Conversion of units is the conversion of theunit of measurement in which aquantity is expressed, typically through a multiplicativeconversion 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 ametric system such as theSI than in others, due to the system'scoherence and itsmetric prefixes that act as power-of-10 multipliers.
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 publishedstandards. 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. Anadaptive conversion may not produce an exactly equivalent expression.Nominal values are sometimes allowed and used.
Thefactor–label method, also known as theunit–factor method or theunity bracket method,[1] is a widely used technique for unit conversions that uses the rules ofalgebra.[2][3][4]
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, 10miles per hour can be converted tometres per second by using a sequence of conversion factors as shown below:
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, "mile" will need to be the denominator in the conversion factor. Dividing both sides of the equation by 1 mile yields, which when simplified results in the dimensionless. Because of the identity property of multiplication, multiplying any quantity (physical or not) by the dimensionless 1 does not change that quantity.[5] Once this and the conversion factor for seconds per hour have been multiplied by the original fraction to cancel out the unitsmile andhour, 10 miles per hour converts to 4.4704 metres per second.
As a more complex example, theconcentration ofnitrogen oxides (NOx) in theflue gas from an industrialfurnace can be converted to amass flow rate expressed in grams per hour (g/h) of NOx by using the following information as shown below:
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.
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 theuniversal gas law equation ofPV =nRT, when:
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, thePlanck constant, a fundamental physical constant, was 'discovered' as a purely mathematical abstraction or representation that built on theRayleigh–Jeans law for preventing theultraviolet catastrophe. It was assigned and ascended to its quantum physical significance either in tandem or post mathematical dimensional adjustment – not earlier.
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 theCelsius scale and theKelvin scale (or theFahrenheit 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, anaffine transform (, rather than alinear transform) 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 temperatureT[F] in degrees Fahrenheit to a numerical quantity valueT[C] in degrees Celsius, this formula may be used:
To convertT[C] in degrees Celsius toT[F] in degrees Fahrenheit, this formula may be used:
Starting with:
replace the original unit with its meaning in terms of the desired unit, e.g. if, then:
Now and are both numerical values, so just calculate their product.
Or, which is just mathematically the same thing, multiplyZ by unity, the product is stillZ:
For example, we have an expression for a physical valueZ involving the unitfeet per second () and we want it in terms of the unitmiles per hour ():
Or as an example using the metric system, we have a value of fuel economy in the unitlitres per 100 kilometres and we want it in terms of the unitmicrolitres per metre:
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 ofBose–Einstein condensate,[6]atomic massm is usually given indaltons, instead ofkilograms, andchemical potentialμ is often given in theBoltzmann constant timesnanokelvin. The condensate'shealing length is given by:
For a23Na condensate with chemical potential of (the Boltzmann constant times) 128 nK, the calculation of healing length (in micrometres) can be done in two steps:
Assume that, this giveswhich is our factor.
Now, make use of the fact that. With,.
This method is especially useful for programming and/or making aworksheet, 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 of174Yb with chemical potential 20.3 nK is
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, thefree software movement offers a command line utilityGNU units for GNU and Windows.[7] TheUnified Code for Units of Measure is also a popular option.
