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# International System of Units

## Discussion

### introduction

The creation of the metric system following the complete destruction of the traditional-imperial French unit system marks the beginning of a series of events that eventually lead to the currently accepted International System of Units. The great German mathematician Carl Friedrich Gauss (1777–1855) was the first to promote the idea of combining metric units with the second to form a complete and consistent unit system for mechanics. With the help of the German physicist Wilhelm Weber (1804–1891), he was able to extend this concept to include the units for electricity and magnetism. What came to be known as the Gaussian System of Units arose from this proposal. Its organization served as a model for the International System.

The International System of Units (called Le Système international d’unités in French and abbreviated as SI by international convention) arose during the Eleventh General Conference of Weights and Measures (Conférence générale des poids et mesures or CGPM) conducted by the International Bureau of Weights and Measures (Bureau international des poids et mesures or BIPM) in Paris in 1960. The SI model has three major components.

1. Seven well-defined, dimensionally independent, fundamental units (or base units) that are assumed irreducible by convention (meter, kilogram, second, ampère, kelvin, mole, and candela).
2. A large number of derived units formed by combining fundamental units according to the algebraic relations of the corresponding quantities (some of which are assigned special names and symbols and which themselves can be further combined to form even more derived units).
• The derived units are coherent in the sense that they are all mutually related only by the rules of multiplication and division with no numerical factor other than 1 needed.
• The derived units are also complete in the sense that one and only one unit exists for every defined physical quantity. Although it is possible to express many units in more than one way, they are all equivalent. (The converse statement is not necessarily true, however. Some units are used for more than one physical quantity.)
3. Twenty currently agreed upon prefixes that can be attached to any of the fundamental units or derived units with special names creating multiples and division as needed. (The exception to this rule is the kilogram, which is already itself a multiple of the gram. In this case, prefixes should be added to the word gram.)
• The first three named multiples are the first three powers of ten (101, 102, 103).
Subsequent named multiples are larger than the previous named multiple by three orders of magnitude (106, 109, 1012,… ).
• The first three named divisions are the first three negative powers of ten (10−1, 10−2, 10−3).
Subsequent named divisions are smaller than the previous named division by three orders of magnitude (10−6, 10−9, 10−12,… ).

Each of the three components are described in greater detail below.

### fundamental units

While it is possible for many units to serve as the fundamental building blocks of a system, the following seven were chosen for historical and practical reasons to serve as the fundamental units of the International System of Units.

The meter is the fundamental unit of length in the International System of Units. It was originally defined in 1799 as one ten millionth of the distance from the equator to the north pole as measured on a line of longitude passing through Paris (so that the circumference of the Earth would be 40 million meters). This would then fix the circumference of the Earth at the reasonably convenient value of 40 million meters. There are two problems with this statement, however. First, although nearly spherical to the eye, the Earth is slightly flattened and deviates from the geometric ideal by about one part in three hundred. A trip around the equator would be 134 km longer that a trip around the poles. Second, a small mistake was made when measuring the equator to pole distance through Paris. The length of the actual meter thus differs from its expected length by about one part in four thousand.

In 1889, the BIPM in Paris constructed a precisely machined, platinum-iridium bar with two lines etched at opposite ends. The meter was then defined as the distance between these two lines when stored under specified conditions. Regionally authorized metrologists (measurement technicians) would then travel to BIPM headquarters in France and copy the etched lines of the international prototype on to their own platinum-iridium bars and create regional prototypes. These would then be used to create local prototypes, which would then be used to create individual prototypes, which would then be used to calibrate measuring devices.

Access to the international prototype was strictly controlled so as to keep the wear and tear on it to a minimum and to lessen the possibility of a catastrophic handling error. Given the inaccessibility of the prototype and the possibility of its accidental or intentional destruction a new kind of standard was called for. The definition of the meter based on the international prototype was replaced by a series of definitions based on experiment. The meter is now effectively indestructible as it is possible to reproduced these experiments anywhere in the universe at any time. The original international prototype meter is still kept at the BIPM under conditions specified in 1889, but it is unlikely that it will ever have any official duties again in the future

From 1960 to 1983, the meter was defined as the length equal to 1,650,763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton 86 atom. After 1983, the definition was changed so that the meter [m] became the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. The change made in 1983 meant that lengths could now be measured as accurately as times are measured — time being the most accurately measured of all physical quantities. The current definition had the additional effect of fixing the speed of light in a vacuum at exactly 299,792,458 m/s.

The gram was the fundamental unit of mass in the metric system, but it is the kilogram (one thousand grams) that plays this role in the International System. (A gram was deemed too small to be of much practical use, I suspect.) The kilogram was originally defined in 1799 as the mass of one liter (one thousand cubic centimeters) of pure liquid water at 0 ℃.

In 1889, the BIPM in Paris constructed a precisely machined, platinum-iridium cylinder from the 1799 definition to serve as the international prototype of the kilogram [kg]. Like the international prototype meter (which was constructed at the same time) access to the international prototype kilogram is strictly controlled to reduce wear and tear on it from normal use and to prevent its accidental or intentional destruction. Unlike the meter, however, no reproducible experiment has yet been designed to define the kilogram. If this prototype should somehow become damaged, the definition of the kilogram would itself become damaged. A number of secondary prototypes exist, scattered around the world in regional standards offices. They differ from the international prototype by no more than one part in 109 and could be pressed into service should something happen to the original.

