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# Metric System -- International System of Units (SI)

In 1960, SI was selected as a specific subset of the existing Metre-Kilogram-Second systems of units (MKS), rather than the older Centimetre-Gram-Second system (CGS). Various new units were added with the introduction of the SI and at later times. SI is sometimes referred to as the metric system, especially in the United States, which has not widely adopted it, although it has been used more commonly in recent years, and in the UK, where conversion is only partial. SI is a specific canon of measurements derived and extended from the Metric system; however, not all metric units of measurement are accepted as SI units.

## History

Scientists, chiefly in France, had been advocating and discussing a decimal system of measurement based on natural units at least since 1640, but the first official adoption of such a system was after the French Revolution of 1789. The metric system tried to choose units which were non-arbitrary, while practical, merging well with the revolution's official ideology of "pure reason"; it was proposed as a considerable improvement over the inconsistent customary units which existed before, whose value often depended on the region.

The most important unit is that of length: one metre was originally defined to be equal to 1/10 000 000th of the distance from the pole to the equator along the meridian through Paris. (Prior discussions had often suggested the length of a seconds pendulum in some standard gravity, which would have been only slightly shorter, and perhaps easier to determine.) This is approximately 10% longer than one yard. Later on, a platinum rod with a rigid, X-shaped cross section was produced to serve as the easy-to-check standard for one metre's length. Due to the difficulty of actually measuring the length of a meridian quadrant in the 18th century, the first platinum prototype was short by 0.2 millimetres. More recently, the metre was redefined as a certain multiple of a specific radiation wavelength, and currently it is defined as the distance travelled by light in a vacuum in a specific period of time. Attempts to relate an integer multiple of the metre to any meridian have been abandoned.

The original base unit of mass in the metric system was the gram, but this was soon changed to the kilogram, which was defined as the mass of 0.001 m3 of distilled pure water at its densest (+3.98 degrees Celsius). One kilogram is about 2.2 pounds. 0.001 m3 was also defined as one litre, so that volumes could easily be compared using a more convenient unit than the very large cubic metre. By 1799, a platinum cylinder was manufactured to serve as the standard for a kilogram, so no water-based standard ever served as the primary standard when the metric system was actually used anywhere. In 1890, this was replaced by a cylinder of a 90% platinum, 10% iridium alloy which has served as the standard ever since and is stored in a Paris vault. The kilogram is the only base unit not to have been redefined in terms of an unchanging natural phenomenon. However, at meeting of the Royal Society in London on 15 February 2005, scientists called for the mass of the standard kilogramme in Paris to be replaced as the official definition arguing that "an invariable property of nature" should be used (rather than a material object whose mass can change slightly), but no decision on redefinition can be taken before 2007.

The unit of temperature became the centigrade or inverted Celsius grade, which means the mercury scale is divided into 100 equal length parts between the water-ice mixture and the boiling point of pure, distilled water. Boiling water thus becomes one hundred degrees Celsius and freezing is zero degrees Celsius. This is the metric unit of temperature in everyday use. A hundred years later, scientists discovered absolute zero. This prompted the establishment of a new temperature scale, an absolute scale which relocates the zero point at absolute zero. It was named the Kelvin scale and was scaled with the aim of maintaining 100 kelvins between the freezing point and boiling point of water.

The metric unit of time became the second, originally defined as 1/86 400th of a mean solar day. The formal definition of the second has been changed several times for enhanced scientific requirements (astronomic observations, tuning fork clock, quartz clock and then caesium atomic clock) but wristwatch users remain relatively unaffected.

The swift worldwide adoption of the metric system as a tool of economy and everyday commerce was based mainly on the lack of customary systems in many countries to adequately describe some concepts, or as a result of an attempt to standardize the many regional variations in the customary system. International factors also affected the adoption of the metric system, as many countries increased their trade. Scientifically, it provides ease when dealing with very large and small quantities because it lines up so well with our decimal numeral system.

