
| = {{nowrap|0.044 048 837 7086}} m<sup>3</sup>
|-
| bushel (US dry level) || bu (US lvl)
| ≡ 2 150.42 cu in
| = {{nowrap|0.035 239 070 166 88}} m<sup>3</sup>
|-
| ], pipe || &nbsp;
| ≡ 126 gal (wine)
| = {{nowrap|0.476 961 884 784}} m<sup>3</sup>
|-
| ] || &nbsp;
| ≡ 4 bu (imp)
| = {{nowrap|0.145 474 88}} m<sup>3</sup>
|-
| cord (]) || &nbsp;
| ≡ 8&nbsp;ft × 4&nbsp;ft × 4&nbsp;ft
| = {{nowrap|3.624 556 363 776}} m<sup>3</sup>
|-
| cord-foot || &nbsp;
| ≡ 16 cu ft
| = {{nowrap|0.453 069 545 472}} m<sup>3</sup>
|-
| cubic ] || cu fm
| ≡ 1 fm × 1 fm × 1 fm
| = {{nowrap|6.116 438 863 872}} m<sup>3</sup>
|-
| ] || cu ft
| ≡ 1&nbsp;ft × 1&nbsp;ft × 1&nbsp;ft
| ≡ {{nowrap|0.028 316 846 592}} m<sup>3</sup>
|-
| cubic ] || cu in
| ≡ 1 in × 1 in × 1 in
| ≡ {{nowrap|16.387 064{{e|−6}}}} m<sup>3</sup>
|- style="background:#dfd;"
| ] (SI unit) || m<sup>3</sup>
| ≡ 1 m × 1 m × 1 m
| ≡ 1 m<sup>3</sup>
|-
| cubic ] || cu mi
| ≡ 1&nbsp;mi × 1&nbsp;mi × 1&nbsp;mi
| ≡ {{nowrap|4 168 181 825.440 579 584}} m<sup>3</sup>
|-
| cubic ] || cu yd
| ≡ 27 cu ft
| ≡ {{nowrap|0.764 554 857 984}} m<sup>3</sup>
|-
| ] (breakfast) || &nbsp;
| ≡ 10 fl oz (imp)
| = {{nowrap|284.130 625{{e|−6}}}} m<sup>3</sup>
|-
| cup (Canadian) || c (CA)
| ≡ 8 fl oz (imp)
| = 227.3045{{e|−6}} m<sup>3</sup>
|-
| cup (metric) || c
| ≡ 250.0{{e|−6}} m<sup>3</sup>
| = 250.0{{e|−6}} m<sup>3</sup>
|-
| cup (US customary) || c (US)
| ≡ 8 US fl oz ≡ 1/16 gal (US)
| = {{nowrap|236.588 2365{{e|−6}}}} m<sup>3</sup>
|-
| cup (US food nutrition labeling) || c (US)
| ≡ 240&nbsp;mL<ref name=cfr21>{{citation | url=http://ecfr.gpoaccess.gov/cgi/t/text/text-idx?c=ecfr&rgn=div8&view=text&node=21:2.0.1.1.2.1.1.6&idno=21 | title=US Code of Federal Regulations, Title 21, Section 101.9, Paragraph (b)(5)(viii) | accessdate=August 29, 2009}}</ref>
| = {{val|fmt=commas|2.4|e=-4|u=m<sup>3</sup>}}
|-
| dash (imperial) || &nbsp;
| ≡ 1/384 gi (imp) = ½ pinch (imp)
| = {{nowrap|369.961 751 302 08 {{overline|3}}{{e|−9}}}} m<sup>3</sup>
|-
| dash (US) || &nbsp;
| ≡ 1/96 US fl oz = ½ US pinch
| = {{nowrap|308.057 599 609 375{{e|−9}}}} m<sup>3</sup>
|-
| ] (imperial) || &nbsp;
| ≡ 1/12 gi (imp)
| = {{nowrap|11.838 776 041{{overline|6}}{{e|−6}}}} m<sup>3</sup>
|-
| ] (imperial) || gtt
| ≡ 1/288 fl oz (imp)
| = {{nowrap|98.656 467 013 {{overline|8}}{{e|−9}}}} m<sup>3</sup>
|-
| drop (imperial) (alt) || gtt
| ≡ {{nowrap|1/1 824}} gi (imp)
| ≈ {{nowrap|77.886 684{{e|−9}}}} m<sup>3</sup>
|-
| drop (medical) || &nbsp;
| ≡ 0.9964/12 ml
| = 83.0{{overline|3}}{{e|−9}} m<sup>3</sup>
|-
| drop (medical) || &nbsp;
| ≡ 1/12 ml
| = 83.{{overline|3}}{{e|−9}} m<sup>3</sup>
|-
| drop (metric) || &nbsp;
| ≡ 1/20 mL
| = 50.0{{e|−9}} m<sup>3</sup>
|-
| drop (US) || gtt
| ≡ 1/360 US fl oz
| = {{nowrap|82.148 693 2291{{overline|6}}{{e|−9}}}} m<sup>3</sup>
|-
| drop (US) (alt) || gtt
| ≡ 1/456 US fl oz
| ≈ {{nowrap|64.854 231 496 71{{e|−9}}}} m<sup>3</sup>
|-
| drop (US) (alt) || gtt
| ≡ 1/576 US fl oz
| ≈ {{nowrap|51.342 933 268 23{{e|−9}}}} m<sup>3</sup>
|-
| fifth || &nbsp;
| ≡ 1/5 US gal
| = {{nowrap|757.082 3568{{e|−6}}}} m<sup>3</sup>
|-
| ] || &nbsp;
| ≡ 9 gal (imp)
| = {{nowrap|0.040 914 81}} m<sup>3</sup>
|-
| ] (imperial) || fl dr
| ≡ ⅛ fl oz (imp)
| = {{nowrap|3.551 632 8125{{e|−6}}}} m<sup>3</sup>
|-
| ] (US); US fluidram || fl dr
| ≡ ⅛ US fl oz
| = {{nowrap|3.696 691 195 3125{{e|−6}}}} m<sup>3</sup>
|-
| ] (imperial) || fl s
| ≡ 1/24 fl oz (imp)
| = {{nowrap|1.183 877 6041{{overline|6}}{{e|−6}}}} m<sup>3</sup>
|-
| ] (beer) || beer gal
| ≡ 282 cu in
| = {{nowrap|4.621 152 048{{e|−3}}}} m<sup>3</sup>
|-
| gallon (imperial) || gal (imp)
| ≡ {{nowrap|4.546  09}} L
| ≡ {{nowrap|4.546 09{{e|−3}}}} m<sup>3</sup>
|-
| gallon (US dry) || gal (US)
| ≡ ⅛ bu (US lvl)
| = {{nowrap|4.404 883 770 86{{e|−3}}}} m<sup>3</sup>
|-
| gallon (US fluid; Wine) || gal (US)
| ≡ 231 cu in
| ≡ {{nowrap|3.785 411 784{{e|−3}}}} m<sup>3</sup>
|-
| ] (imperial); Noggin || gi (imp); nog
| ≡ 5 fl oz (imp)
| = {{nowrap|142.065 3125{{e|−6}}}} m<sup>3</sup>
|-
| gill (US) || gi (US)
| ≡ 4 US fl oz
| = {{nowrap|118.294 118 25{{e|−6}}}} m<sup>3</sup>
|-
| ] (imperial) || hhd (imp)
| ≡ 2 bl (imp)
| = {{nowrap|0.327 318 48}} m<sup>3</sup>
|-
| hogshead (US) || hhd (US)
| ≡ 2 fl bl (US)
| = {{nowrap|0.238 480 942 392}} m<sup>3</sup>
|-
| ] || &nbsp;
| ≡ 1½ US fl oz
| ≈ 44.36{{e|−6}} m<sup>3</sup>
|-
| ] || &nbsp;
| ≡ 18 gal (imp)
| = {{nowrap|0.081 829 62}} m<sup>3</sup>
|-
| ] || λ
| ≡ 1&nbsp;mm<sup>3</sup>
| = 1{{e|−9}} m<sup>3</sup>
|-
| ] || &nbsp;
| ≡ 80 bu (imp)
| = {{nowrap|2.909 4976}} m<sup>3</sup>
|-
| ]<br /><small>(liter)</small> || L
| ≡ 1 dm<sup>3</sup> <ref name=specpub330>Barry N. Taylor, Ed., (2001 Edition), Washington: US Government Printing Office, 43,"The 12th Conference Generale des Poids et Mesures (CGPM)…declares that the word “litre” may be employed as a special name for the cubic decimetre".</ref>
| ≡ 0.001 m<sup>3</sup>
|-
| load || &nbsp;
| ≡ 50 cu ft
| = {{nowrap|1.415 842 3296}} m<sup>3</sup>
|-
| ] (imperial) || min
| ≡ 1/480 fl oz (imp) = 1/60 fl dr (imp)
| = {{nowrap|59.193 880 208 {{overline|3}}{{e|−9}}}} m<sup>3</sup>
|-
| minim (US) || min
| ≡ 1/480 US fl oz = 1/60 US fl dr
| = {{nowrap|61.611 519 921 875{{e|−9}}}} m<sup>3</sup>
|-
| ] (fluid imperial) || fl oz (imp)
| ≡ 1/160 gal (imp)
| ≡ {{nowrap|28.413 0625{{e|−6}}}} m<sup>3</sup>
|-
| ] (fluid US customary) || US fl oz
| ≡ 1/128 gal (US)
| ≡ {{nowrap|29.573 529 5625{{e|−6}}}} m<sup>3</sup>
|-
| ounce (fluid US food nutrition labeling) || US fl oz
| ≡ 30&nbsp;mL<ref name="cfr21"/>
| ≡ {{val|fmt=commas|3|e=-5|u=m<sup>3</sup>}}
|-
| ] (imperial) || pk
| ≡ 2 gal (imp)
| = {{nowrap|9.092 18{{e|−3}}}} m<sup>3</sup>
|-
| peck (US dry) || pk
| ≡ ¼ US lvl bu
| = {{nowrap|8.809 767 541 72{{e|−3}}}} m<sup>3</sup>
|-
| ] || per
| ≡ 16½ ft × 1½ ft × 1&nbsp;ft
| = {{nowrap|0.700 841 953 152}} m<sup>3</sup>
|-
| pinch (imperial) || &nbsp;
| ≡ 1/192 gi (imp) = ⅛ tsp (imp)
| = {{nowrap|739.923 502 6041{{overline|6}}{{e|−9}}}} m<sup>3</sup>
|-
| pinch (US) || &nbsp;
| ≡ 1/48 US fl oz = ⅛ US tsp
| = {{nowrap|616.115 199 218 75{{e|−9}}}} m<sup>3</sup>
|-
| ] (imperial) || pt (imp)
| ≡ ⅛ gal (imp)
| = {{nowrap|568.261 25{{e|−6}}}} m<sup>3</sup>
|-
| pint (US dry) || pt (US dry)
| ≡ 1/64 bu (US lvl) ≡ ⅛ gal (US dry)
| = {{nowrap|550.610 471 3575{{e|−6}}}} m<sup>3</sup>
|-
| pint (US fluid) || pt (US fl)
| ≡ ⅛ gal (US)
| = {{nowrap|473.176 473{{e|−6}}}} m<sup>3</sup>
|-
| pony || &nbsp;
| ≡ 3/4 US fl oz
| = {{nowrap|22.180 147 171 875{{e|−6}}}} m<sup>3</sup>
|-
| pottle; quartern || &nbsp;
| ≡ ½ gal (imp) = 80 fl oz (imp)
| = {{nowrap|2.273 045{{e|−3}}}} m<sup>3</sup>
|-
| ] (imperial) || qt (imp)
| ≡ ¼ gal (imp)
| = {{nowrap|1.136 5225{{e|−3}}}} m<sup>3</sup>
|-
| quart (US dry) || qt (US)
| ≡ 1/32 bu (US lvl) = ¼ gal (US dry)
| = {{nowrap|1.101 220 942 715{{e|−3}}}} m<sup>3</sup>
|-
| quart (US fluid) || qt (US)
| ≡ ¼ gal (US fl)
| = {{nowrap|946.352 946{{e|−6}}}} m<sup>3</sup>
|-
| quarter; pail || &nbsp;
| ≡ 8 bu (imp)
| = {{nowrap|0.290 949 76}} m<sup>3</sup>
|-
| register ton || &nbsp;
| ≡ 100 cu ft
| = {{nowrap|2.831 684 6592}} m<sup>3</sup>
|-
| sack (imperial); bag || &nbsp;
| ≡ 3 bu (imp)
| = {{nowrap|0.109 106 16}} m<sup>3</sup>{{Citation needed|date=April 2010}}
|-
| sack (US) || &nbsp;
| ≡ 3 bu (US lvl)
| = {{nowrap|0.105 717 210 500 64}} m<sup>3</sup>
|-
| seam || &nbsp;
| ≡ 8 bu (US lvl)
| = {{nowrap|0.281 912 561 335 04}} m<sup>3</sup>{{Citation needed|date=April 2010}}
|-
| shot (US) || &nbsp;
| usually 1.5 US fl oz<ref name=howmany/>
| ≈ 44{{e|−6}} m<sup>3</sup>
|-
| strike (imperial) || &nbsp;
| ≡ 2 bu (imp)
| = {{nowrap|0.072 737 44}} m<sup>3</sup>
|-
| strike (US) || &nbsp;
| ≡ 2 bu (US lvl)
| = {{nowrap|0.070 478 140 333 76}} m<sup>3</sup>
|-
| ] (Australian metric) || &nbsp;
|
| ≡ 20.0{{e|−6}} m<sup>3</sup>
|-
| tablespoon (Canadian) || tbsp
| ≡ ½ fl oz (imp)
| = {{nowrap|14.206 531 25{{e|−6}}}} m<sup>3</sup>
|-
| tablespoon (imperial) || tbsp
| ≡ 5/8 fl oz (imp)
| = {{nowrap|17.758 164 0625{{e|−6}}}} m<sup>3</sup>
|-
| tablespoon (metric) || &nbsp;
|
| ≡ 15.0{{e|−6}} m<sup>3</sup>
|-
| tablespoon (US customary) || tbsp
| ≡ ½ US fl oz
| = {{nowrap|14.786 764 7825{{e|−6}}}} m<sup>3</sup>
|-
| tablespoon (US food nutrition labeling) || tbsp
| ≡ 15&nbsp;mL<ref name="cfr21"/>
| = {{val|fmt=commas|1.5|e=-5|u=m<sup>3</sup>}}
|-
| ] (Canadian) || tsp
| ≡ 1/6 fl oz (imp)
| = {{nowrap|4.735 510 41{{overline|6}}{{e|−6}}}} m<sup>3</sup>
|-
| teaspoon (imperial) || tsp
| ≡ 1/24 gi (imp)
| = {{nowrap|5.919 388 0208{{overline|3}}{{e|−6}}}} m<sup>3</sup>
|-
| teaspoon (metric) || &nbsp;
| ≡ 5.0{{e|−6}} m<sup>3</sup>
| = 5.0{{e|−6}} m<sup>3</sup>
|-
| teaspoon (US customary) || tsp
| ≡ 1/6 US fl oz
| = {{nowrap|4.928 921 595{{e|−6}}}} m<sup>3</sup>
|-
| teaspoon (US food nutrition labeling) || tsp
| ≡ 5&nbsp;mL<ref name="cfr21"/>
| = {{val|fmt=commas|5|e=-6|u=m<sup>3</sup>}}
|-
| ] || &nbsp;
| ≡ 1 cu ft
| = {{nowrap|0.028 316 846 592}} m<sup>3</sup>
|-
| ] (displacement) || &nbsp;
| ≡ 35 cu ft
| = {{nowrap|0.991 089 630 72}} m<sup>3</sup>
|-
| ton (freight) || &nbsp;
| ≡ 40 cu ft
| = {{nowrap|1.132 673 863 68}} m<sup>3</sup>
|-
| ton (water) || &nbsp;
| ≡ 28 bu (imp)
| = {{nowrap|1.018 324 16}} m<sup>3</sup>
|-
| ] || &nbsp;
| ≡ 252 gal (wine)
| = {{nowrap|0.953 923 769 568}} m<sup>3</sup>
|-
| ] (US) || &nbsp;
| ≡ 40 bu (US lvl)
| = {{nowrap|1.409 562 806 6752}} m<sup>3</sup>
|}


