Measurement (LINEAR) – CONVERSION

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 $\xi ={\frac {15.574}{\sqrt {174\cdot 20.3}}}\,\mu m=0.262\,\mathrm {\mu } m$ .

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. Conversions between units in the metric system are defined by their prefixes (for example, 1 kilogram = 1000 grams, 1 milligram = 0.001 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 = 10⁻⁶ metre). Within each table, the units are listed alphabetically, and the SI units (base or derived) are highlighted.

Legend
Symbol	Definition
≡	exactly equal
≈	approximately equal to
≘	(exactly) corresponds to (different types of quantity describing the same phenomenon)
${\overline {\text{digits}}}$	indicates that `digits` repeat infinitely (e.g. $8.294{\overline {369}}$ corresponds to 8.294 369 369 369 369 ...)
(H)	of chiefly historical interest

Length

Length
Name of unit	Symbol	Definition	Relation to SI units
ångström	Å	≡ 1 x 10^-10 m	≡ 0.1 nm
astronomical unit	au	≡ 149 597 870 700 m ≈ Distance from Earth to Sun	≡ 149 597 870 700 m
attometre	am	≡ 1 x 10^-18 m	≡ 1 x 10^-18 m
barleycorn (H)		= ${\tfrac {1}{3}}$ in (see note above about rounding)	= $8.4{\overline {6}}\times 10^{-3}$ m
bohr, atomic unit of length	a₀	= Bohr radius of hydrogen	(17) x 10^-11 m
cable length (imperial)		≡ 608 ft	≈ 185.3184 m
cable length (International)		≡ ${\tfrac {1}{10}}$ nmi	≡ 185.2 m
cable length (US)		≡ 720 ft	= 219.456 m
chain (Gunter's; Surveyor's)	ch	≡ 66 ft (US) ≡ 4 rods	≈ 20.116 84 m
cubit (H)		≡ Distance from fingers to elbow ≈ 18 in	≈ 0.5 m
ell (H)	ell	≡ 45 in (In England usually)	= 1.143 m
fathom	ftm	≡ 6 ft	= 1.8288 m
femtometre	fm	≡ 1 x 10^-15 m	≡ 1 x 10^-15 m
fermi	fm	≡ 1 x 10^-15 m	≡ 1 x 10^-15 m
finger		≡ ${\tfrac {7}{8}}$ in	= 0.022 225 m
finger (cloth)		≡ $4{\tfrac {1}{2}}$ in	= 0.1143 m
foot (Benoît) (H)	ft (Ben)		≈ 0.304 799 735 m
foot (Cape) (H)		Legally defined as 1.033 English feet in 1859	≈ 0.314 858 m
foot (Clarke's) (H)	ft (Cla)		≈ 0.304 797 2654 m
foot (Indian) (H)	ft Ind		≈ 0.304 799 514 m
foot, metric	mf	≡ 300 mm	≡ 0.3 m
foot, metric (Mesures usuelles) (H)		≡ ${\tfrac {1}{3}}$ m	≡ $0.{\overline {3}}$ m
foot (International)	ft	≡ 0.3048 m ≡ ${\tfrac {1}{3}}$ yd ≡ 12 inches	≡ 0.3048 m
foot (Sear's) (H)	ft (Sear)		≈ 0.304 799 47 m
foot (US Survey)	ft (US)	≡ ${\tfrac {1200}{3937}}$ m	≈ 0.304 800 610 m
french; charriere	F	≡ ${\tfrac {1}{3}}$ mm	= $0.{\overline {3}}$ x 10^-3 m
furlong	fur	≡ 10 chains = 660 ft = 220 yd	= 201.168 m
hand		≡ 4 in	≡ 0.1016 m
inch (International)	in	≡ 2.54 cm ≡ ${\tfrac {1}{36}}$ yd ≡ ${\tfrac {1}{12}}$ ft	≡ 0.0254 m
league (land)	lea	≈ 1 hour walk, Currently defined in US as 3 Statute miles, but historically varied from 2 to 9 km	≈ 4828 m
light-day		≡ 24 light-hours	≡ 2.590 206 837 12 × 10¹³ m