Cultural differences can be represented in the local everyday uses of metric units. For example, bread is sold in one-half, one or two kilogram sizes in many countries, but you buy them by multiples of one hundred grams in the former USSR. In some countries, the informal cup measurement has become 250 mL, and prices for items are sometimes given per 100 g rather than per kilogram. A profound cultural difference between physicists and engineers, especially radio engineers, existed prior to the adoption of the MKS system and hence its descendent, SI. Engineers work with volts, amperes, ohms, farads, and coulombs, which are of great practical utility, while the cgs units, fine for theoretical physics (because of the absence of fabricated "constants" like the "permittivity of the vacuum") are quite inconvenient for engineering usage and totally unfamiliar to householders using appliances rates in volts and watts.

Non-scientific people should not be put off by the fine-tuning that has happened to the metric base units over the past two hundred years, as experts regularly tried to refine the metric system to fit the best scientific researcher (e.g. CGS to MKS to SI system changes or the invention of Kelvin scale). These changes do not affect the everyday use of metric units. The presence of these adjustments has been one reason advocates of the U.S. customary units have used against metrication; these customary units, however, are nowadays defined in terms of SI units, thus any difference in the definition of the SI units results in a difference of the definition of the customary units.

The SI was first given its name in 1960, and last added to in 1971. It is administered by the standards organization: the Bureau International des Poids et Mesures (International Bureau of Weights and Measures).

## Basis

SI consists of seven base units and several derived units, together with a set of prefixes.

The seven SI base units are: the kilogram, metre, second, ampere, kelvin, mole, and candela.

SI also defines a number of SI prefixes to be used with the units: these combine with any unit name to give subdivisions and multiples. For example, the prefix kilo denotes a multiple of a thousand, so the kilometre is 1000 metres, the kilogram 1000 grams, and so on. The prefixes are never chained, so a millionth of a kilogram is a milligram, and not a 'microkilogram'.

## SI writing style

• Symbols are written in lower case, except for symbols derived from the name of a person. For example, the unit of pressure is named after Blaise Pascal, so its symbol is written "Pa" whereas the unit itself is written "pascal". The one exception is the litre, whose offical abbreviation is "L". "l" is commonly used, but is deprecated for being too similar to "1".
• Symbols are written in singular form: i.e. "25 kg", not "25 kgs".
• Symbols do not have an appended period (.).
• It is preferable to write symbols in upright Roman type (m for metres, L for litres), so as to differentiate from the italic type used for mathematical variables (m for mass, l for length).
• A space should separate the number and the symbol, e.g. "2.21 kg", "7.3·102 m2". Exceptions are the symbols for plane angular degrees, minutes and seconds (°, 2 and 3), which are placed immediately after the number with no intervening space.
• Spaces should be used to group decimal digits in threes, e.g. 1 000 000 or 342 142 (in contrast to the commas or dots used in other systems, e.g. 1,000,000 or 1.000.000).
• In English, the decimal point should be written as the full stop, i.e. the number "twenty four point five one" would be written as "24.51". (This was introduced by the CIPM in 1997). In all other languages, the comma is used instead, (i.e. "24,51").
• Symbols for derived units formed from multiple units by multiplication are joined with a space or centre dot (·), e.g. N m or N·m.
• Symbols formed by division of two units are joined with a solidus (/), or given as a negative exponent. For example, the "metre per second" can be written "m/s", "m s-1", "m·s-1" or $\frac{\mbox{m}}{\mbox{s}}$. A solidus should not be used if the result is ambiguous, i.e. "kg·m-1·s-2" is preferable to "kg/m/s2".

With a few exceptions (such as draught beer sales in the United Kingdom) the system can legally be used in every country in the world and many countries do not maintain definitions of other units. Those countries that still give official recognition to non-SI units (e.g. the US and UK) have defined many of the modern units in terms of SI units; for example, the common yard is defined to be exactly 0.9144 metres. In the US, survey distances are also defined in terms of metric units, but differently: 1 survey yard = 3600/3937 m. They have, however, not been redefined due to the accumulation of error it would entail and the survey foot and survey mile remain as separate units. (This was not a problem for the United Kingdom, as the Ordnance Survey has been metric since before World War II.) (See weights and measures for a history of the development of units of measurement.)

