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Conversion of units is the conversion between different
units of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a mul ...
for the same
quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a u ...
, typically through multiplicative conversion factors which change the measured quantity value without changing its effects.

# Overview

The process of conversion depends on the specific situation and the intended purpose. This may be governed by regulation,
contract A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to ...
,
technical specifications A specification often refers to a set of documented requirements to be satisfied by a material, design, product, or service. A specification is often a type of technical standard. There are different types of technical or engineering specificati ...
or other published
standard Standard may refer to: Symbols * Colours, standards and guidons, kinds of military signs * Standard (emblem), a type of a large symbol or emblem used for identification Norms, conventions or requirements * Standard (metrology), an object ...
s. Engineering judgment may include such factors as: * The precision and accuracy of measurement and the associated uncertainty of measurement. * The statistical
confidence interval In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as ...
or tolerance interval of the initial measurement. * The number of
significant figures Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something. If a number expre ...
of the measurement. * The intended use of the measurement including the
engineering tolerance Engineering tolerance is the permissible limit or limits of variation in: # a physical dimension; # a measured value or physical property of a material, manufactured object, system, or service; # other measured values (such as temperature, ...
s. * Historical definitions of the units and their derivatives used in old measurements; e.g.,
international foot The foot ( feet), standard symbol: ft, is a unit of length in the British imperial and United States customary systems of measurement. The prime symbol, , is a customarily used alternative symbol. Since the International Yard and ...
vs. US
survey foot The foot ( feet), standard symbol: ft, is a unit of length in the British imperial and United States customary systems of measurement. The prime symbol, , is a customarily used alternative symbol. Since the International Yard and ...
. 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. 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. Nominal values are sometimes allowed and used.

# Factor-label method

The factor-label method, also known as the unit-factor method or the unity bracket method, is a widely used technique for unit conversions using the rules of
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
. 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 Miles per hour (mph, m.p.h., MPH, or mi/h) is a British imperial and United States customary unit of speed expressing the number of miles travelled in one hour. It is used in the United Kingdom, the United States, and a number of smaller countri ...
can be converted to
metres per second The metre per second is the unit of both speed (a scalar quantity) and velocity (a vector quantity, which has direction and magnitude) in the International System of Units (SI), equal to the speed of a body covering a distance of one metre in ...
by using a sequence of conversion factors as shown below: $\frac \times \frac \times \frac = \mathrm.$ 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 re-arranged to create a factor that cancels out the original unit. For example, as "mile" is the numerator in the original fraction and $\mathrm = \mathrm$, "mile" will need to be the denominator in the conversion factor. Dividing both sides of the equation by 1 mile yields $\frac = \frac$, which when simplified results in the dimensionless $1 = \frac$. 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 In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', '' molar concentration'', '' number concentration'', ...
of
nitrogen oxides Nitrogen oxide may refer to a binary compound of oxygen and nitrogen, or a mixture of such compounds: Charge-neutral *Nitric oxide (NO), nitrogen(II) oxide, or nitrogen monoxide * Nitrogen dioxide (), nitrogen(IV) oxide * Nitrogen trioxide (), o ...
( NO''x'') in the
flue gas Flue gas is the gas exiting to the atmosphere via a flue, which is a pipe or channel for conveying exhaust gases from a fireplace, oven, furnace, boiler or steam generator. Quite often, the flue gas refers to the combustion exhaust gas produc ...
from an industrial furnace can be converted to a
mass flow rate In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units. The common symbol is \dot (''á ...
expressed in grams per hour (g/h) of NO''x'' by using the following information as shown below: ; NO''x'' concentration := 10
parts per million In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, th ...
by volume = 10 ppmv = 10 volumes/106 volumes ; NO''x'' molar mass := 46 kg/kmol = 46 g/mol ; Flow rate of flue gas := 20 cubic metres per minute = 20 m3/min : The flue gas exits the furnace at 0 Â°C temperature and 101.325 kPa absolute pressure. : The
molar volume In chemistry and related fields, the molar volume, symbol ''V''m, or \tilde V of a substance is the ratio of the volume occupied by a substance to the amount of substance, usually given at a given temperature and pressure. It is equal to the mola ...
of a gas at 0 Â°C temperature and 101.325 kPa is 22.414 m3/
kmol The mole, symbol mol, is the unit of amount of substance in the International System of Units (SI). The quantity amount of substance is a measure of how many elementary entities of a given substance are in an object or sample. The mole is define ...
. :$\frac \times \frac \times \frac \times \frac \times \frac \times \frac = 24.63\ \frac$ After canceling out any dimensional units that appear both in the numerators and denominators of the fractions in the above equation, the NO''x'' 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 , when: * the pressure ''P'' is in pascals (Pa) * the volume ''V'' is in cubic metres (m3) * the amount of substance ''n'' is in moles (mol) * the
universal gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
''R'' is 8.3145 Paâ‹…m3/(molâ‹…K) * the temperature ''T'' is in kelvins (K) $\mathrm = \frac \times \frac \times \frac$ 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 hitherto unknown or overlooked properties of matter, in the form of left-over dimensions â€“ dimensional adjusters â€“ 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 The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, a fundamental physical constant, was 'discovered' as a purely mathematical abstraction or representation that built on the
Rayleighâ€“Jeans law In physics, the Rayleighâ€“Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength Î», it is: B_ (T) = \ ...
for preventing the ultraviolet catastrophe. It was assigned and ascended to its quantum physical significance either in tandem or post mathematical dimensional adjustment â€“ 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 units fit this paradigm. An example for which it cannot be used is the conversion between
degrees Celsius The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The d ...
and
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
s (or
degrees Fahrenheit The Fahrenheit scale () is a temperature scale based on one proposed in 1724 by the physicist Daniel Gabriel Fahrenheit (1686â€“1736). It uses the degree Fahrenheit (symbol: Â°F) as the unit. Several accounts of how he originally defined h ...
). 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 ($x \mapsto ax+b$, rather than a linear transform $x \mapsto ax$) between them. For example, the freezing point of water is 0 Â°C and 32 Â°F, and a 5 Â°C change is the same as a 9 Â°F change. Thus, to convert from units of Fahrenheit to units of Celsius, one subtracts 32 Â°F (the offset from the point of reference), divides by 9 Â°F and multiplies by 5 Â°C (scales by the ratio of units), and adds 0 Â°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 Â°C and 212 Â°F, though this would yield the same formula at the end. 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'' âˆ’ 32) Ã— 5/9. To convert ''T'' in degrees Celsius to ''T'' in degrees Fahrenheit, this formula may be used: :''T'' = (''T'' Ã— 9/5) + 32.