===Plane angle=== === Limitations ===
The factor–label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0 (] in Stevens's typology). Most conversions fit this paradigm. An example for which it cannot be used is the conversion between the ] and the ] (or the ]). Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between degrees Celsius and degrees Fahrenheit there is neither a constant difference nor a constant ratio. There is, however, an ] ({{tmath|1= x \mapsto ax+b }}, rather than a ] {{tmath|1= x \mapsto ax }}) between them.
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] || µ
| ≡ {{nowrap|2π/6400}} rad
| ≈ {{nowrap|0.981 748{{e|-3}} }} rad
|-
| ]; MOA|| '
| ≡ 1°/60
| ≈ {{nowrap|0.290 888{{e|-3}} }} rad
|-
| ] || "
| ≡ {{nowrap|1°/3600}}
| ≈ {{nowrap|4.848 137{{e|-6}} }} rad
|-
| ] ] || '
| ≡ 1 grad/100
| ≈ {{nowrap|0.157 080{{e|-3}} }} rad
|-
| ] ] || "
| ≡ {{nowrap|1 grad/(10 000)}}
| ≈ {{nowrap|1.570 796{{e|-6}} }} rad
|-
| ] || °
| ≡ 1/360 of a revolution ≡ π/180 rad
| ≈ {{nowrap|17.453 293{{e|-3}} }} rad
|-
| ]; gradian; gon || grad
| ≡ 1/400 of a revolution ≡ 2π/400 rad ≡ 0.9°
| {{nowrap|≈ 15.707 963{{e|-3}} rad}} <!-- Extended nowrap here to prevent wrapping of units for entire column -->
|-
| ] || &nbsp;
| ≡ 45°
| ≈ {{nowrap|0.785 398}} rad
|-
| ] || &nbsp;
| ≡ 90°
| ≈ {{nowrap|1.570 796}} rad
|- style="background:#dfd;"
| ] (SI unit) || rad
| The angle subtended at the center of a circle by an arc whose length is equal to the circle's radius. One full revolution encompasses 2π radians.
| = 1 rad
|-
| sextant || &nbsp;
| ≡ 60°
| ≈ {{nowrap|1.047 198}} rad
|-
| sign || &nbsp;
| ≡ 30°
| ≈ {{nowrap|0.523 599}} rad
|}


For example, the freezing point of water is 0&nbsp;°C and 32&nbsp;°F, and a 5&nbsp;°C change is the same as a 9&nbsp;°F change. Thus, to convert from units of Fahrenheit to units of Celsius, one subtracts 32&nbsp;°F (the offset from the point of reference), divides by 9&nbsp;°F and multiplies by 5&nbsp;°C (scales by the ratio of units), and adds 0&nbsp;°C (the offset from the point of reference). Reversing this yields the formula for obtaining a quantity in units of Celsius from units of Fahrenheit; one could have started with the equivalence between 100&nbsp;°C and 212&nbsp;°F, which yields the same formula.
===Solid angle===
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] || deg²; sq.deg.; (°)²
| {{nowrap|≡ (π/180)² sr}}
| {{nowrap|≈ 0.30462{{e|-3}} sr}}
|- style="background:#dfd;"
| ] (SI unit) || sr
| The solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere having an area r<sup>2</sup>. A sphere encompasses 4π sr.<ref name="howmany"/>
| = 1 sr
|}


Hence, to convert the numerical quantity value of a temperature ''T'' in degrees Fahrenheit to a numerical quantity value ''T'' in degrees Celsius, this formula may be used:
===Mass===
: ''T'' = (''T'' − 32) × 5/9.
Notes:
* See ] for detail of mass/weight distinction and conversion.
* ] is a system of mass based on a pound of 16 ounces, while ] is the system of mass where 12 troy ounces equals one troy pound.
* In this table, the unit ''gee'' is used to denote ] in order to avoid confusion with the "g" symbol for grams.
* In ], the ] is sometimes written '''lbm''' to distinguish it from the ] ('''lbf'''). It should not be read as the mongrel unit "pound metre".
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] || u; AMU
|
| ≈ {{nowrap|1.660 538 73{{e|−27}}}} ± 1.3{{e|−36}} kg
|-
| ], ] rest mass || m<sub>e</sub>
|
| ≈ {{nowrap|9.109 382 15{{e|−31}}}} ± 45{{e|−39}} kg <ref> (2006). ]. Retrieved 16 September 2008.</ref>
|-
| bag (]) || &nbsp;
| ≡ 60&nbsp;kg
| = 60&nbsp;kg
|-
| bag (])|| &nbsp;
| ≡ 94&nbsp;lb av
| = {{nowrap|42.637 682 78}} kg
|-
| barge || &nbsp;
| ≡ 22½ sh tn
| = {{nowrap|20 411.656 65}} kg
|-
| ] || kt
| ≡ 3 1/6 gr
| ≈ {{nowrap|205.196 548 333}} mg
|-
| ] (metric) || ct
| ≡ 200&nbsp;mg
| = 200&nbsp;mg
|-
| ] || &nbsp;
| ≡ 8&nbsp;lb av
| = {{nowrap|3.628 738 96}} kg
|-
| ] || &nbsp;
|
| ≈ 89.9349&nbsp;mg
|-
| ] || Da
|
| ≈ {{nowrap|1.660 902 10{{e|−27}}}} ± 1.3{{e|−36}} kg
|-
| ] (apothecary; ]) || dr t
| ≡ 60 gr
| = {{nowrap|3.887 9346}} g
|-
| ] (avoirdupois) || dr av
| ≡ {{nowrap|27 11/32}} gr
| = {{nowrap|1.771 845 195 3125}} g
|-
| ] || eV
| ≡ 1 eV (energy unit) / c<sup>2</sup>
| = 1.7826{{e|-36}} kg
|-
| ] || γ
| ≡ 1 μg
| = 1 μg
|-
| ] || gr
| ≡ 1/7000&nbsp;lb av
| ≡ {{nowrap|64.798 91}} mg
|-
| ] || G
| grave was the original name of the kilogram
| ≡ 1&nbsp;kg
|-
| ] (long) || long cwt or cwt
| ≡ 112&nbsp;lb av
| = {{nowrap|50.802 345 44}} kg
|-
| ] (short); cental || sh cwt
| ≡ 100&nbsp;lb av
| = {{nowrap|45.359 237}} kg
|- style="background:#dfd;"
| ]<br /><small>(kilogramme)</small>|| kg
| ≡ mass of the prototype near Paris (≈ mass of 1L of water)
| ≡ 1&nbsp;kg (])<ref name="sibaseunits"/>
|-
| ] || kip
| ≡ {{nowrap|1000}} lb av<!--KIloPound-->
| = {{nowrap|453.592 37}} kg
|-
| ] || &nbsp;
| ≡ 8 oz t
| = {{nowrap|248.827 8144}} g
|-
| ] || &nbsp;
| ≡ 1/20 gr
| = {{nowrap|3.239 9455}} mg
|-
| ] (metric) || &nbsp;
| ≡ 1/20 g
| = 50&nbsp;mg
|-
| ] || oz t
| ≡ 1/12&nbsp;lb t
| = {{nowrap|31.103 4768}} g
|-
| ] (]) || oz av
| ≡ 1/16&nbsp;lb
| = {{nowrap|28.349 523 125}} g
|-
| ounce (US food nutrition labeling) || oz
| ≡ 28&nbsp;g<ref name="cfr21"/>
| = 28&nbsp;g
|-
| ] || dwt; pwt
| ≡ 1/20 oz t
| = {{nowrap|1.555 173 84}} g
|-
| ] || &nbsp;
| ≡ 1/100 ct
| = 2&nbsp;mg
|-
| ] || lb
| ≡ slug·ft/s<sup>2</sup>
| = {{nowrap|0.45359237 }} kg
|-
| ] || lb av
| ≡ {{nowrap|0.453 592 37}} kg = 7000 grains
| ≡ {{nowrap|0.453 592 37}} kg
|-
| ] || &nbsp;
| ≡ 500 g
| = 500 g
|-
| ] || lb t
| ≡ {{nowrap|5 760}} grains
| = {{nowrap|0.373 241 7216}} kg
|-
| quarter (imperial) || &nbsp;
| ≡ 1/4 long cwt = 2 st = 28&nbsp;lb av
| = {{nowrap|12.700 586 36}} kg
|-
| quarter (informal)|| &nbsp;
| ≡ ¼ short tn
| = {{nowrap|226.796 185}} kg
|-
| quarter, long (informal)|| &nbsp;
| ≡ ¼ long tn
| = {{nowrap|254.011 7272}} kg
|-
| ] (metric) || q
| ≡ 100&nbsp;kg
| = 100&nbsp;kg
|-
| ] (]) || s ap
| ≡ 20 gr
| = {{nowrap|1.295 9782}} g
|-
| ] || &nbsp;
| ≡ 1/700&nbsp;lb av
| = 647.9891&nbsp;mg
|-
| ]; geepound; hyl || slug
| ≡ 1 gee × 1&nbsp;lb av × 1 s<sup>2</sup>/ft
| ≈ {{nowrap|14.593 903}} kg
|-
| ] || st
| ≡ 14&nbsp;lb av
| = {{nowrap|6.350 293 18}} kg
|-
| ] (long) || AT
| ≡ 1&nbsp;mg × 1 long tn ÷ 1 oz t
| ≈ {{nowrap|32.666 667}} g
|-
| ] (short) || AT
| ≡ 1&nbsp;mg × 1 sh tn ÷ 1 oz t
| ≈ {{nowrap|29.166 667}} g
|-
| ]|| long tn or ton
| ≡ {{nowrap|2 240}} lb
| = {{nowrap|1 016.046 9088}} kg
|-
| ] || sh tn
| ≡ {{nowrap|2 000}} lb
| = {{nowrap|907.184 74}} kg
|-
| ] (] unit) || t
| ≡ {{nowrap|1 000}} kg
| = {{nowrap|1 000}} kg
|-
| ] || &nbsp;
| ≡ 252&nbsp;lb = 18 st
| = {{nowrap|114.305 277 24}} kg (variants exist)
|-
| Zentner || Ztr.
| Definitions vary; see <ref>The Swiss Federal Office for Metrology gives ''Zentner'' on a German language web page and ''quintal'' on the English translation of that page ; the unit is marked "spécifiquement suisse !"</ref> and.<ref name="howmany"/>
|
|}


