Metric system
Definition
The metric system is an internationally adopted decimal system of measurement. It is in widespread use, and where it is used, it is the only or most common system of weights and measures. It is now known as the International System of Units (SI). It is used to measure everyday things such as the mass of a sack of flour, the height of a person, the speed of a car, and the volume of fuel in its tank. It is also used in science, industry and trade.
In its modern form, it consists of a set of electromechanical base units including metre for length, kilogram for mass, second for time and ampere as an electrical unit, and a few others, which together with their derived units, can measure any useful quantity. Metric system may also refer to other systems of related base and derived units defined before the middle of the 20th century, some of which are still in limited use today.
The metric system was designed to have properties that make it easy to use and widely applicable, including units based on the natural world, decimal ratios, prefixes for multiples and sub-multiples, and a structure of base and derived units. It is also a coherent system, which means that its units do not introduce conversion factors not already present in equations relating quantities. It has a property called rationalisation that eliminates certain constants of proportionality in equations of physics.
The units of the metric system, originally taken from observable features of nature, are now realised by synthetic phenomena such as the microwave frequency of a caesium atomic clock which accurately measures seconds. One unit, the kilogram, remains defined in terms of a man-made artefact.
While there are numerous named derived units of the metric system, such as watt and lumen, other common quantities such as velocity and acceleration do not have their own unit, but are defined in terms of existing base and derived units such as metres per second for velocity.
Though other currently or formerly widespread systems of weights and measures continue to exist, such as the British imperial system and the US customary system of weights and measures, in those systems some or all of the units are now defined in terms of the metric system, such as the US foot which is now a defined decimal fraction of a metre.
The metric system is also extensible, and new base and derived units are defined as needed in fields such as radiology and chemistry. The most recent derived unit was added in 1999. Recent changes are directed toward defining base units in terms of invariant constants of physics to provide more precise realisations of units for advances in science and industry.
Units
Base units
The modern metric system consists of four electromechanical base units representing four fundamental dimensions of measure: length, mass, time and electromagnetism. The units are:
- the metre for length
- kilogram for mass
- second for time
- ampere for electromagnetism
Together they are sufficient for measuring any known quantity, without reference to further quantities or phenomena.
Three supplemental base units have been defined, but these are not independent since they can be specified entirely in terms of the above four base units: the kelvin, a thermodynamic measure; the candela, a measure of irradiance; and the mole, representing a quantity of substance.
Derived units with special names
There are currently 22 derived units with special names in the metric system, these are defined in terms of the base units or other named derived units.
Eight of these units are electromagnetic quantities:
- volt, a unit of electrical potential
- ohm, a unit of electrical resistance
- tesla, a unit of magnetic flux density
- weber, a unit of magnetic flux
- farad, a unit of electrical capacitance
- henry, a unit of electrical inductance
- siemens, a unit of electrical conductance (the inverse of ohm)
- coulomb, a unit of electrical charge
Four of these units are mechanical quantities:
- watt, a unit of mechanical or electrical power
- newton, a unit of mechanical force
- joule, a unit of mechanical, electrical or thermodynamic energy
- pascal, a unit of pressure
Five units represent measures of electromagnetic radiation:
- becquerel, a unit of radioactive decay
- sievert, a unit of absorbed ionising radiation
- gray, a unit of ionising radiation
- lux, a unit of luminous flux
- lumen, a unit of luminous intensity
Two units are measures of circular arcs and spherical surfaces:
- radian, a unit of circular arc
- steradian, a unit of spherical surface area
Three units are miscellaneous:
- degree Celsius, a unit of thermodynamic temperature
- katal, a unit of catalytic activity (enzymatic)
- hertz, a unit of cycles per second (inverse of second)
Auxiliary and accessory units
Although SI, as published by the CGPM, should, in theory, meet all the requirements of commerce, science, and technology, certain customary units of measure have acquired established positions within the world community. In order that such units are used consistently around the world, the CGPM catalogued such units in Tables 6 to 9 of the SI brochure. These categories are:
- Non-SI units accepted for use with the International System of Units (Table 6). This list includes the hour and minute, the angular measures (degree, minute and second of arc), and the historic [non-coherent] metric units, the litre, tonne and hectare (originally agreed by the CGPM in 1879)
- Non-SI units whose values in SI units must be obtained experimentally (Table 7). This list includes various units of measure used in atomic and nuclear physics and in astronomy such as the dalton, the electron mass, the electron volt, the astronomical unit, the solar mass, and a number of other units of measure that are well-established, but dependent on experimentally-determined physical quantities.
