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
- Chinese unit

Table of contents |

2 Basic and derived units 3 Conversion of units 4 Prefixes in the SI system 5 Calculations with units |

## Units as dimensions

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

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

## 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

(Note: for the purposes of this section, variables are written in*italics*, while units are normal text.)

Guidelines:

- Treat units like variables. Only add like terms. When a unit is divided by itself, the division yields a unitless one. When two different units are multiplied, the result is a new unit, referred to by the combination of the units. For instance, in SI, the unit of speed is meters per second (m/s). See dimensional analysis. A unit can be multiplied by itself, creating a unit with an exponent (ex: m
^{2}). - Some units have special names, however these should be treated like their equivalents. For example, one Newton (N) is equivalent to one kg m/s
^{2}. This creates the possiblity for units with multiple designations, for example: the unit for surface tension can be referred to as either N/m (Newtons per meter) or kg/s^{2}(kilograms per seconds squared). - Don't let definitions like
*density is mass per unit volume*obscure your understanding of units. It sounds as if it says:

*m*/ V (mass divided by the

*unit*of volume) (WRONG)

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

- D =
*m*/*V*(mass divided by volume, both variables)

- Expressing a physical value in terms of another unit:

- Q = n_i * [Q]_i

*e.g.*if [Q]_i = c_ij * [Q]_f, then:

- Q = n_i * c_ij * [Q]_f

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 )

*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

- 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)

- 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.

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