To convert ''T'' in degrees Celsius to ''T'' in degrees Fahrenheit, this formula may be used:
=== Density ===
: ''T'' = (''T'' × 9/5) + 32.
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| gram per millilitre
| g/mL
| ≡ g/mL
| = {{val|fmt=commas|1000|u=kg/m<sup>3</sup>}}
|- style="background:#dfd;"
| kilogram per cubic metre (SI unit)
| kg/m<sup>3</sup>
| ≡ kg/m<sup>3</sup>
| = 1&nbsp;kg/m<sup>3</sup>
|-
| kilogram per litre
| kg/L
| ≡ kg/L
| = {{val|fmt=commas|1000|u=kg/m<sup>3</sup>}}
|-
| ounce (avoirdupois) per cubic foot
| oz/ft<sup>3</sup>
| ≡ oz/ft<sup>3</sup>
| ≈ {{val|fmt=commas|1.001153961|u=kg/m<sup>3</sup>}}
|-
| ounce (avoirdupois) per cubic inch
| oz/in<sup>3</sup>
| ≡ oz/in<sup>3</sup>
| ≈ {{val|fmt=commas|1.729994044|e=3|u=kg/m<sup>3</sup>}}
|-
| ounce (avoirdupois) per gallon (imperial)
| oz/gal
| ≡ oz/gal
| ≈ {{val|fmt=commas|6.236023291|u=kg/m<sup>3</sup>}}
|-
| ounce (avoirdupois) per gallon (US fluid)
| oz/gal
| ≡ oz/gal
| ≈ {{val|fmt=commas|7.489151707|u=kg/m<sup>3</sup>}}
|-
| pound (avoirdupois) per cubic foot
| lb/ft<sup>3</sup>
| ≡ lb/ft<sup>3</sup>
| ≈ {{val|fmt=commas|16.01846337|u=kg/m<sup>3</sup>}}
|-
| pound (avoirdupois) per cubic inch
| lb/in<sup>3</sup>
| ≡ lb/in<sup>3</sup>
| ≈ {{val|fmt=commas|2.767990471|e=4|u=kg/m<sup>3</sup>}}
|-
| pound (avoirdupois) per gallon (imperial)
| lb/gal
| ≡ lb/gal
| ≈ {{val|fmt=commas|99.77637266|u=kg/m<sup>3</sup>}}
|-
| pound (avoirdupois) per gallon (US fluid)
| lb/gal
| ≡ lb/gal
| ≈ {{val|fmt=commas|119.8264273|u=kg/m<sup>3</sup>}}
|-
| ] per cubic foot
| slug/ft<sup>3</sup>
| ≡ slug/ft<sup>3</sup>
| ≈ {{val|fmt=commas|515.3788184|u=kg/m<sup>3</sup>}}
|}


===Time=== === Example ===
Starting with:
{| class="wikitable"
: <math>Z = n_i \times _i</math>
|+ ]
replace the original unit {{tmath|1= _i }} with its meaning in terms of the desired unit {{tmath|1= _j }}, e.g. if {{tmath|1= _i = c_{ij} \times _j }}, then:
!Name of unit
: <math>Z = n_i \times (c_{ij} \times _j) = (n_i \times c_{ij}) \times _j</math>
!Symbol
!Definition
!Relation to SI units
|-
| ] || au
| ≡ ]/(]·c)
| ≈ {{nowrap|2.418 884 254{{e|−17}}}} s
|-
| ] || &nbsp;
| ≡ 441 mo (hollow) + 499 mo (full) = 76 a of 365.25 d
| = {{nowrap|2.398 3776{{e|9}}}} s
|-
| ] || c
| ≡ 100 a (see below for definition of year length)
| = 100 years
|-
| ] || d
| = 24 h
| = 1440 min = {{gaps|86|400|u=s}}
|-
| ] (sidereal) || d
| ≡ Time needed for the Earth to rotate once around its axis, determined from successive transits of a very distant astronomical object across an observer's meridian (])
| ≈ {{nowrap|86 164.1}} s
|-
| ] || dec
| ≡ 10 a (see below for definition of year length)
| = 10 years
|-
| ] || fn
| ≡ 2 wk
| = {{nowrap|1 209 600}} s
|-
| ] ||
| ≡ {{nowrap|1/1 080}} h
| = 3.{{overline|3}} s
|-
| ] || &nbsp;
| ≡ 4 Callippic cycles - 1 d
| = {{nowrap|9.593 424{{e|9}}}} s
|-
| ] || h
| ≡ 60 min
| = {{nowrap|3 600}} s
|-
| ] || j
| ≡ 1/60 s
| = .01{{overline|6}} s
|-
| jiffy (alternate) || ja
| ≡ 1/100 s
| = 10 ms
|-
| ] (quarter of an hour) || &nbsp;
| ≡ ¼ h = 1/96 d
| = 60 × 60 / 4 s = 900 s = 60 / 4 min = 15 min
|-
| ] (traditional) || &nbsp;
| ≡ 1/100 d
| = 24 × 60 × 60 / 100 s = 864 s = 24 * 60 / 100 min = 14.4 min
|-
| lustre; lustrum || &nbsp;
| ≡ 5 a of 365 d
| = 1.5768{{e|8}} s
|-
| ]; enneadecaeteris || &nbsp;
| ≡ 110 mo (hollow) + 125 mo (full) = 6940 d ≈ 19 a
| = {{nowrap|5.996 16{{e|8}}}} s
|-
| ] || &nbsp;
| ≡ {{nowrap|1 000}} a (see below for definition of year length)
| = 1000 years
|-
| ] || md
| ≡ 1/{{nowrap|1 000}} d
| = 24 × 60 × 60 / {{nowrap|1 000}} s = 86.4 s
|-
| ] || min
| ≡ 60 s, due to ]s sometimes 59 s or 61 s,
| = 60 s
|-
| ] || &nbsp;
| ≡ 90 s
| = 90 s
|-
| ] (full) || mo
| ≡ 30 d<ref name="PedersenGloss">Pedersen O. (1983). "Glossary" in ], Hoskin, M., and Pedersen, O. ''Gregorian Reform of the Calendar: Proceedings of the Vatican Conference to Commemorate its 400th Anniversary''. Vatican Observatory. Available from ].</ref>
| = {{nowrap|2 592 000}} s
|-
| ] (Greg. av.) || mo
| Average Gregorian month = 1 a (Gregorian average) / 12 = {{nowrap|365.242 5}} d / 12 = {{nowrap|30.436 875}} d
| ≈ {{nowrap|2.6297{{e|6}}}} s
|-
| ] (hollow) || mo
| ≡ 29 d<ref name="PedersenGloss"/>
| = {{nowrap|2 505 600}} s
|-
| ] (synodic) || mo
| Cycle time of moon phases ≈ {{nowrap|29.530 589}} d (Average)
| ≈ {{nowrap|2.551{{e|6}}}} s
|-
| ] || &nbsp;
| = 48 mo (full) + 48 mo (hollow) + 3 mo (full)<ref>{{Citation | last=Richards | first=E.G. | title=Mapping Time | year=1998 | pages=94–95 | publisher=Oxford University Press | isbn=0-19-850413-6}}</ref><ref>{{Citation | last=Steel | first=Duncan | title=Marking Time | year=2000 | page=46 | publisher=John Wiley & Sons | isbn=0-471-29827-1}}</ref> = 8 a of 365.25 d = 2922 d
| = {{nowrap|2.524 608{{e|8}}}} s
|-
| ] || &nbsp;
| ≡ (]]/'']''<sup>5</sup>)<sup>½</sup>
| ≈ {{nowrap|1.351 211 868{{e|−43}}}} s
|- style="background:#dfd;"
| ] || s
| time of {{nowrap|9 192 631 770}} periods of the radiation corresponding to the transition between the 2 hyperfine levels of the ground state of the caesium 133 atom at 0 K<ref name="sibaseunits"/> (but other seconds are sometimes used in astronomy). Also that time it takes for light to travel a distance of 299,792,458 meters.
| (])
|-
| ] || &nbsp;
| ≡ 10<sup>−8</sup> s
| = 10 ns
|-
| sigma || &nbsp;
| ≡ 10<sup>−6</sup> s
| = 1 μs
|-
| ] || &nbsp;
| ≡ {{nowrap|1 461}} a of 365 d
| = {{nowrap|4.607 4096{{e|10}}}} s
|-
| ] || S
| ≡ 10<sup>−13</sup> s
| = 100 fs
|-
| ] || wk
| ≡ 7 d
| = 168 h = {{nowrap|10 080}} min = {{nowrap|604 800}} s
|-
| ] (common) || a, y, ''or'' yr || 365 d || = {{nowrap|31 536 000}} s<ref name=Richards>Richards, E. G. (2013). "Calendars" in S. E. Urban & P. K. Seidelmann, eds. ''Explanatory Supplement to the Astronomical Almanac''. Mill Valley, CA: University Science Books.</ref>
|-
| ] (Gregorian) || a, y, ''or'' yr
| = 365.2425 d average, calculated from common years (365 d) plus leap years (366 d) on most years divisible by 4. See ] for details.
| = {{nowrap|31 556 952}} s
|-
| year (Julian) || a, y, ''or'' yr
| = 365.25 d average, calculated from common years (365 d) plus one leap year (366 d) every four years
| = {{nowrap|31 557 600}} s
|-
| ]|| a, y, ''or'' yr || 366 d || = {{nowrap|31 622 400}} s<ref name = Richards/>
|-
| ] || a, y, ''or'' yr
| Conceptually, length of time it takes for the Sun to return to the same position in the cycle of seasons<ref group=Converter>The technical definition of tropical year is the period of time for the ecliptic longitude of the Sun to increase 360 degrees. (Urban & Seidelmann 2013, Glossary, s.v. year, tropical)</ref>
| ≈ {{nowrap|365.24219}} d, each day being 86,400 SI seconds<ref>Richards, E. G. (2013). "Calendars" in S. E. Urban & P. K. Seidelmann, eds. ''Explanatory Supplement to the Astronomical Almanac''. Mill Valley, CA: University Science Books. p. 587.</ref>
≈ {{nowrap|31 556 925}} s
|-
| ] (sidereal) || a, y, ''or'' yr
| ≡ time taken for Sun to return to the same position with respect to the stars of the celestial sphere
| ≈ {{nowrap|365.256 363}} d ≈ {{nowrap|31 558 149.7632}} s
|-
|COLSPAN="4"|Where ] is observed, the length of time units longer than 1 s may increase or decrease by 1 s if a ] occurs during the time interval of interest.
|}


Now {{tmath|1= n_i }} and {{tmath|1= c_{ij} }} are both numerical values, so just calculate their product.
===Frequency===
{| class="wikitable"
|+ Frequency
!Name of unit
!Symbol
!Definition
!Relation to SI units
|- style="background:#dfd;"
| ] (SI unit) || Hz
| ≡ Number of cycles per second
| = 1&nbsp;Hz = 1/s
|-
| ] || rpm
| ≡ One unit rpm equals one rotation completed around a fixed axis in one minute of time.
| ≈ {{val|fmt=commas|0.104719755|u=rad/s}}
|}


Or, which is just mathematically the same thing, multiply ''Z'' by unity, the product is still ''Z'':
===Speed or velocity===
: <math>Z = n_i \times _i \times ( c_{ij} \times _j/_i )</math>
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] per ] || fph
| ≡ 1&nbsp;ft/h
| ≈ {{nowrap|8.466 667{{e|−5}}}} m/s
|-
| ] per ] || fpm
| ≡ 1&nbsp;ft/min
| = 5.08{{e|−3}} m/s
|-
| ] per ] || fps
| ≡ 1&nbsp;ft/s
| = 3.048{{e|−1}} m/s
|-
| ] per ] || &nbsp;
| ≡ furlong/fortnight
| ≈ {{nowrap|1.663 095{{e|−4}}}} m/s
|-
| ] per ] || iph
| ≡ 1 in/hr
| ≈ {{nowrap|7.05 556{{e|−6}}}} m/s
|-
| ] per ] || ipm
| ≡ 1 in/min
| ≈ {{nowrap|4.23 333{{e|−4}}}} m/s
|-
| ] per ] || ips
| ≡ 1 in/s
| = 2.54{{e|−2}} m/s
|-
| ] || km/h
| ≡ 1&nbsp;km/h
| = {{nowrap|1/3.6}} m/s {{nowrap|≈ 2.777 778{{e|−1}}}} m/s
|-
| ] || kn
| ≡ 1 ]/h = 1.852&nbsp;km/h
| ≈ {{nowrap|0.514 444}} m/s
|-
| ] (Admiralty) || kn
| ≡ 1 NM (Adm)/h = {{nowrap|1.853 184}} km/h {{Citation needed|date=September 2008}}
| = {{nowrap|0.514 77{{overline|3}}}} m/s
|-
| ] || ''M''
| Ratio of the speed to the speed of sound in the medium. Varies especially with temperature. About 1225&nbsp;km/h (761&nbsp;mph) in air at sea level to about 1062&nbsp;km/h (660&nbsp;mph) at jet altitudes ({{convert|12200|m|ft}}).<ref>Tom Benson. (2010.) in ''Beginner's Guide to Aeronautics''. ].</ref> Unitless
| ≈ 340 to 295&nbsp;m/s for aircraft
|- style="background:#dfd;"
| ] (SI unit)|| m/s
| ≡ 1&nbsp;m/s
| = 1&nbsp;m/s
|-
| ] || mph
| ≡ 1&nbsp;mi/h
| = {{nowrap|0.447 04}} m/s
|-
| ] per ] || mpm
| ≡ 1&nbsp;mi/min
| = 26.8224&nbsp;m/s
|-
| ] per ] || mps
| ≡ 1&nbsp;mi/s
| = {{nowrap|1 609.344}} m/s
|-
| ] in vacuum || ''c''
| ≡ {{nowrap|299 792 458}} m/s
| = {{nowrap|299 792 458}} m/s
|-
| ] in air || ''s''
| Varies especially with temperature. About 1225&nbsp;km/h (761&nbsp;mph) in air at sea level to about 1062&nbsp;km/h (660&nbsp;mph) at jet altitudes.
| ≈ 340 to 295&nbsp;m/s at aircraft altitudes
|}
A ] consists of a speed combined with a direction; the speed part of the velocity takes units of speed.