light-hour		≡ 60 light-minutes	≡ 1.079 252 8488 × 10¹² m
light-minute		≡ 60 light-seconds	≡ 1.798 754 748 × 10¹⁰ m
light-second		≡ Distance light travels in one second in vacuum	≡ 299 792 458 m
light-year	ly	≡ Distance light travels in vacuum in 365.25 days	≡ 9.460 730 472 5808 × 10¹⁵ m
line	ln	≡ ${\tfrac {1}{12}}$ in	= 0.002 11 ${\overline {6}}$ m
link (Gunter's; Surveyor's)	lnk	≡ ${\tfrac {1}{100}}$ ch ≡ 0.66 ft (US) ≡ 7.92 in	≈ 0.201 168 4 m
link (Ramsden's; Engineer's)	lnk	≡ 1 ft	= 0.3048 m
metre (SI base unit) (meter)	m	≡ Distance light travels in ${\tfrac {1}{299792458}}$ of a second in vacuum.	(SI base unit)
mickey		≡ ${\tfrac {1}{200}}$ in	= 1.27 x 10^-4 m
micrometre (old: micron)	μ; μm	≡ 1 x 10^-6 m	≡ 1 x 10^-6 m
mil; thou	mil	≡ 1 x 10^-3 in	= 2.54 x 10^-5 m
mil (Sweden and Norway)	mil	≡ 10 km	= 10 000 m
mile (geographical) (H)		≡ 6082 ft	= 1 853.7936 m
mile (international)	mi	≡ 80 chains ≡ 5280 ft ≡ 1760 yd	≡ 1 609.344 m
mile (tactical or data)		≡ 6000 ft	≡ 1 828.8 m
mile (telegraph) (H)	mi	≡ 6087 ft	= 1 855.3176 m
mile (US Survey)	mi	≡ 5280 US Survey feet ≡ (5280 × ${\tfrac {1200}{3937}}$ ) m	≈ 1 609.347 219 m
nail (cloth)		≡ $2{\tfrac {1}{4}}$ in	= 0.057 15 m
nanometre	nm	≡ 1 x 10^-9 m	≡ 1 x 10^-3 m
nautical league	NL; nl	≡ 3 nmi	= 5556 m
nautical mile (Admiralty)	NM (Adm); nmi (Adm)	= 6080 ft	= 1 853.184 m
nautical mile (international)	NM; nmi	≡ 1852 m	≡ 1852 m
nautical mile (US pre 1954)		≡ 1853.248 m	≡ 1853.248 m
pace		≡ 2.5 ft	= 0.762 m
palm		≡ 3 in	= 0.0762 m
parsec	pc	Distant point with a parallax shift of one arc second from a base of one astronomical unit. ≡ ${\tfrac {648000}{\pi }}$ au	≈ 30 856 775 814 913 700 m
pica		≡ 12 points	Dependent on point measures.
picometre	pm	≡ 1 x 10^-12 m	≡ 1 x 10^-12 m
point (American, English)	pt	≡ ${\tfrac {1}{72.272}}$ in	≈ 0.000 351 450 m
point (Didot; European)	pt	≡ ${\tfrac {1}{12}}\times {\tfrac {1}{72}}$ of pied du roi; After 1878: ≡ ${\tfrac {5}{133}}$ cm	≈ 0.000 375 97 m; After 1878: ≈ 0.000 375 939 85 m
point (PostScript)	pt	≡ ${\tfrac {1}{72}}$ in	= $0.000352{\overline {7}}$ m
point (TeX)	pt	≡ ${\tfrac {1}{72.27}}$ in	= $0.00{\overline {0}}\ {\overline {351}}\ {\overline {4598}}$ m
quarter		≡ ${\tfrac {1}{4}}$ yd	= 0.2286 m
rod; pole; perch (H)	rd	≡ $16{\tfrac {1}{2}}$ ft	= 5.0292 m
rope (H)	rope	≡ 20 ft	= 6.096 m
shaku (Japan)		≡ 10/33 m	≈ 0.303 0303 m
span (H)		≡ 9 in	= 0.2286 m
spat			≡ 1 x 10¹² m
stick (H)		≡ 2 in	= 0.0508 m
toise (French, post 1667) (H)	T	≡ 27000/13853 m	≈ 1.949 0363 m
twip	twp	≡ ${\tfrac {1}{1440}}$ in	= $1.763{\overline {8}}$ x 10^-5 m
x unit; siegbahn	xu		≈ 1.0021 x 10^-13 m
yard (International)	yd	≡ 0.9144 m ≡ 3 ft ≡ 36 in	≡ 0.9144 m
yoctometre	ym	≡ 1 x 10^-24 m	≡ 1 x 10^-24 m
zeptometre	zm	≡ 1 x 10^-21 m	≡ 1 x 10^-21 m