# Calculation involving non-SI Units

In the cases where non-
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
are used, the numerical calculation of a formula can be done by first working out the pre-factor, and then plug in the numerical values of the given/known quantities. For example, in the study of
Boseâ€“Einstein condensate In condensed matter physics, a Boseâ€“Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (âˆ’273.15 Â°C or âˆ’459.67&n ...
,
atomic mass The atomic mass (''m''a or ''m'') is the mass of an atom. Although the SI unit of mass is the kilogram (symbol: kg), atomic mass is often expressed in the non-SI unit dalton (symbol: Da) â€“ equivalently, unified atomic mass unit (u). 1&nb ...
is usually given in daltons, instead of
kilogram The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquiall ...
s, and
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
is often given in the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas con ...
times nanokelvin. The condensate's healing length is given by: $\xi=\frac\,.$ For a 23Na 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 pre-factor

Assume that $m=1 \,\text,\mu = k_\text\cdot 1\,\text\,,$ this gives $\xi=\frac = 15.574 \,\mathrm\,,$ which is our pre-factor.

## Calculate the numbers

Now, make use of the fact that $\xi\propto\frac$. With $m=23 \,\text,\mu=128\,k_\text\cdot\text$, $\xi=\frac \,\text=0.287\,\text$. This method is especially useful for programming and/or making a worksheet, where input quantities are taking multiple different values; For example, with the pre-factor calculated above, it is very easy to see that the healing length of 174Yb with chemical potential 20.3 nK is $\xi=\frac \,\text=0.262\,\text$.

# 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 The free software movement is a social movement with the goal of obtaining and guaranteeing certain freedoms for software users, namely the freedoms to run the software, to study the software, to modify the software, and to share copies of the ...
offers a command line utility
GNU units GNU Units is a cross-platform computer program for conversion of units of quantities. It has a database of measurement units, including esoteric and historical units. This for instance allows conversion of velocities specified in furlongs per fo ...
for Linux and Windows.

*
Accuracy and precision Accuracy and precision are two measures of '' observational error''. ''Accuracy'' is how close a given set of measurements ( observations or readings) are to their '' true value'', while ''precision'' is how close the measurements are to each o ...
*
Conversion of units of temperature This is a collection of temperature conversion formulas and comparisons among eight different temperature scales, several of which have long been obsolete. Temperatures on scales that either do not share a numeric zero or are nonlinearly related ...
*
Dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such a ...
*
English units English units are the units of measurement used in England up to 1826 (when they were replaced by Imperial units), which evolved as a combination of the Anglo-Saxon and Roman systems of units. Various standards have applied to English units at ...
* False precision *
Imperial units The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units first defined in the British Weights and Measures Act 1824 and continued to be developed th ...
*
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
*
Mesures usuelles Mesures usuelles (, ''customary measurements'') were a French system of measurement introduced by Napoleon I in 1812 to act as compromise between the metric system and traditional measurements. The system was restricted to use in the retail indus ...
*
Metric prefix A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pr ...
(e.g. "kilo-" prefix) *
Metric system The metric system is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the Intern ...
*
Natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ...
*
Orders of Magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic d ...
*
Rounding Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $with$, the fraction 312/937 with 1/3, or the expression with . Rounding is often done to ob ...
*
Significant figures Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something. If a number expre ...
* Unified Code for Units of Measure *
United States customary units United States customary units form a system of measurement units commonly used in the United States and U.S. territories since being standardized and adopted in 1832. The United States customary system (USCS or USC) developed from English uni ...
*
Unit of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric units, used in every country globally. In the United States the U.S. customary unit ...
* Units (software) * Units conversion by factor-label *
Units of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a mul ...

# Notes and references

;Notes

* *
NIST Guide to SI Units
Many conversion factors listed.
Units, Symbols, and Conversions XML Dictionary
* *
dÃ©duites de la grandeur de la terre,