For example, you have an expression for a physical value ''Z'' involving the unit ''feet per second'' ({{tmath|1= _i }}) and you want it in terms of the unit ''miles per hour'' ({{tmath|1= _j }}):
===Flow (volume)===
{| class="wikitable"
|+ Flow
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| cubic foot per minute
| CFM{{Citation needed|date=August 2011}}
| ≡ 1&nbsp;ft<sup>3</sup>/min
| = {{val|fmt=commas|4.719474432|e=-4|u=m<sup>3</sup>/s}}
|-
| cubic foot per second
| ft<sup>3</sup>/s
| ≡ 1&nbsp;ft<sup>3</sup>/s
| = {{val|fmt=commas|0.028316846592|u=m<sup>3</sup>/s}}
|-
| cubic inch per minute
| in<sup>3</sup>/min
| ≡ 1 in<sup>3</sup>/min
| = {{val|fmt=commas|2.731177}}{{overline|3}}{{e|-7}}&nbsp;m<sup>3</sup>/s
|-
| cubic inch per second
| in<sup>3</sup>/s
| ≡ 1 in<sup>3</sup>/s
| = {{val|fmt=commas|1.6387064|e=-5|u=m<sup>3</sup>/s}}
|- style="background:#dfd;"
| cubic metre per second (SI unit)
| m<sup>3</sup>/s
| ≡ 1 m<sup>3</sup>/s
| = 1 m<sup>3</sup>/s
|-
| gallon (US fluid) per day
| GPD{{Citation needed|date=August 2011}}
| ≡ 1 gal/d
| = {{val|fmt=commas|4.38126363}}{{overline|8}}{{e|-8}}&nbsp;m<sup>3</sup>/s
|-
| gallon (US fluid) per hour
| GPH{{Citation needed|date=August 2011}}
| ≡ 1 gal/h
| = {{val|fmt=commas|1.05150327}}{{overline|3}}{{e|-6}}&nbsp;m<sup>3</sup>/s
|-
| gallon (US fluid) per minute
| GPM{{Citation needed|date=August 2011}}
| ≡ 1 gal/min
| = {{val|fmt=commas|6.30901964|e=-5|u=m<sup>3</sup>/s}}
|-
| litre per minute
| LPM{{Citation needed|date=August 2011}}
| ≡ 1 L/min
| = 1.{{overline|6}}{{e|-5}} m<sup>3</sup>/s
|}


{{ordered list
===Acceleration===
|1= Find facts relating the original unit to the desired unit:
{| class="wikitable"
: 1 mile = 5280 feet and 1 hour = 3600 seconds
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] per ] per ] || fph/s
| ≡ 1&nbsp;ft/(h·s)
| ≈ {{nowrap|8.466 667{{e|−5}}}} m/s<sup>2</sup>
|-
| ] per ] per ] || fpm/s
| ≡ 1&nbsp;ft/(min·s)
| = 5.08{{e|−3}} m/s<sup>2</sup>
|-
| ] per ] squared || fps<sup>2</sup>
| ≡ 1&nbsp;ft/s<sup>2</sup>
| = 3.048{{e|−1}} m/s<sup>2</sup>
|-
| ]; galileo || Gal
| ≡ 1&nbsp;cm/s<sup>2</sup>
| = 10<sup>−2</sup> m/s<sup>2</sup>
|-
| ] per ] per ] || ipm/s
| ≡ 1 in/(min·s)
| ≈ {{nowrap|4.233 333{{e|−4}}}} m/s<sup>2</sup>
|-
| ] per ] squared || ips<sup>2</sup>
| ≡ 1 in/s<sup>2</sup>
| = 2.54{{e|−2}} m/s<sup>2</sup>
|-
| ] per ] || kn/s
| ≡ 1 kn/s
| ≈ {{nowrap|5.144 444{{e|−1}}}} m/s<sup>2</sup>
|- style="background:#dfd;"
| ] (SI unit)|| m/s<sup>2</sup>
| ≡ 1&nbsp;m/s<sup>2</sup>
| = 1&nbsp;m/s<sup>2</sup>
|-
| ] per ] per ] || mph/s
| ≡ 1&nbsp;mi/(h·s)
| = 4.4704{{e|−1}} m/s<sup>2</sup>
|-
| ] per ] per ] || mpm/s
| ≡ 1&nbsp;mi/(min·s)
| = 26.8224&nbsp;m/s<sup>2</sup>
|-
| ] per ] squared || mps<sup>2</sup>
| ≡ 1&nbsp;mi/s<sup>2</sup>
| = {{nowrap|1.609 344{{e|3}}}} m/s<sup>2</sup>
|-
| ] || ''g''
| ≡ {{nowrap|9.806 65}} m/s<sup>2</sup>
| = {{nowrap|9.806 65}} m/s<sup>2</sup>
|}


|2= Next use the above equations to construct a fraction that has a value of unity and that contains units such that, when it is multiplied with the original physical value, will cancel the original units:
===Force===
: <math>1 = \frac{1\,\mathrm{mi}}{5280\,\mathrm{ft}}\quad \mathrm{and}\quad 1 = \frac{3600\,\mathrm{s}}{1\,\mathrm{h}}</math>
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] ||
| ≡ m<sub>e</sub>·]<sup>2</sup>·'']''<sup>2</sup>/]
| ≈ {{nowrap|8.238 722 06{{e|−8}}}} N <ref>. (2006). ]. Retrieved September 14, 2008.</ref>
|-
| ] (]) || dyn
| ≡ g·cm/s<sup>2</sup>
| = 10<sup>−5</sup> N
|-
| ]; kilopond; ]-force || kgf; kp; Gf
| ≡ ''g'' × 1&nbsp;kg
| = {{nowrap|9.806 65}} N
|-
| ]; kip-force || kip; kipf; klbf
| ≡ ''g'' × 1 000&nbsp;lb
| = {{nowrap|4.448 221 615 2605{{e|3}}}} N
|-
| ]-force, gravet-force || mGf; gf
| ≡ '']'' × 1 g
| = {{nowrap|9.806 65}} mN
|- style="background:#dfd;"
| ] (SI unit) || N
| A force capable of giving a mass of one kg an acceleration of one metre per second, per second.<ref name=cipm1946>{{citation | title=Comité International des Poids et Mesures, Resolution 2 | url=http://www.bipm.org/en/CIPM/db/1946/2/ | year=1946 | accessdate=August 26, 2009}}</ref>
| = 1&nbsp;N = 1&nbsp;kg·m/s<sup>2</sup>
|-
| ] || ozf
| ≡ ''g'' × 1 oz
| = {{nowrap|0.278 013 850 953 781 25}} N
|-
| ] || ]
| ≡ ''g'' × 1&nbsp;lb
| = {{nowrap|4.448 221 615 2605}} N
|-
| ] || pdl
| ≡ 1&nbsp;lb·ft/s<sup>2</sup>
| = {{nowrap|0.138 254 954 376}} N
|-
| sthene (] unit) || sn
| ≡ 1 t·m/s<sup>2</sup>
| = 1{{e|3}} N
|-
| ]-force || tnf
| ≡ ''g'' × 1 sh tn
| = {{nowrap|8.896 443 230 521{{e|3}}}} N
|}
''See also:'' ]


|3= Last, multiply the original expression of the physical value by the fraction, called a ''conversion factor'', to obtain the same physical value expressed in terms of a different unit. Note: since valid conversion factors are ] and have a numerical value of ], multiplying any physical quantity by such a conversion factor (which is 1) does not change that physical quantity.
===Pressure or mechanical stress===
: <math> 52.8\,\frac{\mathrm{ft}}{\mathrm{s}} =
{| class="wikitable"
52.8\,\frac{\mathrm{ft}}{\mathrm{s}}
|+ ]
\frac{1\,\mathrm{mi}}{5280\,\mathrm{ft}}
!Name of unit
\frac{3600\,\mathrm{s}}{1\,\mathrm{h}} =
!Symbol
\frac {52.8 \times 3600}{5280}\,\mathrm{mi/h}
!Definition
= 36\,\mathrm{mi/h}</math>
!Relation to SI units
}}
|-
| ] (standard) || atm
|
| ≡ {{nowrap|101 325}} Pa <ref name=press811>Barry N. Taylor, (April 1 995), (NIST Special Publication 811), Washington, DC: US Government Printing Office, pp. 57&ndash;68.</ref>
|-
| ] (technical) || at
| ≡ 1 kgf/cm<sup>2</sup>
| = {{nowrap|9.806 65{{e|4}}}} Pa <ref name=press811/>
|-
| ] || bar
|
| ≡ 10<sup>5</sup> Pa
|-
| barye (]) || &nbsp;
| ≡ 1 dyn/cm<sup>2</sup>
| = 0.1 Pa
|-
| centimetre of mercury || cmHg
| ≡ 13 595.1&nbsp;kg/m<sup>3</sup> × 1&nbsp;cm × ''g''
| ≈ {{nowrap|1.333 22{{e|3}}}} Pa <ref name=press811/>
|-
| centimetre of ] (4&nbsp;°C) || cmH<sub>2</sub>O
| ≈ 999.972&nbsp;kg/m<sup>3</sup> × 1&nbsp;cm × ''g''
| ≈ 98.063 8 Pa <ref name=press811/>
|-
| ] of mercury (conventional)|| ftHg
| ≡ {{nowrap|13 595.1}} kg/m<sup>3</sup> × 1&nbsp;ft × ''g''
| ≈ {{nowrap|40.636 66{{e|3}}}} Pa <ref name=press811/>
|-
| ] of ] (39.2&nbsp;°F) || ftH<sub>2</sub>O
| ≈ 999.972&nbsp;kg/m<sup>3</sup> × 1&nbsp;ft × ''g''
| ≈ {{nowrap|2.988 98{{e|3}}}} Pa <ref name=press811/>
|-
| ] of mercury (conventional) || inHg
| ≡ {{nowrap|13 595.1}} kg/m<sup>3</sup> × 1 in × ''g''
| ≈ {{nowrap|3.386 389{{e|3}}}} Pa <ref name=press811/>
|-
| ] of ] (39.2&nbsp;°F) || inH<sub>2</sub>O
| ≈ 999.972&nbsp;kg/m<sup>3</sup> × 1 in × ''g''
| ≈ 249.082 Pa <ref name=press811/>
|-
| kilogram-force per square millimetre || kgf/mm<sup>2</sup>
| ≡ 1 kgf/mm<sup>2</sup>
| = {{nowrap|9.806 65{{e|6}}}} Pa <ref name=press811/>
|-
| ] per square ] || ksi
| ≡ 1 kipf/sq in
| ≈ {{nowrap|6.894 757{{e|6}}}} Pa <ref name=press811/>
|-
| micron (micrometre) of mercury || <math>\mu</math>mHg
| ≡ {{nowrap|13 595.1}} kg/m<sup>3</sup> × 1 <math>\mu</math>m × ''g'' ≈ 0.001 torr
| ≈ {{nowrap|0.133 322 4}} Pa <ref name=press811/>
|-
| ] || ]
| ≡ {{nowrap|13 595.1}} kg/m<sup>3</sup> × 1&nbsp;mm × ''g'' ≈ 1 torr
| ≈ 133.3224 Pa <ref name=press811/>
|-
| millimetre of ] (3.98&nbsp;°C) || mmH<sub>2</sub>O
| ≈ 999.972&nbsp;kg/m<sup>3</sup> × 1&nbsp;mm × ''g'' = {{nowrap|0.999 972}} kgf/m<sup>2</sup>
| = {{nowrap|9.806 38}} Pa
|- style="background:#dfd;"
| ] (SI unit) || Pa
| ≡ N/m<sup>2</sup> = kg/(m·s<sup>2</sup>)
| = 1 Pa <ref>Barry N. Taylor, (April 1995), ''Guide for the Use of the International System of Units (SI)'' (NIST Special Publication 811), Washington, DC: US Government Printing Office, p. 5.</ref>
|-
| pièze (] unit) || pz
| ≡ {{nowrap|1 000}} kg/m·s<sup>2</sup>
| = 1{{e|3}} Pa = 1 kPa
|-
| ] per square ] || psf
| ≡ 1&nbsp;lbf/ft<sup>2</sup>
| ≈ {{nowrap|47.880 26}} Pa <ref name=press811/>
|-
| ] || psi
| ≡ 1&nbsp;lbf/in<sup>2</sup>
| ≈ {{nowrap|6.894 757{{e|3}}}} Pa <ref name=press811/>
|-
| ] per square ] || pdl/sq ft
| ≡ 1 pdl/sq ft
| ≈ {{nowrap|1.488 164}} Pa <ref name=press811/>
|-
| short ] per square ] || &nbsp;
| ≡ 1 sh tn × ''g'' / 1 sq ft
| ≈ {{nowrap|95.760 518{{e|3}}}} Pa
|-
| ] || torr
| ≡ {{nowrap|101 325/760}} Pa
| ≈ 133.322 4 Pa <ref name=press811/>
|}