## Units

### Base units

The following are the fundamental units from which all others are derived, they are dimensionally independent. The definitions stated below are widely accepted.

SI Base units

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NameSymbolQuantityDefinition
kilogramkgMassThe unit of mass is equal to the mass of the international prototype kilogram (a platinum-iridium cylinder) kept at the Bureau International des Poids et Mesures (BIPM), Sèvres, Paris (1st CGPM (1889), CR 34-38). Note that the kilogram is the only base unit with a prefix; the gram is defined as a derived unit, equal to 1/1000 of a kilogram; prefixes such as mega are applied to the gram, not the kg; e.g. Gg, not Mkg. It is also the only unit still defined by a physical prototype instead of a measurable natural phenomenon (see the kilogram article for an alternate definition).
secondsTimeThe unit of time is the duration of exactly 9 192 631 770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the caesium-133 atom at a temperature of 0 K (13th CGPM (1967-1968) Resolution 1, CR 103).
metremLengthThe unit of length is equal to the length of the path travelled by light in a vacuum during the time interval of 1/299 792 458 of a second (17th CGPM (1983) Resolution 1, CR 97).
ampereAElectrical currentThe unit of electrical current is the constant current which, if maintained in two straight parallel conductors, of infinite length and negligible cross-section, placed 1 metre apart in a vacuum, would produce a force between these conductors equal to 2×10 "7 newtons per metre of length (9th CGPM (1948) Resolution 7, CR 70).
kelvinKThermodynamic temperatureThe unit of thermodynamic temperature (or absolute temperature) is the fraction 1/273.16 (exactly) of the thermodynamic temperature at the triple point of water (13th CGPM (1967) Resolution 4, CR 104).
molemolAmount of substanceThe unit of amount of substance is the amount of substance which contains as many elementary entities as there are atoms in 0.012 kilograms of pure carbon-12 (14th CGPM (1971) Resolution 3, CR 78). (Elementary entities may be atoms, molecules, ions, electrons, or particles.) It is approximately equal to 6.02214199×1023 units (Avogadro's number).
candelacdLuminous intensityThe unit of luminous intensity is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian (16th CGPM (1979) Resolution 3, CR 100).

### Dimensionless units

The following SI units are dimensionless, and therefore do not require the definition of a base unit.

SI Dimensionless units

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NameSymbolQuantityDefinition
radianradAngleThe unit of angle is the angle subtended at the centre of a circle by an arc of the circumference equal in length to the radius of the circle. There are 2À radians in a circle.
steradiansrSolid angleThe unit of solid angle is the solid angle subtended at the centre of a sphere of radius r by a portion of the surface of the sphere having an area r2. There are 4À steradians on a sphere.

### Derived units with special names

Base units can be put together to derive units of measurement for other quantities. Some have been given names.

SI derived units with special names

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NameSymbolQuantityExpressed in base units
hertzHzFrequencys"1
newtonNForcekg m s "2
jouleJEnergyN m = kg m2 s"2
wattWPowerJ/s = kg m2 s"3
pascalPaPressure, StressN/m2 = kg m "1 s"2
lumenlmLuminous fluxcd sr
luxlxIlluminancelm/m2 = cd sr m"2
coulombCElectric chargeA s
voltVElectrical potential differenceW/A = J/C = kg m2 A"1 s"3
ohmElectric resistanceV/A = kg m2 A"2 s"3
faradFElectric capacitanceC/V = A2 s4 kg"1 m"2
weberWbMagnetic fluxkg m2 s"2 A"1
teslaTMagnetic flux densityWb/m2 = kg s"2 A"1
henryHInductanceWb/A = kg m2 A"2 s"2
siemensSElectric conductance"1 = kg"1 m"2 A2 s3
grayGyAbsorbed dose (of ionising radiation)J/kg = m2 s"2
sievertSvEquivalent dose (of ionising radiation)J/kg = m2 s"2
katalkatCatalytic activitymol/s = mol s"1
degree Celsius°CThermodynamic temperatureK (0 °C = 273.15 K, 0 K = "273.15 °C)

### Non-SI units accepted for use with SI

The following units are not SI units but are "accepted for use with the International System."