- Other non-SI units (Table 8). This list catalogues a number of units of measure that have been used internationally in certain well-defined spheres including the bar for pressure, the ångström for atomic physics, the nautical mile and the knot in navigation.
- Non-SI units associated with the CGS and the CGS-Gaussian system of units (Table 9). This table catalogues a number of units of measure based on the CGS system and dating from the nineteenth century. They appear frequently in the literature, but their continued use is discouraged by the CGPM.
The SI symbols for the metric units are intended to be identical, regardless of the language used but unit names are ordinary nouns and use the character set and follow the grammatical rules of the language concerned. For example, the SI unit symbol for kilometre is "km" everywhere in the world, even though the local language word for the unit name may vary. Language variants for the kilometre unit name include: chilometro (Italian), Kilometer (German), kilometer (Dutch), kilomètre (French), χιλιόμετρο (Greek), quilómetro/quilômetro (Portuguese), kilómetro (Spanish) and километр (Russian).
Variations are also found with the spelling of unit names in countries using the same language, including differences in American English and British spelling. For example, meter and liter are used in the United States whereas metre and litre are used in other English-speaking countries. In addition, the official US spelling for the rarely used SI prefix for ten is deka. In American English the term metric ton is the normal usage whereas in other varieties of English tonne is common. Gram is also sometimes spelled gramme in English-speaking countries other than the United States, though this older usage is declining.
In SI, which is a coherent system, the unit of power is the "watt", which is defined as "one joule per second". In the US customary system of measurement, which is non-coherent, the unit of power is the "horsepower", which is defined as "550 foot-pounds per second" (the pound in this context being the pound-force). Similarly, neither the US gallon nor the imperial gallon is one cubic foot or one cubic yard— the US gallon is 231 cubic inches and the imperial gallon is 277.42 cubic inches.
The concept of coherence was only introduced into the metric system in the third quarter of the 19th century; in its original form the metric system was non-coherent—in particular the litrewas 0.001 m and the are (from which the hectare derives) was 100 m. However the units of mass and length were related to each other through the physical properties of water, the gram having been designed as being the mass of one cubic centimetre of water at its freezing point.
Realisation of units
The base units used in the metric system must be realisable. Each of the definitions of the base units in SI is accompanied by a defined mise en pratique[practical realisation] that describes in detail at least one way in which the base unit can be measured. Where possible, definitions of the base units were developed so that any laboratory equipped with proper instruments would be able to realise a standard without reliance on an artefact held by another country. In practice, such realisation is done under the auspices of a mutual acceptance arrangement (MAA).
The standard metre is defined as exactly 1/299,792,458 of the distance that light travels in a second. The realisation of the metre depends in turn on precise realisation of the second. There are both astronomical observation methods and laboratory measurement methods that are used to realise units of the standard metre. Because the speed of light is now exactly defined in terms of the metre, more precise measurement of the speed of light does not result in a more accurate figure for its velocity in standard units, but rather a more accurate definition of the metre. The accuracy of the measured speed of light is considered to be within 1 m/s, and the realisation of the metre is within about 3 parts in 1,000,000,000, or an order of 10 parts.
The kilogram is defined by the mass of a man-made artefact of platinum-iridium held in a laboratory in France. Replicas made in 1879 at the time of the artefact's fabrication and distributed to signatories of the Metre Convention serve as de facto standards of mass in those countries. Additional replicas have been fabricated since as additional countries have joined the convention. The replicas are subject to periodic validation by comparison to the original, called the IPK. It has become apparent that either the IPK or the replicas or both are deteriorating, and are no longer comparable: they have diverged by 50 μg since fabrication, so figuratively, the accuracy of the kilogram is no better than 5 parts in a hundred million or within an order of 10 parts.