Or as an example using the metric system, you have a value of fuel economy in the unit ''litres per 100 kilometres'' and you want it in terms of the unit ''microlitres per metre'':
===Torque or moment of force===<!-- not equivalent to energy -->
: <math> \mathrm{\frac{9\,\rm{L}}{100\,\rm{km}}} =
{| class="wikitable"
\mathrm{\frac{9\,\rm{L}}{100\,\rm{km}}}
|+ ]
\mathrm{\frac{1000000\,\rm{\mu L}}{1\,\rm{L}}}
!Name of unit
\mathrm{\frac{1\,\rm{km}}{1000\,\rm{m}}} =
!Symbol
\frac {9 \times 1000000}{100 \times 1000}\,\mathrm{\mu L/m} =
!Definition
90\,\mathrm{\mu L/m}</math>
!Relation to SI units
|-
| ] || ft lbf
| ≡ ''g'' × 1&nbsp;lb × 1&nbsp;ft
| = {{nowrap|1.355 817 948 331 4004}} N·m
|-
| ]-poundal || ft pdl
| ≡ 1&nbsp;lb·ft<sup>2</sup>/s<sup>2</sup>
| = {{nowrap|4.214 011 009 380 48{{e|−2}}}} N·m
|-
| ] || in lbf
| ≡ ''g'' × 1&nbsp;lb × 1 in
| = {{nowrap|0.112 984 829 027 6167}} N·m
|-
| ] ] || m kg
| ≡ N × m / ''g''
| ≈ {{nowrap|0.101 971 621}} N·m
|- style="background:#dfd;"
| ] (SI unit) || N·m
| ≡ N × m<!-- ≠ J --> = kg·m<sup>2</sup>/s<sup>2</sup>
| = 1 N·m
|}


== Calculation involving non-SI Units ==
===Energy===<!--work, heat, etc. not torque -->
In the cases where non-] are used, the numerical calculation of a formula can be done by first working out the factor, and then plug in the numerical values of the given/known quantities.
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] || boe
| ≈ 5.8{{e|6}} BTU<sub>59&nbsp;°F</sub>
| ≈ 6.12{{e|9}} J
|-
| ] (ISO) || BTU<sub>ISO</sub>
| ≡ 1.0545{{e|3}} J
| = 1.0545{{e|3}} J
|-
| British thermal unit (International Table) || BTU<sub>IT</sub>
|
| = {{nowrap|1.055 055 852 62{{e|3}}}} J
|-
| British thermal unit (mean) || BTU<sub>mean</sub>
|
| ≈ {{nowrap|1.055 87{{e|3}}}} J
|-
| British thermal unit (thermochemical) || BTU<sub>th</sub>
|
| ≈ {{nowrap|1.054 350{{e|3}}}} J
|-
| British thermal unit (39&nbsp;°F) || BTU<sub>39&nbsp;°F</sub>
|
| ≈ {{nowrap|1.059 67{{e|3}}}} J
|-
| British thermal unit (59&nbsp;°F) || BTU<sub>59&nbsp;°F</sub>
| ≡ {{nowrap|1.054 804{{e|3}}}} J
| = {{nowrap|1.054 804{{e|3}}}} J
|-
| British thermal unit (60&nbsp;°F) || BTU<sub>60&nbsp;°F</sub>
|
| ≈ {{nowrap|1.054 68{{e|3}}}} J
|-
| British thermal unit (63&nbsp;°F) || BTU<sub>63&nbsp;°F</sub>
|
| ≈ 1.0546{{e|3}} J
|-
| ] (International Table) || cal<sub>IT</sub>
| ≡ 4.1868 J
| = 4.1868 J
|-
| calorie (mean) || cal<sub>mean</sub>
| {{frac|100}} of the energy required to warm one gram of air-free water from 0&nbsp;°C to 100&nbsp;°C @ 1&nbsp;atm
| ≈ {{nowrap|4.190 02}} J
|-
| calorie (thermochemical) || cal<sub>th</sub>
| ≡ 4.184 J
| = 4.184 J
|-
| Calorie (US; ])
| Cal
| ≡ 1 kcal
| = 1000 cal = 4184 J
|-
| calorie (3.98&nbsp;°C) || cal<sub>3.98&nbsp;°C</sub>
|
| ≈ 4.2045 J
|-
| calorie (15&nbsp;°C) || cal<sub>15&nbsp;°C</sub>
| ≡ 4.1855 J
| = 4.1855 J
|-
| calorie (20&nbsp;°C) || cal<sub>20&nbsp;°C</sub>
|
| ≈ 4.1819 J
|-
| ] heat unit (International Table) || CHU<sub>IT</sub>
| ≡ 1 BTU<sub>IT</sub> × 1 K/°R
| = {{nowrap|1.899 100 534 716{{e|3}}}} J
|-
| cubic centimetre of ]; standard cubic centimetre || cc atm; scc
| ≡ 1 atm × 1&nbsp;cm<sup>3</sup>
| = {{nowrap|0.101 325}} J
|-
| cubic ] of atmosphere; standard cubic foot || cu ft atm; scf
| ≡ 1 atm × 1&nbsp;ft<sup>3</sup>
| = {{nowrap|2.869 204 480 9344{{e|3}}}} J
|-
| cubic foot of natural gas || &nbsp;
| ≡ {{nowrap|1 000}} BTU<sub>IT</sub>
| = {{nowrap|1.055 055 852 62{{e|6}}}} J
|-
| cubic ] of atmosphere; standard cubic yard || cu yd atm; scy
| ≡ 1 atm × 1 yd<sup>3</sup>
| = {{nowrap|77.468 520 985 2288{{e|3}}}} J
|-
| ] || eV
| ≡ '']'' × 1 V
| ≈ {{nowrap|1.602 177 33{{e|−19}}}} ± 4.9{{e|-26}} J
|-
| ] (]) || erg
| ≡ 1 g·cm<sup>2</sup>/s<sup>2</sup>
| = 10<sup>−7</sup> J
|-
| ] || ft lbf
| ≡ ''g'' × 1&nbsp;lb × 1&nbsp;ft
| = {{nowrap|1.355 817 948 331 4004}} J
|-
| foot-poundal || ft pdl
| ≡ 1&nbsp;lb·ft<sup>2</sup>/s<sup>2</sup>
| = {{nowrap|4.214 011 009 380 48{{e|−2}}}} J
|-
| ]-atmosphere (imperial) || imp gal atm
| ≡ 1 atm × 1 gal (imp)
| = {{nowrap|460.632 569 25}} J
|-
| gallon-atmosphere (US) || US gal atm
| ≡ 1 atm × 1 gal (US)
| = {{nowrap|383.556 849 0138}} J
|-
| ], ] || E<sub>h</sub>
| ≡ m<sub>e</sub>·]<sup>2</sup>·'']''<sup>2</sup> (= 2 Ry)
| ≈ {{nowrap|4.359 744{{e|−18}}}} J
|-
| ] || hp·h
| ≡ 1&nbsp;hp × 1 h
| = {{nowrap|2.684 519 537 696 172 792{{e|6}}}} J
|-
| ] || in lbf
| ≡ ''g'' × 1&nbsp;lb × 1 in
| = {{nowrap|0.112 984 829 027 6167}} J
|- style="background:#dfd;"
| ] (SI unit) || J
| The work done when a force of one newton moves the point of its application a distance of one metre in the direction of the force.<ref name="cipm1946"/>
| = 1&nbsp;J = 1&nbsp;m·N = 1&nbsp;kg·m<sup>2</sup>/s<sup>2</sup> = 1&nbsp;C·V = 1&nbsp;W·s
|-
| kilocalorie; large ] || kcal; Cal
| ≡ {{nowrap|1 000}} cal<sub>IT</sub>
| = 4.1868{{e|3}} J
|-
| ]; Board of Trade Unit || kW·h; B.O.T.U.
| ≡ 1&nbsp;kW × 1 h
| = 3.6{{e|6}} J
|-
| ]-] || l atm; sl
| ≡ 1 atm × 1 L
| = 101.325 J
|-
| ] || &nbsp;
| ≡ 10<sup>15</sup> BTU<sub>IT</sub>
| = {{nowrap|1.055 055 852 62{{e|18}}}} J
|-
| ] || Ry
| ≡ '']''·]·'']''
| ≈ {{nowrap|2.179 872{{e|−18}}}} J
|-
| ] (E.C.) || &nbsp;
| ≡ {{nowrap|100 000}} BTU<sub>IT</sub>
| = {{nowrap|105.505 585 262{{e|6}}}} J
|-
| ] (US) || &nbsp;
| ≡ {{nowrap|100 000}} BTU<sub>59&nbsp;°F</sub>
| = 105.4804{{e|6}} J
|-
| thermie || th
| ≡ 1 Mcal<sub>IT</sub>
| = 4.1868{{e|6}} J
|-
| ] || TCE
| ≡ 7 Gcal<sub>th</sub>
| = 29.288{{e|9}} J
|-
| ] || TOE
| ≡ 10 Gcal<sub>th</sub>
| = 41.84{{e|9}} J
|-
| ] of ] || tTNT
| ≡ 1 Gcal<sub>th</sub>
| = 4.184{{e|9}} J
|}


For example, in the study of ],<ref>{{Cite book |last=Foot |first=C. J. |url=https://books.google.com/books?id=kXYpAQAAMAAJ|title=Atomic physics |date=2005|publisher=Oxford University Press |isbn=978-0-19-850695-9|language=en}}</ref> ] {{math|''m''}} is usually given in ], instead of ]s, and ] {{math|''μ''}} is often given in the ] times ]. The condensate's ] is given by:
===Power or heat flow rate===
<math display="block">\xi=\frac{\hbar}{\sqrt{2m\mu}}\,.</math>
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ]-cubic centimetre per ] || atm ccm
| ≡ 1 atm × 1&nbsp;cm<sup>3</sup>/min
| = {{nowrap|1.688 75{{e|−3}}}} W
|-
| atmosphere-cubic centimetre per ] || atm ccs
| ≡ 1 atm × 1&nbsp;cm<sup>3</sup>/s
| = {{nowrap|0.101 325}} W
|-
| atmosphere-cubic ] per ] || atm cfh
| ≡ 1 atm × 1 cu ft/h
| = {{nowrap|0.797 001 244 704}} W
|-
| atmosphere-cubic foot per minute || atm·cfm
| ≡ 1 atm × 1 cu ft/min
| = {{nowrap|47.820 074 682 24}} W
|-
| atmosphere-cubic foot per second || atm cfs
| ≡ 1 atm × 1 cu ft/s
| = {{nowrap|2.869 204 480 9344{{e|3}}}} W
|-
| ] (International Table) per hour || BTU<sub>IT</sub>/h
| ≡ 1 BTU<sub>IT</sub>/h
| ≈ {{nowrap|0.293 071}} W
|-
| BTU (International Table) per minute || BTU<sub>IT</sub>/min
| ≡ 1 BTU<sub>IT</sub>/min
| ≈ {{nowrap|17.584 264}} W
|-
| BTU (International Table) per second || BTU<sub>IT</sub>/s
| ≡ 1 BTU<sub>IT</sub>/s
| = {{nowrap|1.055 055 852 62{{e|3}}}} W
|-
| ] (International Table) per second || cal<sub>IT</sub>/s
| ≡ 1 cal<sub>IT</sub>/s
| = 4.1868 W
|-
| erg per second || erg/s
| ≡ 1 erg/s
| = 10<sup>−7</sup> W
|-
| foot-] per hour || ft lbf/h
| ≡ 1&nbsp;ft lbf/h
| ≈ {{nowrap|3.766 161{{e|−4}}}} W
|-
| foot-pound-force per minute || ft lbf/min
| ≡ 1&nbsp;ft lbf/min
| = {{nowrap|2.259 696 580 552 334{{e|−2}}}} W
|-
| foot-pound-force per second || ft lbf/s
| ≡ 1&nbsp;ft lbf/s
| = {{nowrap|1.355 817 948 331 4004}} W
|-
| ] (boiler) || bhp
| ≈ 34.5&nbsp;lb/h × 970.3 BTU<sub>IT</sub>/lb
| ≈ {{nowrap|9.810 657{{e|3}}}} W
|-
| horsepower (European electrical) || hp
| ≡ 75 kp·m/s
| = 736 W
|-
| horsepower (imperial electrical) || hp
| ≡ 746 W
| = 746 W
|-
| horsepower (imperial mechanical) || hp
| ≡ 550&nbsp;ft lbf/s
| = {{nowrap|745.699 871 582 270 22}} W
|-
| horsepower (metric) || hp
| ≡ 75 m kgf/s
| = {{nowrap|735.498 75}} W
|-
| ]-atmosphere per minute || L·atm/min
| ≡ 1 atm × 1 L/min
| = {{nowrap|1.688 75}} W
|-
| litre-atmosphere per second || L·atm/s
| ≡ 1 atm × 1 L/s
| = 101.325 W
|-
| lusec || lusec
| ≡ 1 L·µmHg/s <ref name="howmany"/>
| ≈ 1.333{{e|−4}} W
|-
| ] || p
| ≡ 100 m kgf/s
| = 980.665 W
|-
| square foot equivalent direct radiation || sq ft EDR
| ≡ 240 BTU<sub>IT</sub>/h
| ≈ {{nowrap|70.337 057}} W
|-
| ] of air conditioning || &nbsp;
| ≡ 2000&nbsp;lbs of ice melted / 24 h
| ≈ {{nowrap|3 504 }} W
|-
| ton of refrigeration (imperial) || &nbsp;
| ≡ 2240&nbsp;lb × ice<sub>IT</sub> / 24 h: ice<sub>IT</sub> = 144&nbsp;°F × 2326 J/kg.°F
| ≈ {{nowrap|3.938 875{{e|3}}}} W
|-
| ton of refrigeration (IT) || &nbsp;
| ≡ 200&nbsp;lbs × ice<sub>IT</sub> / 24 h: ice<sub>IT</sub> = 144° × 2326 J/kg.°F
| ≈ {{nowrap|3.516 853{{e|3}}}} W
|- style="background:#dfd;"
| ] (SI unit) || W
| The power which in one second of time gives rise to one joule of energy.<ref name="cipm1946"/>
| = 1&nbsp;W = 1&nbsp;J/s = 1&nbsp;N·m/s = 1&nbsp;kg·m<sup>2</sup>/s<sup>3</sup>
|}


