Physical unit

In physics and metrology, units are standards for measurement of physical quantities that need clear definitions to be useful. Reproducibility of experimental results is central to the scientific method. To facilitate this we need standards, and to get convenient measures of the standards we need a system of units. Scientific systems of units are a formalization of the concept of weights and measures, initially developed for commercial purposes.

Different systems of units are based on different choices of a set of fundamental units. The most widely used system of units is the international system, or SI system, of units derived from the seven SI base units. All other SI units can be derived from these base units.

Other systems of units that have been used for various purposes include:

• the centimeter-gram-second system of units
• the Planck units
• U.S. customary units
• Imperial units

Units as dimensions

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 Q is written as the product of a unit [Q] and a numerical factor:

Q = n * [Q] = n [Q]

The multiplication sign is usually left out, just as it is left out between variables in scientific notation of formulas. In formulas the unit [Q] can be treated as if it was a kind of physical dimension: see dimensional analysis for more on this treatment.

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 one metre long only at a certain temperature.

Basic and derived units

For most quantities a unit is absolutely necessary to communicate values of that physical quantity. Try for example to tell someone the value of a length without the use of a unit. That is not possible because you can't verbally describe a length.

But not all quantities require a unit of their own. Using physical laws, units of quantities can be expressed as combinations of units of other quantities. Thus only a small set of units is required. These units are taken as the basic units. Other units are derived units. Derived units are a matter of convenience, as they can be expressed in terms of basic units. Which units are considered basic is a matter of choice.

The basic units of SI are actually not the smallest set. Smaller sets have been defined. There are sets in which the electric and magnetic field have the same unit. This is based on physical laws that show that electric and magnetic field are actually different manifestations of the same phenomenon. In some fields of science such systems of units are highly favored over the SI system.

Conversion of units

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.

Thus conversion factors between units are always imprecise to some level and improved values may be found when a more precise comparison is performed.

Prefixes in the SI system

In the SI system some letters denoting conveniently chosen numerical values can be used as prefixes to any of the units.

For example, c = 0.01, and thus cm = 0.01 * m and cN = 0.01 * N

There is one exception: for historical reasons, the unit of mass, kg, already contains a prefix and prefixes are not to be added to it but to g. Thus: mg and not µkg (with "µ" = "micro-"). To many this is a source of mistakes and frustration; see Talk.

Use of prefixes does not involve any unit conversion, as the prefixes are just defined as numerical values. They can not be imprecise.

For example, the expressions 'cm' and '0.01 m' mean mathematically exactly the same thing. It is not a unit conversion, just a mathematical conversion, just like '4 * 5' and '20' are mathematical expressions with the same meaning.

Calculations with units

Hints:

• Use formulas involving physical values whenever possible, be reluctant to split up physical values into units and numerical values, as you increase the complexity by a factor of two!

• If you calculate the value of a physical quantity A from a formula involving a combination of other physical quantities (B, C, D), you don't have to calculate the resulting unit: if you just convert all values of B, C, D so as to be expressed in SI units (no prefixes), the resulting unit is the SI unit of the quantity A. The SI system is set up to ensure this convenience. Don't use the gram instead of the kilogram, because naturally that will not work!

• Don't let definitions like density is mass per unit volume obscure your understanding of units. It sounds as if it says:

  D = m / [V] (WRONG)

  This is not true. The correct statement is that density is mass divided by volume:

  D = m / V

  The sentence `density is mass per unit volume' uses another way of perceiving the concept. It says that the density D_s of system s is the mass m_u of a subsystem u of s, divided by the volume V_u of subsystem u, given that the volume of subsystem u is unit volume:

  D_s = m_u / V_u

  V_u = 1 [V]

Mathematical rules for calculations with units follow from the formula for physical values, Q = n * [Q]

• Values of the same quantity can of course always be added, but not by just adding their numerical values. The numerical value is not all of the value of the physical quantity.

  The units in the physical values have to be converted so that they are the same. Then the numerical values can be added. The same principle is known from adding fractions: you have to make the denominators the same and then you can add the numerators.

• When a unit is divided by itself, the division yields a unitless 1.

• When two different units are multiplied by each other, the result is a new unit. For instance, in SI, the unit of momentum is one kilogram multiplied by one meter divided by one second. See also dimensional analysis.

• Expressing a physical value in terms of another unit:

  Starting with:

  Q = n_i * [Q]_i

  just replace the origional unit [Q]_i with its meaning in terms of the desired unit [Q]_f, e.g. if [Q]_i = c_ij * [Q]_f, then:

  Q = n_i * c_ij * [Q]_f

  Now n_i and c_ij are both numerical values, so just calculate their product.

  Or, which is just mathematically the same thing, multiply Q by unity, the product is still Q:

  Q = n_i * [Q]_i * ( c_ij * [Q]_f/[Q]_i )

  For example, you have an expression for a physical value Q involving the unit feet per second ([Q]_i) and you want it in terms of the unit miles per hour ([Q]_f):

  1. Find facts relating the original unit to the desired unit:

     1 mile = 5280 feet and 1 hour = 3600 seconds

  2. Next use the above equations to construct a fraction that has a value of unity and that contains units such that, when it is multiplied with the original physical value, will cancel the original units:

     1 = (1 mile) / (5280 feet) and 1 = (3600 seconds) / (1 hour)

  3. Last, multiply the original expression of the physical value by the fraction, called a conversion factor, to obtain the same physical value expressed in terms of a different unit. Note: since the conversion factors have a numerical value of unity, multiplying any physical value by them will not change that value.

See also: units unit conversion computer program