Properties as a system
Although the metric system has changed and developed since its inception, its basic concepts have hardly changed. Designed for transnational use, it consisted of a basic set of units of measurement, now known as base units. Derived units were built up from the base units using logical rather than empirical relationships while multiples and submultiples of both base and derived units were decimal-based and identified by a standard set of prefixes.
Units based on the natural world
Like most units of measure, the units of the metric system were based on perceptual quantities of the natural world. But they also had definitions in terms of stable relationships in that world: a metre was defined not by the span of a man's arms like a toise, but on a quantitative measure of the earth. A kilogram was defined by a volume of water, whose linear dimensions were fractions of the unit of length. The earth was not easy to measure, nor was it uniformly shaped, but the principle that units of measure were to be based on quantitative relationships among invariant facets of the physical world was established. The units of the metric system today still adhere to that principle, but the relationships used are based on the physics of nature, rather than its sensory dimensions.
Base and derived unit structure
The metric system base units were originally adopted because they represented fundamental orthogonal dimensions of measurement corresponding to how we perceive nature: a spacial dimension, a time dimension, one for the effect of gravitation, and later, a more subtle one for the dimension of an "invisible substance" known as electricity or more generally, electromagnetism. One and only one unit in each of these dimensions was defined, unlike older systems where multiple perceptual quantities with the same dimension were prevalent, like inches, feet and yards or ounces, pounds and tons. Units for other quantities like area and volume, which are also spacial dimensional quantities, were derived from the fundamental ones by logical relationships, so that a unit of square area for example, was the unit of length squared.
Many derived units were already in use before and during the time the metric system evolved, because they represented convenient abstractions of whatever base units were defined for the system, especially in the sciences. So analogous units were scaled in terms of the metric units, and their names adopted into the system. Many of these were associated with electromagnetism. Other perceptual units, like volume, which were not defined in terms of base units, were incorporated into the system with definitions in the metric base units, so that the system remained simple. It grew in number of units, but the system retained a uniform structure.
Decimal ratios
Some customary systems of weights and measures had duodecimal ratios, which meant quantities were conveniently divisible by 2, 3, 4, and 6. But it was difficult to do arithmetic with things like ⁄4 pound or ⁄3 foot. There was no system of notation for successive fractions: for example, ⁄3 of ⁄3 of a foot was not an inch or any other unit. But the system of counting in decimal ratios did have notation, and the system had the algebraic property of multiplicative closure: a fraction of a fraction, or a multiple of a fraction was a quantity in the system, like ⁄10 of ⁄10which is ⁄100. So a decimal radix became the ratio between unit sizes of the metric system.
Prefixes for multiples and submultiples
In the metric system, multiples and submultiples of units follow a decimal pattern.
| Metric prefixes in everyday use | |||
|---|---|---|---|
| Text | Symbol | Factor | Power |
| yotta | Y | 1000000000000000000000000 | 10 |
| zetta | Z | 1000000000000000000000 | 10 |
| exa | E | 1000000000000000000 | 10 |
| peta | P | 1000000000000000 | 10 |
| tera | T | 1000000000000 | 10 |
| giga | G | 1000000000 | 10 |
| mega | M | 1000000 | 10 |
| kilo | k | 1000 | 10 |
| hecto | h | 100 | 10 |
| deca | da | 10 | 10 |
| (none) | (none) | 1 | 10 |
| deci | d | 0.1 | 10 |
| centi | c | 0.01 | 10 |
| milli | m | 0.001 | 10 |
| micro | μ | 0.000001 | 10 |
| nano | n | 0.000000001 | 10 |
| pico | p | 0.000000000001 | 10 |
| femto | f | 0.000000000000001 | 10 |
| atto | a | 0.000000000000000001 | 10 |
| zepto | z | 0.000000000000000000001 | 10 |
| yocto | y | 0.000000000000000000000001 | 10 |
A common set of decimal-based prefixes that have the effect of multiplication or division by an integer power of ten can be applied to units that are themselves too large or too small for practical use. The concept of using consistent classical (Latin or Greek) names for the prefixes was first proposed in a report by the French Revolutionary Commission on Weights and Measures in May 1793. The prefix kilo, for example, is used to multiply the unit by 1000, and the prefix milli is to indicate a one-thousandth part of the unit. Thus the kilogram and kilometre are a thousand grams and metres respectively, and a milligram and millimetre are one thousandth of a gram and metre respectively. These relations can be written symbolically as:
In the early days, multipliers that were positive powers of ten were given Greek-derived prefixes such as kilo- and mega-, and those that were negative powers of ten were given Latin-derived prefixes such as centi- and milli-. However, 1935 extensions to the prefix system did not follow this convention: the prefixes nano- and micro-, for example have Greek roots. During the 19th century the prefix myria-, derived from the Greek word μύριοι (mýrioi), was used as a multiplier for 10000.