For a <sup>23</sup>Na condensate with chemical potential of (the Boltzmann constant times) 128&nbsp;nK, the calculation of healing length (in micrometres) can be done in two steps:
===Action===
{| class="wikitable"
|+ Action
!Name of unit
!Symbol
!Definition
!Relation to SI units
<!--- previous entry was:
|-----
| ] || au
| ] = ]/2] 1.054 571 596{{e|−34}} J·s
--->
|-
| ] || au
| ≡ ] ≡ ]/2]
| ≈ {{nowrap|1.054 571 68{{e|−34}}}} J·s<ref> 8th ed. (2006), ], Section 4.1 Table 7.</ref>
|}


=== Calculate the factor ===
===Dynamic viscosity===
Assume that {{tmath|1= m=1 \,\text{Da},\mu = k_\text{B}\cdot 1\,\text{nK} }}, this gives
{| class="wikitable"
<math display="block">\xi=\frac{\hbar}{\sqrt{2m\mu}} = 15.574 \,\mathrm{\mu m}\,,</math>
|+ Dynamic ]
which is our factor.
!Name of unit
!Symbol
!Definition
!Relation to SI units
|- style="background:#dfd;"
| ] (SI unit) || Pa·s
| ≡ N·s/m<sup>2</sup>, kg/(m·s)
| = 1 Pa·s
|-
| ] (]) || P
| ≡ 1 barye·s
| = 0.1 Pa·s
|-
| pound per foot hour || lb/(ft·h)
| ≡ 1&nbsp;lb/(ft·h)
| ≈ {{gaps|4.133 789{{e|-4}} }} Pa·s
|-
| pound per foot second || lb/(ft·s)
| ≡ 1&nbsp;lb/(ft·s)
| ≈ {{val|fmt=commas|1.488164}}&nbsp;Pa·s
|-
| pound-force second per square foot || lbf·s/ft<sup>2</sup>
| ≡ 1&nbsp;lbf·s/ft<sup>2</sup>
| ≈ {{val|fmt=commas|47.88026}}&nbsp;Pa·s
|-
| pound-force second per square inch || lbf·s/in<sup>2</sup>
| ≡ 1&nbsp;lbf·s/in<sup>2</sup>
| ≈ {{val|fmt=commas|6894.757}}&nbsp;Pa·s
|}


=== Calculate the numbers ===
===Kinematic viscosity===
Now, make use of the fact that {{tmath|1= \xi\propto\frac{1}{\sqrt{m\mu} } }}. With {{tmath|1= m=23 \,\text{Da},\mu=128\,k_\text{B}\cdot\text{nK} }}, {{tmath|1= \xi=\frac{15.574}{\sqrt{23 \cdot 128} } \,\text{μm}=0.287\,\text{μm} }}.
{| class="wikitable"
|+ Kinematic ]
!Name of unit
!Symbol
!Definition
!Relation to ]
|-
| square foot per second || ft<sup>2</sup>/s
| ≡ 1&nbsp;ft<sup>2</sup>/s
| = {{val|fmt=commas|0.09290304|u=m<sup>2</sup>/s}}
|- style="background:#dfd;"
| square metre per second (SI unit) || m<sup>2</sup>/s
| ≡ 1&nbsp;m<sup>2</sup>/s
| = 1&nbsp;m<sup>2</sup>/s
|-
| ] (]) || St
| ≡ 10<sup>−4</sup>&nbsp;m<sup>2</sup>/s
| = 10<sup>−4</sup>&nbsp;m<sup>2</sup>/s
|}


This method is especially useful for programming and/or making a ], where input quantities are taking multiple different values; For example, with the factor calculated above, it is very easy to see that the healing length of <sup>174</sup>Yb with chemical potential 20.3&nbsp;nK is
===Electric current===
:{{tmath|1= \xi=\frac{15.574}{\sqrt{174\cdot20.3} } \,\text{μm}=0.262\,\text{μm} }}.
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|- style="background:#dfd;"
| ] (]) || A
| ≡ The constant current needed to produce a force of 2 {{e|-7}} newton per metre between two straight parallel conductors of infinite length and negligible circular cross-section placed one metre apart in a vacuum.<ref name="sibaseunits"/>
| = 1 A = 1 C/s
|-
| ]; abampere (]) || abamp
| ≡ 10 A
| = 10 A
|-
| ]; statampere (]) || esu/s
| ≡ (0.1 A·m/s) / '']''
| ≈ {{val|fmt=commas|3.335641|e=-10|u=A}}
|}

===Electric charge===
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ]; electromagnetic unit (]) || abC; emu
| ≡ 10 C
| = 10 C
|-
| ] || au
| ≡ '']''
| ≈ {{nowrap|1.602 176 462{{e|−19}}}} C
|- style="background:#dfd;"
| ] || C
| ≡ The amount of electricity carried in one second of time by one ampere of current.<ref name="cipm1946"/>
| = 1 C = 1 A·s
|-
| ] || F
| ≡ 1&nbsp;mol × '']''·'']''
| ≈ {{nowrap|96 485.3383}} C
|-
| ] || mA·h
| ≡ 0.001 A × 1 h
| = 3.6 C
|-
| ]; ]; electrostatic unit (]) || statC; Fr; esu
| ≡ (0.1 A·m) / '']''
| ≈ {{nowrap|3.335 641{{e|−10}}}} C
|}

===Electric dipole===
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| atomic unit of electric dipole moment || '']'']
| &nbsp;
| ≈ {{nowrap|8.478 352 81{{e|-30}} }} C·m<ref>{{citation | url=http://physics.nist.gov/cgi-bin/cuu/Value?auedm | year=2006 | title=The NIST Reference on Constants, Units, and Uncertainty | accessdate=August 26, 2009}}</ref>
|- style="background:#dfd;"
| coulomb meter || C·m
| &nbsp;
| = 1 C · 1 m
|-
| ] || D
| = 10<sup>−10</sup> esu·Å
| = 3.33564095{{e|-30}} C·m <ref>Robert G. Mortimer ''Physical chemistry'',Academic Press, 2000 ISBN 0-12-508345-9, page 677</ref>
|}

===Electromotive force, electric potential difference===
{| class="wikitable"
|+ ], electromotive force
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] (]) || abV
| ≡ 1{{e|-8}} V
| = 1{{e|-8}} V
|-
| ] (]) || statV
| ≡ '']''· (1 μJ/A·m)
| = {{nowrap|299.792 458}} V
|- style="background:#dfd;"
| ] (SI unit) || V
| The difference in electric potential across two points along a conducting wire carrying one ampere of constant current when the power dissipated between the points equals one watt.<ref name="cipm1946"/>
| = 1 V = 1 W/A = {{nowrap|1 kg·m<sup>2</sup>/(A·s<sup>3</sup>)}}
|}

===Electrical resistance===
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|- style="background:#dfd;"
| ] (SI unit) || Ω
| The resistance between two points in a conductor when one volt of electric potential difference, applied to these points, produces one ampere of current in the conductor.<ref name="cipm1946"/>
| = 1 Ω = 1 V/A = {{nowrap|1 kg·m<sup>2</sup>/(A<sup>2</sup>·s<sup>3</sup>)}}
|}

===Capacitance===
{| class="wikitable"
|+ ]'s ability to store ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|- style="background:#dfd;"
|] (SI unit)
|F
|The capacitance between two parallel plates that results in one volt of potential difference when charged by one coulomb of electricity.<ref name="cipm1946"/>
| = 1 F = 1 C/V = {{nowrap|1 A<sup>2</sup>·s<sup>4</sup>/(kg·m<sup>2</sup>)}}
|}

===Magnetic flux===
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] (CGS unit)
| Mx
| ≡ 10<sup>−8</sup> Wb<ref name=nistguide>{{citation | title=NIST Guide to SI Units, Appendix B.9 | url=http://physics.nist.gov/Pubs/SP811/appenB9.html | accessdate=August 27, 2009}}</ref>
| = 1{{e|-8}} Wb
|- style="background:#dfd;"
| ] (SI unit)
| Wb
| Magnetic flux which, linking a circuit of one turn, would produce in it an electromotive force of 1 volt if it were reduced to zero at a uniform rate in 1 second.<ref name="cipm1946"/>
| = 1 Wb = 1 V·s = {{nowrap|1 kg·m<sup>2</sup>/(A·s<sup>2</sup>)}}
|}

===Magnetic flux density===
{| class="wikitable"
|+ What physicists call ] is called ] density by electrical engineers and magnetic induction by applied mathematicians and electrical engineers.
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] (CGS unit) || G
| ≡ ]/cm<sup>2</sup> = 10<sup>−4</sup> T
| = 1{{e|-4}} T <ref name="SI-10">''Standard for the Use of the International System of Units (SI): The Modern Metric System'' IEEE/ASTM SI 10-1997. (1997). New York and West Conshohocken, PA: ] and ]. Tables A.1 through A.5.</ref>
|- style="background:#dfd;"
| ] (SI unit) || T
| ≡ ]/]
| = 1 T = 1 Wb/m<sup>2</sup> = {{nowrap|1 kg/(A·s<sup>2</sup>)}}
|}

===Inductance===
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|- style="background:#dfd;"
| ] (SI unit) || H
| The inductance of a closed circuit that produces one volt of electromotive force when the current in the circuit varies at a uniform rate of one ampere per second.<ref name="cipm1946"/>
| = 1 H = 1 Wb/A = {{nowrap|1 kg·m<sup>2</sup>/(A·s)<sup>2</sup>}}
|}

===Temperature===
{{details|Conversion of units of temperature}}
{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| degree ] || °C
| °C ≡ K − 273.15
| ≡ + 273.15
|-
| degree ] || °De
|
| = 373.15 − × 2/3
|-
| degree ] || °F
| °F ≡ °C × 9/5 + 32
| ≡ ( + 459.67) × 5/9
|-
| degree ] || °N
|
| = × 100/33 + 273.15
|-
| degree ] || °R;
| °R ≡ K × 9/5
| ≡ × 5/9
|-
| degree ] || °Ré
|
| = × 5/4 + 273.15
|-
| degree ] || °Rø
|
| = ( − 7.5) × 40/21 + 273.15
|-
| Regulo ] || GM;
| °F ≡ GM × 25 + 300
| ≡ × 125/9 + 422.038
|- style="background:#dfd;"
| ] (SI base unit) || K
| ≡ 1/273.16 of the ] of the ].<ref name="sibaseunits"/>
| ≡ 1 K
|}

===Information entropy===

{| class="wikitable"
|+ ]
!Name of unit
!Symbol
!Definition
!Relation to SI units
!Relation to bits
|-
| SI unit || J/K
| ≡ J/K
| = 1 J/K
|
|-
| ]; nip; nepit || nat
| ≡ ''k<sub>B</sub>''
| = {{nowrap|1.380 650 5(23){{e|-23}}}} J/K
|
|-
| ]; shannon || bit; b; Sh
| ≡ ln(2) × ''k<sub>B</sub>''
| = {{nowrap|9.569 940 (16){{e|-24}}}} J/K
| = 1 bit
|-
| ]; hartley || ban; Hart
| ≡ ln(10) × ''k<sub>B</sub>''
| = {{nowrap|3.179 065 3(53){{e|-23}}}} J/K
|
|-
| ] ||
| ≡ 4 bits
| = {{nowrap|3.827 976 0(64){{e|-23}}}} J/K
| = 2<sup>2</sup> bit
|-
| ] || B
| ≡ 8 bits
| = {{nowrap|7.655 952 (13){{e|-23}}}} J/K
| = 2<sup>3</sup> bit
|-
| ] (decimal) || kB
| ≡ {{nowrap|1 000}} B
| = {{nowrap|7.655 952 (13){{e|-20}}}} J/K
| = 8{{e|3}} bit = 8000 bit
|-
| ] (]) || KB; KiB
| ≡ {{nowrap|1 024}} B
| = {{nowrap|7.839 695 (13){{e|-20}}}} J/K
| = 2<sup>13</sup> bit = 8192 bit
|}
Often, information entropy is measured in ]s, whereas the (discrete) storage space of digital devices is measured in bits. Thus, uncompressed redundant data occupy more than one bit of storage per shannon of information entropy. The multiples of a bit listed above are usually used with this meaning. Other times the bit is used as a measure of information entropy and is thus a synonym of shannon.