Non-SI units accepted for use with SI

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NameSymbolQuantityEquivalent SI unit
minutemintime1 min = 60 s
hourhtime1 h = 60 min = 3600 s
daydtime1 d = 24 h = 1440 min = 86400 s
degree of arc°angle1° = (À/180) rad
minute of arc 2angle1 2 = (1/60)° = (À/10800) rad
second of arc 3angle1 3 = (1/60) 2 = (1/3600)° = (À/648000) rad
litrel or Lvolume0.001 m3
tonnetmass1 t = 103 kg

Non-SI units not formally adopted by the CGPM

neper, field quantityNpratio (dimensionless)LF = ln(F/F0) Np
neper, power quantityNpratio (dimensionless)LP = ½ ln(P/P0) Np
bel, field quantityBratio (dimensionless)LF = 2 log10(F/F0) B
bel, power quantityBratio (dimensionless)LP = log10(P/P0) B

Empirical non-SI units accepted for use with SI

electronvolteVenergy1 eV = 1.60217733 (49) × 10"19 J
atomic mass unitumass1 u = 1.6605402 (10) × 10"27 kg
astronomical unitAUlength1 AU = 1.49597870691 (30) × 1011 m

Other Non-SI units currently accepted for use with SI

nautical milenautical milelength1 nautical mile = 1852 m
knotknotspeed1 knot = 1 nautical mile per hour = (1852/3600) m/s
areaarea1 a = 1 dam2 = 100 m2
hectarehaarea1 ha = 100 a = 10000 m2
barbarpressure1 bar = 105 Pa
ångström, angstromÅlength1 Å = 0.1 nm = 10"10 m
barnbarea1 b = 10"28 m2

### SI prefixes

The following SI prefixes can be used to prefix any of the above units to produce a multiple or submultiple of the original unit.

SI prefixes
10nPrefixSymbolShort scaleLong scaleDecimal Equivalent
1024yottaYSeptillionQuadrillion1 000 000 000 000 000 000 000 000
1021zettaZSextillionTrilliard (thousand trillion)1 000 000 000 000 000 000 000
1018exaEQuintillionTrillion1 000 000 000 000 000 000
1015petaPQuadrillionBilliard (thousand billion)1 000 000 000 000 000
1012teraTTrillionBillion1 000 000 000 000
109gigaGBillionMilliard (thousand million)1 000 000 000
106megaMMillion1 000 000
103kilokThousand1 000
102hectohHundred100
100nonenoneOne1
10"1decidTenth0.1
10"2centicHundredth0.01
10"3millimThousandth0.001
10"6microµMillionth0.000 001
10"9nanonBillionthMilliardth0.000 000 001
10"12picopTrillionthBillionth0.000 000 000 001
10"18attoaQuintillionthTrillionth0.000 000 000 000 000 001
10"21zeptozSextillionthTrilliardth0.000 000 000 000 000 000 001
10"24yoctoySeptillionthQuadrillionth0.000 000 000 000 000 000 000 001

### Obsolete metric prefixes

The following metric prefixes are no longer in use: myria-, myrio-, and any double prefixes such as those formerly used in micromicrofarads, hectokilometres, millimicrons.

See: Obsolete metric prefixes

## Spelling variations

Several nations, notably the United States, typically use the spellings 'meter' and 'liter' instead of 'metre' and 'litre'. This is in keeping with standard American English spelling (for example, Americans also use 'center' rather than 'centre,' using the latter only rarely for its stylistic implications; see also American and British English differences). In addition, the official US spelling for the SI prefix 'deca' is 'deka'.

The US government has approved these spellings for official use, but the BIPM only recognizes the British English spellings as official names for the units. In scientific contexts only the symbols are used; since these are universally the same, the differences do not arise in practice in scientific use.

The unit 'gram' is also sometimes spelled 'gramme' in English-speaking countries other than the United States, though that is an older spelling and use is declining.

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