Conversions

Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors.

Techniques

Process overview

The process of conversion depends 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:

• The precision and accuracy of measurement and the associated uncertainty of measurement.
• The statistical confidence interval or tolerance interval of the initial measurement.
• The number of significant figures of the measurement.
• The intended use of the measurement including the engineering tolerances.
• Historical definitions of the units and their derivatives used in old measurements; e.g., international foot vs. US survey foot.

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.

Conversion factors

A conversion factor is used to change the units of a measured quantity without changing its value. The unity bracket method of unit conversion consists of a fraction in which the denominator is equal to the numerator, but they are in different units. Because of the identity property of multiplication, the value of a quantity will not change as long as it is multiplied by one. Also, if the numerator and denominator of a fraction are equal to each other, then the fraction is equal to one. So as long as the numerator and denominator of the fraction are equivalent, they will not affect the value of the measured quantity.

The following example demonstrates how the unity bracket method is used to convert the rate 5 kilometers per second to meters per second. The symbols km, m, and s represent kilometer, meter, and second, respectively.

${\frac {5{\cancel {\text{km}}}}{\text{s}}}\cdot$${\frac {{1000}{\text{ m}}}{{1}{\cancel {\text{ km}}}}}$$=$${\frac {5000\cdot {\text{m}}}{{\text{s}}\cdot {1}}}=$${\frac {5000{\text{ m}}}{\text{s}}}$

Thus, it is found that 5 kilometers per second is equal to 5000 meters per second.

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 Linux and Windows.

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 pre-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 Boltzmann constant times nanokelvin. The condensate's healing length is given by:

$\xi ={\frac {\hbar }{\sqrt {2m\mu }}}\,.$

For a ²³Na condensate with chemical potential of (Boltzmann constant times) 128 nK, the calculation of healing length (in micrometres) can be done in two steps:

Calculate the pre-factor

Assume that $m=1\,{\text{Da}},\mu =k_{\text{B}}\cdot 1\,{\text{nK}}\,,$ this gives

$\xi ={\frac {\hbar }{\sqrt {2m\mu }}}=15.574\,\mathrm {\mu } m\,,$

which is our pre-factor.

Calculate the numbers

Now, make use of the fact that $\xi \propto {\frac {1}{\sqrt {m\mu }}}$. With $m=23\,{\text{Da}},\mu =128\,k_{\text{B}}\cdot {\text{nK}}$, $\xi ={\frac {15.574}{\sqrt {23\cdot 128}}}\,\mathrm {\mu } m=0.287\,\mathrm {\mu } m$.

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's very easy to see that the healing length of ¹⁷⁴Yb with chemical potential 20.3 nK is $\xi ={\frac {15.574}{\sqrt {174\cdot 20.3}}}\,\mu m=0.262\,\mathrm {\mu } m$.

Resources

• Conversion Between Metric Units, Khan Academy
• Understanding Conversion Factors, Tyler DeWitt
• Converting Units With Conversion Factors - Metric System Review & Dimensional Analysis, The Organic Chemistry Tutor

Licensing

Content obtained and/or adapted from:

• Conversion of units, Wikipedia under a CC BY-SA license