When applying prefixes to derived units of area and volume that are expressed in terms of units of length squared or cubed, the square and cube operators are applied to the unit of length including the prefix, as illustrated below.
| 1 mm (square millimetre) | = (1 mm) | = (0.001 m) | = 0.000001 m |
| 1 km (square kilometre) | = (1 km) | = (1000 m) | = 1000000 m |
| 1 mm (cubic millimetre) | = (1 mm) | = (0.001 m) | = 0.000000001 m |
| 1 km (cubic kilometre) | = (1 km) | = (1000 m) | = 1000000000 m |
Prefixes are not usually used to indicate multiples of a second greater than 1; the non-SI units of minute, hour and day are used instead. On the other hand, prefixes are used for multiples of the non-SI unit of volume, the litre (l, L) such as millilitres (ml).
Coherence
Each variant of the metric system has a degree of coherence—the derived units are directly related to the base units without the need for intermediate conversion factors. For example, in a coherent system the units of force, energy and power are chosen so that the equations
| force | = | mass | × | acceleration |
| energy | = | force | × | distance |
| energy | = | power | × | time |
hold without the introduction of unit conversion factors. Once a set of coherent units have been defined, other relationships in physics that use those units will automatically be true. Therefore, Einstein's mass–energy equation, E = mc, does not require extraneous constants when expressed in coherent units.
The CGS system had two units of energy, the erg that was related to mechanics and the calorie that was related to thermal energy; so only one of them (the erg) could bear a coherent relationship to the base units. Coherence was a design aim of SI, which resulted in only one unit of energy being defined – the joule.
Rationalisation
Maxwell's equations of electromagnetism contained a factor relating to steradians, representative of the fact that electric charges and magnetic fields may be considered to emanate from a point and propagate equally in all directions, i.e. spherically. This factor appeared awkwardly in many equations of physics dealing with the dimensionality of electromagnetism and sometimes other things.
International System of Units
The International System of Units is the modern metric system. It is based on the Metre-Kilogram-Second-Ampere (MKSA) system of units from early in the 20th century. It also includes numerous coherent derived units for common quantities like power (watt) and irradience (lumen). Electrical units were taken from the International system then in use. Other units like those for energy (joule) were modeled on those from the older CGS system, but scaled to be coherent with MKSA units. Two additional base units, degree Kelvin equivalent to degree Centigrade for thermodynamic temperature, and candela, roughly equivalent to the international candle unit of illumination, were introduced. Later, another base unit, the mole, a unit of mass equivalent to Avogadro's number of specified molecules, was added along with several other derived units.
The system was promulgated by the General Conference on Weights and Measures (French: Conférence générale des poids et mesures – CGPM) in 1960. At that time, the metre was redefined in terms of the wavelength of a spectral line of the krypton-86 atom, and the standard metre artefact from 1889 was retired.
Today, the International system of units consists of 7 base units and innumerable coherent derived units including 22 with special names. The last new derived unit, the katal for catalytic activity, was added in 1999. Some of the base units are now realised in terms of invariant constants of physics. As a consequence, the speed of light has now become an exactly defined constant, and defines the metre as ⁄299,792,458 of the distance light travels in a second. The kilogram remains defined by a man-made artefact of platinum-iridium, and it is deteriorating. The range of decimal prefixes has been extended to those for 10, yotta, and 10, yocto, which are unfamiliar because nothing in our everyday lives is that big or that small.
The International System of Units has been adopted as the official system of weights and measures by all nations in the world except for Myanmar, Liberia, and the United States, while the United States is the only industrialised country where the metric system is not the predominant system of units.