===Luminous intensity===
The candela is the preferred nomenclature for the SI unit.
{| class="wikitable"
|+ ]
|-
!Name of unit
!Symbol
!Definition
!Relation to SI units
|- style="background:#dfd;"
| ] (SI base unit); candle
| cd
| The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540{{e|12}} hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.<ref name="sibaseunits"/>
| = 1&nbsp;cd
|-
| ] (new)
| cp
| ≡ cd The use of ''candlepower'' as a unit is discouraged due to its ambiguity.
| = 1&nbsp;cd
|-
| ] (old, pre-1948)
| cp
| Varies and is poorly reproducible.<ref>{{citation | title=The NIST Reference on Constants, Units, and Uncertainty | url=http://physics.nist.gov/cuu/Units/candela.html | accessdate=August 28, 2009}}</ref> Approximately 0.981&nbsp;cd.<ref name="howmany"/>
| ≈ 0.981&nbsp;cd
|}

===Luminance===
{| class="wikitable"
|+ ]
|-
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| candela per square foot
| cd/ft<sup>2</sup>
| ≡ cd/ft<sup>2</sup>
| ≈ {{val|fmt=commas|10.763910417|u=cd/m<sup>2</sup>}}
|-
| candela per square inch
| cd/in<sup>2</sup>
| ≡ cd/in<sup>2</sup>
| ≈ {{val|fmt=commas|1550.0031|u=cd/m<sup>2</sup>}}
|- style="background:#dfd;"
| ] (SI unit); nit (deprecated<ref name="howmany"/>)
| cd/m<sup>2</sup>
| ≡ cd/m<sup>2</sup>
| = 1&nbsp;cd/m<sup>2</sup>
|-
| ]
| fL
| ≡ (1/π) cd/ft<sup>2</sup>
| ≈ {{val|fmt=commas|3.4262590996|u=cd/m<sup>2</sup>}}
|-
| ]
| L
| ≡ (10<sup>4</sup>/π) cd/m<sup>2</sup>
| ≈ {{val|fmt=commas|3183.0988618|u=cd/m<sup>2</sup>}}
|-
| ] (CGS unit)
| sb
| ≡ 10<sup>4</sup> cd/m<sup>2</sup>
| ≈ 1{{e|4}}&nbsp;cd/m<sup>2</sup>
|}

===Luminous flux===
{| class="wikitable"
|+ ]
|-
!Name of unit
!Symbol
!Definition
!Relation to SI units
|- style="background:#dfd;"
| ] (SI unit)
| lm
| ≡ cd·sr
| = 1&nbsp;lm = 1&nbsp;cd·sr
|}

===Illuminance===
{| class="wikitable"
|+ ]
|-
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ]; lumen per square foot
| fc
| ≡ lm/ft<sup>2</sup>
| = {{val|fmt=commas|10.763910417|u=lx}}
|-
| lumen per square inch
| lm/in<sup>2</sup>
| ≡ lm/in<sup>2</sup>
| ≈ {{val|fmt=commas|1550.0031|u=lx}}
|- style="background:#dfd;"
| ] (SI unit)
| lx
| ≡ lm/m<sup>2</sup>
| = 1&nbsp;lx = 1&nbsp;lm/m<sup>2</sup>
|-
| ] (CGS unit)
| ph
| ≡ lm/cm<sup>2</sup>
| = 1{{e|4}} lx
|}

===Radiation - source activity===
{| class="wikitable"
|+ ]
|-
!Name of unit
!Symbol
!Definition
!Relation to SI units
|- style="background:#dfd;"
| ] (SI unit) || Bq
| ≡ Number of disintegrations per second
| = 1 Bq = 1/s
|-
| ] || Ci
| ≡ 3.7{{e|10}} Bq
| = 3.7{{e|10}} Bq <ref name="SP811-2008">Ambler Thompson & Barry N. Taylor. (2008). Special Publication 811. Gaithersburg, MD: ]. p. 10.</ref>
|-
| ] (H) || rd
| ≡ 1 MBq
| = 1{{e|6}} Bq
|}
Please note that although becquerel (Bq) and hertz (Hz) both ultimately refer to the same SI base unit (s<sup>−1</sup>), Hz is used only for periodic phenomena, and Bq is only used for stochastic processes associated with radioactivity.<ref name=sibrochure222>{{citation | url=http://www.bipm.org/en/si/si_brochure/chapter2/2-2/table3.html | title=The International System of Units, Section 2.2.2., Table 3 | edition=8 | publisher=] | year=2006 | accessdate=August 27, 2009}}</ref>

===Radiation - exposure===
{| class="wikitable"
|+ Radiation - exposure
|-
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] || R
| 1 R ≡ 2.58{{e|-4}} C/kg<ref name="nistguide"/>
| = 2.58{{e|-4}} C/kg
|}
The roentgen is not an SI unit and the ] strongly discourages its continued use.<ref>{{citation | url=http://physics.nist.gov/Pubs/SP811/sec05.html#5.2 | title=The NIST Guide to the SI (Special Publication 811), section 5.2 | year=2008 | accessdate=August 27, 2009}}</ref>

===Radiation - absorbed dose===
{| class="wikitable"
|+ Radiation - ]
|-
!Name of unit
!Symbol
!Definition
!Relation to SI units
|- style="background:#dfd;"
| ] (SI unit) || Gy
| ≡ 1 J/kg = 1 m<sup>2</sup>/s<sup>2</sup> <ref>Ambler Thompson & Barry N. Taylor. (2008). Special Publication 811. Gaithersburg, MD: ]. p. 5.</ref>
| = 1 Gy
|-
| ] || rad
| ≡ 0.01 Gy<ref name="nistguide"/>
| = 0.01 Gy
|}

===Radiation - equivalent dose===
{| class="wikitable"
|+ Radiation - ]
|-
!Name of unit
!Symbol
!Definition
!Relation to SI units
|-
| ] || rem
| ≡ 0.01 Sv
| = 0.01 Sv
|- style="background:#dfd;"
| ] (SI unit) || Sv
| ≡ 1 J/kg<ref name="sibrochure222"/>
| = 1 Sv
|}
Although the definitions for sievert (Sv) and gray (Gy) would seem to indicate that they measure the same quantities, this is not the case. The effect of receiving a certain dose of radiation (given as Gy) is variable and depends on many factors, thus a new unit was needed to denote the biological effectiveness of that dose on the body; this is known as the equivalent dose and is shown in Sv. The general relationship between absorbed dose and equivalent dose can be represented as
:''H = Q · D''
where ''H'' is the equivalent dose, ''D'' is the absorbed dose, and ''Q'' is a dimensionless quality factor. Thus, for any quantity of ''D'' measured in Gy, the numerical value for ''H'' measured in Sv may be different.<ref>{{citation | url=http://www.bipm.org/en/CIPM/db/2002/2/ | title=Comité international des poids et mesures, 2002, Recommendation 2 | accessdate=August 27, 2009}}</ref>


== Software tools == == Software tools ==
There are many conversion tools. They are found in the function libraries of applications such as spreadsheets databases, in calculators, and in macro packages and plugins for many other applications such as the mathematical, scientific and technical applications.
Home and office computers come with converters in bundled spreadsheet applications or can access free converters via the Internet. Units and measurements can be easily converted using these tools, but only if the units are explicitly defined and the conversion is compatible (e.g., cmHg to kPa).


There are many standalone applications that offer the thousands of the various units with conversions. For example, the ] offers a command line utility ] for GNU and Windows.<ref>{{cite web| title = GNU Units |url = https://www.gnu.org/software/units/ |access-date = 2024-09-24}}</ref> The ] is also a popular option.
===General commercial sources of converters===
*] have unit-conversion functionality.
*] programs usually provide conversion ]s or ]s or the user can write their own.
*Commercial mathematical, scientific and technical applications often include converters.


== See also == == See also ==
{{multicol}} {{div col}}
* ]
*]
* ]
*]
*] * ]
*] * ]
* ]
*]
* ]
*]
*] * ]
* ]
*] (e.g. "kilo-" prefix)
* ]
{{multicol-break}}
*] * ]
*] * ]
*] * ]
* ]
*]
* ]
*]
*] * ]
{{div col end}}
*]
*]
{{multicol-end}}


== Notes and references == == Notes and references ==
{{Reflist|2}} {{reflist|30em}}
;Notes ; Notes
{{Reflist|group=Converter}} {{reflist|group=Converter}}


== External links == == External links ==
{{extlinks|date=July 2023}}
{{Wikibooks|FHSST Physics Units:How to Change Units}} {{Wikibooks|FHSST Physics Units:How to Change Units}}
{{Wikivoyage|Metric and Imperial equivalents}} {{Wikivoyage|Metric and Imperial equivalents}}
<!-- ATTENTION! Please do not add links without discussion and consensus on the talk page. Undiscussed links will be removed. --> <!-- ATTENTION! Please do not add links without discussion and consensus on the talk page. Undiscussed links will be removed. -->
* {{UK SI|title=Units of measurement regulations 1995|year=1995|number=1804|showsldlink=yes}}

* {{cite web |url= http://physics.nist.gov/cuu/Document/nonsi_in_1998.pdf |title= NIST: Fundamental physical constants – Non-SI units |access-date= 2004-03-15 |archive-url= https://web.archive.org/web/20161227154531/http://www.physics.nist.gov/cuu/Document/nonsi_in_1998.pdf |archive-date= 2016-12-27 |url-status= dead }}
*{{UK SI|title=Units of measurement regulations 1995|year=1995|number=1804|showsldlink=yes}}
* Many conversion factors listed.
*
*
*{{PDFlink||35.7&nbsp;KB}}
* {{Webarchive|url=https://web.archive.org/web/20230502150911/http://w3.energistics.org/uom/poscUnits22.xml |date=2023-05-02 }}
* Many conversion factors listed.
*
*
*
*
*
*{{dmoz|Science/Reference/Units_of_Measurement/Software/|Units of Measurement Software}}
*
*{{dmoz|Science/Reference/Units_of_Measurement/Online_Conversion/|Units of Measurement Online Conversion}}
*
* ''Chemistry: Concepts and Applications'', Denton independent school District


{{Systems of measurement}} {{Systems of measurement}}
{{SI units}} {{SI units}}


{{DEFAULTSORT:Conversion Of Units}}
] ]
] ]

Latest revision as of 04:21, 29 December 2024

Comparison of various scales

Conversion of units is the conversion of the unit of measurement in which a quantity is expressed, typically through a multiplicative conversion factor that changes the unit without changing the quantity. This is also often loosely taken to include replacement of a quantity with a corresponding quantity that describes the same physical property.

Unit conversion is often easier within a metric system such as the SI than in others, due to the system's coherence and its metric prefixes that act as power-of-10 multipliers.

Overview

The definition and choice of units in which to express a quantity may depend on the specific situation and the intended purpose. This may be governed by regulation, contract, technical specifications or other published standards. Engineering judgment may include such factors as:

For some purposes, conversions from one system of units to another are needed to be exact, without increasing or decreasing the precision of the expressed quantity. An adaptive conversion may not produce an exactly equivalent expression. Nominal values are sometimes allowed and used.

Factor–label method

Further information: Dimensional analysis

The factor–label method, also known as the unit–factor method or the unity bracket method, is a widely used technique for unit conversions that uses the rules of algebra.

The factor–label method is the sequential application of conversion factors expressed as fractions and arranged so that any dimensional unit appearing in both the numerator and denominator of any of the fractions can be cancelled out until only the desired set of dimensional units is obtained. For example, 10 miles per hour can be converted to metres per second by using a sequence of conversion factors as shown below: 10   m i 1   h × 1609.344   m 1   m i × 1   h 3600   s = 4.4704   m s . {\displaystyle {\frac {\mathrm {10~{\cancel {mi}}} }{\mathrm {1~{\cancel {h}}} }}\times {\frac {\mathrm {1609.344~m} }{\mathrm {1~{\cancel {mi}}} }}\times {\frac {\mathrm {1~{\cancel {h}}} }{\mathrm {3600~s} }}=\mathrm {4.4704~{\frac {m}{s}}} .}

Each conversion factor is chosen based on the relationship between one of the original units and one of the desired units (or some intermediary unit), before being rearranged to create a factor that cancels out the original unit. For example, as "mile" is the numerator in the original fraction and ⁠ 1   m i = 1609.344   m {\displaystyle \mathrm {1~mi} =\mathrm {1609.344~m} } ⁠, "mile" will need to be the denominator in the conversion factor. Dividing both sides of the equation by 1 mile yields ⁠ 1   m i 1   m i = 1609.344   m 1   m i {\displaystyle {\frac {\mathrm {1~mi} }{\mathrm {1~mi} }}={\frac {\mathrm {1609.344~m} }{\mathrm {1~mi} }}} ⁠, which when simplified results in the dimensionless ⁠ 1 = 1609.344   m 1   m i {\displaystyle 1={\frac {\mathrm {1609.344~m} }{\mathrm {1~mi} }}} ⁠. Because of the identity property of multiplication, multiplying any quantity (physical or not) by the dimensionless 1 does not change that quantity. Once this and the conversion factor for seconds per hour have been multiplied by the original fraction to cancel out the units mile and hour, 10 miles per hour converts to 4.4704 metres per second.

