Unit of measurement

Historically many of the systems of measurement which had been in use were to some extent based on the dimensions of the human body. As a result, units of measure could vary not only from location to location, but from person to person.

Metric systems of units have evolved since the adoption of the original metric system in France in 1791. The current international standard metric system is the International System of Units (abbreviated to SI). An important feature of modern systems is standardization. Each unit has a universally recognized size.

An example of metrication in 1860 when Tuscany became part of modern Italy (ex. one "libbra" = 339.54 grams)

Both the imperial units and US customary units derive from earlier English units. Imperial units were mostly used in the British Commonwealth and the former British Empire. US customary units are still the main system of measurement used in the United States outside of science, medicine, many sectors of industry, and some of government and military, and despite Congress having legally authorised metric measure on 28 July 1866.[6] Some steps towards US metrication have been made, particularly the redefinition of basic US and imperial units to derive exactly from SI units. Since the international yard and pound agreement of 1959 the US and imperial inch is now defined as exactly 0.0254 m, and the US and imperial avoirdupois pound is now defined as exactly 453.59237 g.[7]

While the above systems of units are based on arbitrary unit values, formalised as standards, some unit values occur naturally in science. Systems of units based on these are called natural units. Similar to natural units, atomic units (au) are a convenient system of units of measurement used in atomic physics.

Also a great number of unusual and non-standard units may be encountered. These may include the solar mass (2×10³⁰ kg), the megaton (the energy released by detonating one million tons of trinitrotoluene, TNT) and the electronvolt.

To reduce the incidence of retail fraud, many national statutes have standard definitions of weights and measures that may be used (hence "statute measure"), and these are verified by legal officers.

In informal settings, a quantity may be described as multiples of that of a familiar entity, which can be easier to contextualize than a value in a formal unit system. For instance, a publication may describe an area in a foreign country as a number of multiples of the area of a region local to the readership. The propensity for certain concepts to be used frequently can give rise to loosely defined "systems" of units.[8][9]

Any value of a physical quantity is expressed as a comparison to a unit of that quantity. For example, the value of a physical quantity Z is expressed as the product of a unit [Z] and a numerical factor:

${\displaystyle Z=n\times [Z]=n[Z].}$ For example, let ${\displaystyle Z}$ be "2 candlesticks", then ${\displaystyle Z=2[Z]=2}$ candlestick.

The multiplication sign is usually left out, just as it is left out between variables in scientific notation of formulas. The conventions used to express quantities is referred to as quantity calculus. In formulas the unit [Z] can be treated as if it were a specific magnitude of a kind of physical dimension: see dimensional analysis for more on this treatment.

Units can only be added or subtracted if they are the same type; however units can always be multiplied or divided, as George Gamow used to explain. Let ${\displaystyle Z_{1}}$ be "2 candlesticks" and ${\displaystyle Z_{2}}$ "3 cabdrivers", then

"2 candlesticks" times "3 cabdrivers" ${\displaystyle =[Z_{1}][Z_{2}]=6}$ candlestick ${\displaystyle \times }$ cabdriver.

A distinction should be made between units and standards. A unit is fixed by its definition, and is independent of physical conditions such as temperature. By contrast, a standard is a physical realization of a unit, and realizes that unit only under certain physical conditions. For example, the metre is a unit, while a metal bar is a standard. One metre is the same length regardless of temperature, but a metal bar will be exactly one metre long only at a certain temperature.

There are certain rules that have to be used when dealing with units:

• Treat units algebraically. Only add like terms. When a unit is divided by itself, the division yields a unitless one. When two different units are multiplied or divided, the result is a new unit, referred to by the combination of the units. For instance, in SI, the unit of speed is metres per second (m/s). See dimensional analysis. A unit can be multiplied by itself, creating a unit with an exponent (e.g. m²/s²). Put simply, units obey the laws of indices. (See Exponentiation.)
• Some units have special names, however these should be treated like their equivalents. For example, one newton (N) is equivalent to 1 kg⋅m/s². Thus a quantity may have several unit designations, for example: the unit for surface tension can be referred to as either N/m (newton per metre) or kg/s² (kilogram per second squared).

Conversion of units involves comparison of different standard physical values, either of a single physical quantity or of a physical quantity and a combination of other physical quantities.

Starting with:

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

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

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

Now ${\displaystyle n_{i}}$ and ${\displaystyle c_{ij}}$ 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:

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

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

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:

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