Historical variants
A number of variants of the metric system evolved, all using the Mètre des Archives and Kilogramme des Archives (or their descendants) as their base units, but differing in the definitions of the various derived units.
| Variants of the metric system | ||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Gaussian second and the first mechanical system of units
In 1832, Gauss used the astronomical second as a base unit in defining the gravitation of the earth, and together with the gram and millimetre, became the first system of mechanical units.
The EMU, ESU and Gaussian systems of electrical units
Several systems of electrical units were defined following discovery of Ohm's law in 1824.
Centimetre–gram–second systems
The centimetre–gram–second system of units (CGS) was the first coherent metric system, having been developed in the 1860s and promoted by Maxwell and Thomson. In 1874, this system was formally promoted by the British Association for the Advancement of Science(BAAS). The system's characteristics are that density is expressed in g/cm, force expressed in dynes and mechanical energy in ergs. Thermal energy was defined in calories, one calorie being the energy required to raise the temperature of one gram of water from 15.5 °C to 16.5 °C. The meeting also recognised two sets of units for electrical and magnetic properties – the electrostatic set of units and the electromagnetic set of units.
International system of electrical units
The CGS units of electricity were cumbersome to work with. This was remedied at the 1893 International Electrical Congress held in Chicago by defining the "international" ampere and ohm using definitions based on the metre, kilogram and second.
MKS and MKSA systems
In 1901, Giovanni Giorgi showed that by adding an electrical unit as a fourth base unit, the various anomalies in electromagnetic systems could be resolved. The metre–kilogram–second–coulomb (MKSC) and metre–kilogram–second–ampere (MKSA) systems are examples of such systems.
The International System of Units (Système international d'unités or SI) is the current international standard metric system and is also the system most widely used around the world. It is an extension of Giorgi's MKSA system—its base units are the metre, kilogram, second, ampere, kelvin, candela and mole. The MKS (Metre, Kilogram, Second) system came into existence in 1889, when artefacts for the metre and kilogram were fabricated according to the convention of the Metre. Early in the 20th century, an unspecified electrical unit was added, and the system was called MKSX. When it became apparent that the unit would be the ampere, the system was referred to as the MKSA system, and was the direct predecessor of the SI.
Metre–tonne–second systems
The metre–tonne–second system of units (MTS) was based on the metre, tonne and second – the unit of force was the sthène and the unit of pressure was the pièze. It was invented in France for industrial use and from 1933 to 1955 was used both in France and in the Soviet Union.
Gravitational systems
Gravitational metric systems use the kilogram-force (kilopond) as a base unit of force, with mass measured in a unit known as the hyl, Technische Mass Einheit (TME), mug or metric slug.Although the CGPM passed a resolution in 1901 defining the standard value of acceleration due to gravity to be 980.665 cm/s, gravitational units are not part of the International System of Units (SI).
Conversion and calculation incidents
The dual usage of or confusion between metric and non-metric units has resulted in a number of serious incidents. These include:
- Flying an overloaded American International Airways aircraft from Miami, Florida to Maiquetia, Venezuela on 26 May 1994. The degree of overloading was consistent with ground crew reading the kilogram markings on the cargo as pounds.
- In 1999 the Institute for Safe Medication Practices reported that confusion between grains and grams led to a patient receiving phenobarbital 0.5 grams instead of 0.5 grains (0.03 grams) after the practitioner misread the prescription.
- The Canadian "Gimli Glider" accident in 1983, when a Boeing 767 jet ran out of fuel in mid-flight because of two mistakes made when calculating the fuel supply of Air Canada's first aircraft to use metric measurements: mechanics miscalculated the amount of fuel required by the aircraft as a result of their unfamiliarity with metric units.
- The root cause of the loss in 1999 of NASA's US$125 million Mars Climate Orbiter was a mismatch of units – the spacecraft engineers calculated the thrust forces required for velocity changes using US customary units (lbf⋅s) whereas the team who built the thrusters were expecting a value in metric units (N⋅s) as per the agreed specification.