As a more complex example, the concentration of nitrogen oxides (NOx) in the flue gas from an industrial furnace can be converted to a mass flow rate expressed in grams per hour (g/h) of NOx by using the following information as shown below:

NOx concentration
= 10 parts per million by volume = 10 ppmv = 10 volumes/10 volumes
NOx molar mass
= 46 kg/kmol = 46 g/mol
Flow rate of flue gas
= 20 cubic metres per minute = 20 m/min
The flue gas exits the furnace at 0 °C temperature and 101.325 kPa absolute pressure.
The molar volume of a gas at 0 °C temperature and 101.325 kPa is 22.414 m/kmol.
1000   g   NO x 1 kg   NO x × 46   kg   NO x 1   kmol   NO x × 1   kmol   NO x 22.414   m 3   NO x × 10   m 3   NO x 10 6   m 3   gas × 20   m 3   gas 1   minute × 60   minute 1   hour = 24.63   g   NO x hour {\displaystyle {\frac {1000\ {\ce {g\ NO}}_{x}}{1{\cancel {{\ce {kg\ NO}}_{x}}}}}\times {\frac {46\ {\cancel {{\ce {kg\ NO}}_{x}}}}{1\ {\cancel {{\ce {kmol\ NO}}_{x}}}}}\times {\frac {1\ {\cancel {{\ce {kmol\ NO}}_{x}}}}{22.414\ {\cancel {{\ce {m}}^{3}\ {\ce {NO}}_{x}}}}}\times {\frac {10\ {\cancel {{\ce {m}}^{3}\ {\ce {NO}}_{x}}}}{10^{6}\ {\cancel {{\ce {m}}^{3}\ {\ce {gas}}}}}}\times {\frac {20\ {\cancel {{\ce {m}}^{3}\ {\ce {gas}}}}}{1\ {\cancel {\ce {minute}}}}}\times {\frac {60\ {\cancel {\ce {minute}}}}{1\ {\ce {hour}}}}=24.63\ {\frac {{\ce {g\ NO}}_{x}}{\ce {hour}}}}

After cancelling any dimensional units that appear both in the numerators and the denominators of the fractions in the above equation, the NOx concentration of 10 ppmv converts to mass flow rate of 24.63 grams per hour.

Checking equations that involve dimensions

The factor–label method can also be used on any mathematical equation to check whether or not the dimensional units on the left hand side of the equation are the same as the dimensional units on the right hand side of the equation. Having the same units on both sides of an equation does not ensure that the equation is correct, but having different units on the two sides (when expressed in terms of base units) of an equation implies that the equation is wrong.

For example, check the universal gas law equation of PV = nRT, when:

  • the pressure P is in pascals (Pa)
  • the volume V is in cubic metres (m)
  • the amount of substance n is in moles (mol)
  • the universal gas constant R is 8.3145 Pa⋅m/(mol⋅K)
  • the temperature T is in kelvins (K)

P a ⋅ m 3 = m o l 1 × P a ⋅ m 3 m o l   K × K 1 {\displaystyle \mathrm {Pa{\cdot }m^{3}} ={\frac {\cancel {\mathrm {mol} }}{1}}\times {\frac {\mathrm {Pa{\cdot }m^{3}} }{{\cancel {\mathrm {mol} }}\ {\cancel {\mathrm {K} }}}}\times {\frac {\cancel {\mathrm {K} }}{1}}}

As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's right hand side are cancelled out, both sides of the equation have the same dimensional units. Dimensional analysis can be used as a tool to construct equations that relate non-associated physico-chemical properties. The equations may reveal undiscovered or overlooked properties of matter, in the form of left-over dimensions – dimensional adjusters – that can then be assigned physical significance. It is important to point out that such 'mathematical manipulation' is neither without prior precedent, nor without considerable scientific significance. Indeed, the Planck constant, a fundamental physical constant, was 'discovered' as a purely mathematical abstraction or representation that built on the Rayleigh–Jeans law for preventing the ultraviolet catastrophe. It was assigned and ascended to its quantum physical significance either in tandem or post mathematical dimensional adjustment – not earlier.

Limitations

The factor–label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0 (ratio scale in Stevens's typology). Most conversions fit this paradigm. An example for which it cannot be used is the conversion between the Celsius scale and the Kelvin scale (or the Fahrenheit scale). Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between degrees Celsius and degrees Fahrenheit there is neither a constant difference nor a constant ratio. There is, however, an affine transform (⁠ x ↦ a x + b {\displaystyle x\mapsto ax+b} ⁠, rather than a linear transform ⁠ x ↦ a x {\displaystyle x\mapsto ax} ⁠) between them.

For example, the freezing point of water is 0 °C and 32 °F, and a 5 °C change is the same as a 9 °F change. Thus, to convert from units of Fahrenheit to units of Celsius, one subtracts 32 °F (the offset from the point of reference), divides by 9 °F and multiplies by 5 °C (scales by the ratio of units), and adds 0 °C (the offset from the point of reference). Reversing this yields the formula for obtaining a quantity in units of Celsius from units of Fahrenheit; one could have started with the equivalence between 100 °C and 212 °F, which yields the same formula.

Hence, to convert the numerical quantity value of a temperature T in degrees Fahrenheit to a numerical quantity value T in degrees Celsius, this formula may be used:

T = (T − 32) × 5/9.

To convert T in degrees Celsius to T in degrees Fahrenheit, this formula may be used:

T = (T × 9/5) + 32.

Example

Starting with:

Z = n i × [ Z ] i {\displaystyle Z=n_{i}\times _{i}}

replace the original unit ⁠ [ Z ] i {\displaystyle _{i}} ⁠ with its meaning in terms of the desired unit ⁠ [ Z ] j {\displaystyle _{j}} ⁠, e.g. if ⁠ [ Z ] i = c i j × [ Z ] j {\displaystyle _{i}=c_{ij}\times _{j}} ⁠, then:

Z = n i × ( c i j × [ Z ] j ) = ( n i × c i j ) × [ Z ] j {\displaystyle Z=n_{i}\times (c_{ij}\times _{j})=(n_{i}\times c_{ij})\times _{j}}

Now ⁠ n i {\displaystyle n_{i}} ⁠ and ⁠ c i j {\displaystyle c_{ij}} ⁠ are both numerical values, so just calculate their product.

Or, which is just mathematically the same thing, multiply Z by unity, the product is still Z:

Z = n i × [ Z ] i × ( c i j × [ Z ] j / [ Z ] i ) {\displaystyle Z=n_{i}\times _{i}\times (c_{ij}\times _{j}/_{i})}

For example, you have an expression for a physical value Z involving the unit feet per second (⁠ [ Z ] i {\displaystyle _{i}} ⁠) and you want it in terms of the unit miles per hour (⁠ [ Z ] j {\displaystyle _{j}} ⁠):

  1. Find facts relating the original unit to the desired unit:
    1 mile = 5280 feet and 1 hour = 3600 seconds
  2. Next use the above equations to construct a fraction that has a value of unity and that contains units such that, when it is multiplied with the original physical value, will cancel the original units:
    1 = 1 m i 5280 f t a n d 1 = 3600 s 1 h {\displaystyle 1={\frac {1\,\mathrm {mi} }{5280\,\mathrm {ft} }}\quad \mathrm {and} \quad 1={\frac {3600\,\mathrm {s} }{1\,\mathrm {h} }}}
  3. Last, multiply the original expression of the physical value by the fraction, called a conversion factor, to obtain the same physical value expressed in terms of a different unit. Note: since valid conversion factors are dimensionless and have a numerical value of one, multiplying any physical quantity by such a conversion factor (which is 1) does not change that physical quantity.
    52.8 f t s = 52.8 f t s 1 m i 5280 f t 3600 s 1 h = 52.8 × 3600 5280 m i / h = 36 m i / h {\displaystyle 52.8\,{\frac {\mathrm {ft} }{\mathrm {s} }}=52.8\,{\frac {\mathrm {ft} }{\mathrm {s} }}{\frac {1\,\mathrm {mi} }{5280\,\mathrm {ft} }}{\frac {3600\,\mathrm {s} }{1\,\mathrm {h} }}={\frac {52.8\times 3600}{5280}}\,\mathrm {mi/h} =36\,\mathrm {mi/h} }

Or as an example using the metric system, you have a value of fuel economy in the unit litres per 100 kilometres and you want it in terms of the unit microlitres per metre:

9 L 100 k m = 9 L 100 k m 1000000 μ L 1 L 1 k m 1000 m = 9 × 1000000 100 × 1000 μ L / m = 90 μ L / m {\displaystyle \mathrm {\frac {9\,{\rm {L}}}{100\,{\rm {km}}}} =\mathrm {\frac {9\,{\rm {L}}}{100\,{\rm {km}}}} \mathrm {\frac {1000000\,{\rm {\mu L}}}{1\,{\rm {L}}}} \mathrm {\frac {1\,{\rm {km}}}{1000\,{\rm {m}}}} ={\frac {9\times 1000000}{100\times 1000}}\,\mathrm {\mu L/m} =90\,\mathrm {\mu L/m} }

Calculation involving non-SI Units

In the cases where non-SI units are used, the numerical calculation of a formula can be done by first working out the factor, and then plug in the numerical values of the given/known quantities.

For example, in the study of Bose–Einstein condensate, atomic mass m is usually given in daltons, instead of kilograms, and chemical potential μ is often given in the Boltzmann constant times nanokelvin. The condensate's healing length is given by: ξ = ℏ 2 m μ . {\displaystyle \xi ={\frac {\hbar }{\sqrt {2m\mu }}}\,.}

For a Na condensate with chemical potential of (the Boltzmann constant times) 128 nK, the calculation of healing length (in micrometres) can be done in two steps:

Calculate the factor

Assume that ⁠ m = 1 Da , μ = k B ⋅ 1 nK {\displaystyle m=1\,{\text{Da}},\mu =k_{\text{B}}\cdot 1\,{\text{nK}}} ⁠, this gives ξ = ℏ 2 m μ = 15.574 μ m , {\displaystyle \xi ={\frac {\hbar }{\sqrt {2m\mu }}}=15.574\,\mathrm {\mu m} \,,} which is our factor.

Calculate the numbers

Now, make use of the fact that ⁠ ξ ∝ 1 m μ {\displaystyle \xi \propto {\frac {1}{\sqrt {m\mu }}}} ⁠. With ⁠ m = 23 Da , μ = 128 k B ⋅ nK {\displaystyle m=23\,{\text{Da}},\mu =128\,k_{\text{B}}\cdot {\text{nK}}} ⁠, ⁠ ξ = 15.574 23 ⋅ 128 μm = 0.287 μm {\displaystyle \xi ={\frac {15.574}{\sqrt {23\cdot 128}}}\,{\text{μm}}=0.287\,{\text{μm}}} ⁠.

This method is especially useful for programming and/or making a worksheet, where input quantities are taking multiple different values; For example, with the factor calculated above, it is very easy to see that the healing length of Yb with chemical potential 20.3 nK is

⁠ ξ = 15.574 174 ⋅ 20.3 μm = 0.262 μm {\displaystyle \xi ={\frac {15.574}{\sqrt {174\cdot 20.3}}}\,{\text{μm}}=0.262\,{\text{μm}}} ⁠.

Software tools

There are many conversion tools. They are found in the function libraries of applications such as spreadsheets databases, in calculators, and in macro packages and plugins for many other applications such as the mathematical, scientific and technical applications.

There are many standalone applications that offer the thousands of the various units with conversions. For example, the free software movement offers a command line utility GNU units for GNU and Windows. The Unified Code for Units of Measure is also a popular option.

See also

Notes and references

  1. Béla Bodó; Colin Jones (26 June 2013). Introduction to Soil Mechanics. John Wiley & Sons. pp. 9–. ISBN 978-1-118-55388-6.
  2. Goldberg, David (2006). Fundamentals of Chemistry (5th ed.). McGraw-Hill. ISBN 978-0-07-322104-5.
  3. Ogden, James (1999). The Handbook of Chemical Engineering. Research & Education Association. ISBN 978-0-87891-982-6.
  4. "Dimensional Analysis or the Factor Label Method". Mr Kent's Chemistry Page.
  5. "Identity property of multiplication". Retrieved 2015-09-09.
  6. Foot, C. J. (2005). Atomic physics. Oxford University Press. ISBN 978-0-19-850695-9.
  7. "GNU Units". Retrieved 2024-09-24.
Notes

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