Conversion table
During its evolution, the metric system has adopted many units of measure. The introduction of SI rationalised both the way in which units of measure were defined and also the list of units in use. These are now catalogued in the official SI Brochure. The table below lists the units of measure in this catalogue and shows the conversion factors connecting them with the equivalent units that were in use on the eve of the adoption of SI.
| Quantity | Dimension | SI unit and symbol | Legacy unit and symbol | Conversion factor |
|---|---|---|---|---|
| Time | T | second (s) | second (s) | 1 |
| Length | L | metre (m) | centimetre (cm) ångström (Å) | 0.01 10 |
| Mass | M | kilogram (kg) | gram (g) | 0.001 |
| Electric current | I | ampere (A) | international ampere abampere or biot statampere | 1.000022 10.0 3.335641×10 |
| Temperature | Θ | kelvin (K) degree Celsius (°C) | centigrade (°C) | [K] = [°C] + 273.15 1 |
| Luminous intensity | J | candela (cd) | international candle | 0.982 |
| Amount of substance | N | mole (mol) | No legacy unit | n/a |
| Area | L | square metre (m) | are (a) | 100 |
| Acceleration | LT | (m⋅s) | gal (gal) | 10 |
| Frequency | T | hertz (Hz) | cycles per second | 1 |
| Energy | LMT | joule (J) | erg (erg) | 10 |
| Power | LMT | watt (W) | (erg/s) horsepower (hp) Pferdestärke (PS) | 10 745.7 735.5 |
| Force | LMT | newton (N) | dyne (dyn) sthene (sn) kilopond (kp) | 10 10 9.80665 |
| Pressure | LMT | pascal (Pa) | barye (Ba) pieze (pz) atmosphere (at) | 0.1 10 1.01325×10 |
| Electric charge | IT | coulomb (C) | abcoulomb statcoulomb or franklin | 10 3.335641×10 |
| Potential difference | LMTI | volt (V) | international volt abvolt statvolt | 1.00034 10 2.997925×10 |
| Capacitance | LMTI | farad (F) | abfarad statfarad | 10 1.112650×10 |
| Inductance | LMTI | henry (H) | abhenry stathenry | 10 8.987552×10 |
| Electric resistance | LMTI | ohm (Ω) | international ohm abohm statohm | 1.00049 10 8.987552×10 |
| Electric conductance | LMTI | siemens (S) | international mho (℧) abmho statmho | 0.99951 10 1.112650×10 |
| Magnetic flux | LMTI | weber (Wb) | maxwell (Mx) | 10 |
| Magnetic flux density | MTI | tesla (T) | gauss (G) | 10 |
| Magnetic field strength | IL | (A/m) | oersted (Oe) | 10⁄4π = 79.57747 |
| Dynamic viscosity | MLT | (Pa⋅s) | poise (P) | 0.1 |
| Kinematic viscosity | LT | (m⋅s) | stokes (St) | 10 |
| Luminous flux | J | lumen (lm) | stilb (sb) | 10 |
| Illuminance | JL | lux (lx) | phot (ph) | 10 |
| [Radioactive] activity | T | becquerel (Bq) | curie (Ci) | 3.70×10 |
| Absorbed [radiation] dose | LT | gray (Gy) | roentgen (R) rad (rad) | ≈0.01 0.01 |
| Radiation dose equivalent | LT | sievert | roentgen equivalent man (rem) | 0.01 |
| Catalytic activity | NT | katal (kat) | enzyme unit(U) | 1/60μkat |
The SI Brochure also catalogues certain non-SI units that are widely used with the SI in matters of everyday life or units that are exactly defined values in terms of SI units and are used in particular circumstances to satisfy the needs of commercial, legal, or specialised scientific interests. These units include:
| Quantity | Dimension | Unit and symbol | Equivalence |
|---|---|---|---|
| Mass | M | tonne (t) | 1000 kg |
| Area | L | hectare (ha) | 0.01 km 10 m |
| Volume | L | litre (L or l) | 0.001 m |
| Time | T | minute (min) hour (h) day (d) | 60 s 3600 s 86400 s |
| Pressure | LMT | bar | 100 kPa |
| Plane angle | none | degree (°) minute (′) second (″) | (⁄180) rad (⁄10800) rad (⁄648000) rad |