Still, it should be evident that a better standard is needed. Despite all the best efforts of the BIPM, the mass of the international prototype increases by approximately 1 part in 109 per year due to the inevitable accumulation of contaminants on its surface. This error is just at the limit of uncertainty in the measurement used to calibrate the secondary prototypes and is therefore of some significance. For this reason, the international prototype has a mass of one kilogram only after it has been cleaned and washed by a specified method.

It's natural to think of the sun as the timekeeper of our lives, but when compared to contemporary wristwatches and wall clocks, the sun is a pretty terrible clock. Variations in the speed of the Earth as it orbits the sun are great enough that a sundial can disagree with a more conventional clock by as much as 16 minutes and would only be considered accurate four times a year. To correct for this deviation from the ideal, astronomers imagine a fictitious sun that moves across the sky at a constant rate, makes the same number of transits across the sky as the real sun, and is in the same place as the real sun once a year. The time it takes this fictitious sun to make one circuit around the Earth is called a mean solar day. Since there are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a mean solar day, the original definition of the second was 1/86,400 of a mean solar day (24 x 60 x 60 = 86,400).

Natural irregularities in the rotation of the Earth limited the accuracy achievable through this definition, however. The rotation of the Earth varies by about 1 part in 108 between May (when it is slowest) and September (when it is fastest). Furthermore, the rate of the Earth's rotation decreases by about 1 part in 109 per year as energy is drained away by the tidal interactions of the Earth and moon.

Greater accuracies could be achieved from the motion of the Earth around the sun, which is less susceptible to change over the years. The time it takes the sun to make one complete circuit across the sky (from summer solstice to summer solstice, for example) is known as a tropical year. In 1960, the CGPM on advice from the International Astronomical Union declared the second to be 1/31,556,925.9747 of the tropical year. This meant that a day, which formerly defined the second, was now defined by it. A unit day is now 86,400 seconds long by definition.

The current standard, adopted in 1967, utilizes the incredible regularity at which certain atomic transitions take place. A second [s] is now defined as 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. Every elementary particle can be thought of as a tiny bar magnet with a north and south pole. The hyperfine transition of the ground state occurs when the outermost electron of an atom changes its magnetic orientation relative to the nucleus from parallel (pointing in the same direction) to antiparallel (pointing in the opposite direction).

A cesium oscillator, commonly referred to as an atomic clock, is a device that counts the hyperfine transitions made by a collection of cesium atoms. Accuracy of 1 part in 1014 are routinely achieved.

The SI unit of electric current is the ampère [A]

The SI unit of thermodynamic temperature is the kelvin [K].

The SI unit of amount of substance is the mole [mol]. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles or specified groups of such particles.

The SI unit of luminous intensity is the candela [cd].

Fundamental units of the International System
length meter m The meter is the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. (1983)
mass kilogram kg The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. (1889)
time second s The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. (1967)
electric
current
ampère A The ampère is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one meter apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 newton per meter of length. (1948)
thermodynamic
temperature
kelvin K The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (1967)
amount of
substance
mole  mol  The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. (1971)
luminous
intensity
candela cd The candela 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. (1979)

### derived units

A large number of derived units formed by combining base units according to the algebraic relations of the corresponding quantities (some of which are assigned special names and symbols and which themselves can be further combined to form even more derived units).

The derived units are coherent in the sense that they are all mutually related only by the rules of multiplication and division with no numerical factor other than 1 needed.

The derived units are also complete in the sense that one and only one unit exists for every defined physical quantity. Although it is possible to express many units in more than one way, they are all equivalent. The converse statement is not necessarily true, however. Some units are used for more than one physical quantity.

• N m is used for energy (where it is called a joule) and torque
• s−1 is used for frequency (hertz), angular frequency (radian per second), becquerel (decays per second)
• J/kg is used in radiology for absorbed dose (gray) and equivalent dose (sievert)
Derived units of the International System with special names
quantity name symbol in terms of…
other units base units
solid angle steradian sr   m2 m−2
frequency hertz Hz   s−1
force newton N   m kg s−2
pressure, stress pascal Pa N/m2 m−1 kg s−2
energy, work, heat joule J N m m2 kg s−2
power, heat flow watt W J/s m2 kg s−3
electric charge coulomb C   s A
electric potential volt V W/A m2 kg s−3 A−1
capacitance farad F C/V m−2 kg−1 s4 A2
resistance ohm Ω V/A m2 kg s−3 A−2
conductance siemens S A/V m−2 kg−1 s3 A2
magnetic flux weber Wb V s m2 kg s−2 A−1
magnetic flux density tesla T Wb/m2 kg s−2 A−1
inductance henry H Wb/A  m2 kg s−2 A−2
celsius temperature degree celsius   K
luminous flux lumen lm cd sr m2 m−2 cd
illuminance lux lx lm/m2 m−2 cd
absorbed dose gray Gy J/kg m2 s−2
equivalent dose sievert Sv J/kg m2 s−2
catalytic activity katal kat   s−1 mol

### prefixes

Twenty currently agreed upon prefixes that can be attached to any of the fundamental units or derived units with special names creating multiples and division as needed. (The exception to this rule is the kilogram, which is already itself a multiple of the gram. In this case, prefixes should be added to the word gram.)

The first three named mutiples are the first three powers of ten (101, 102, 103). Subsequent named multiples are larger than the previous named multiple by three orders of magnitude (106, 109, 1012,… ).

The first three named divisions are the first three negative powers of ten (10−1, 10−2, 10−3). Subsequent named divisions are smaller than the previous named division by three orders of magnitude (10−6, 10−9, 10−12,… ).

Whether one pronounces giga with hard or soft g depends on what one thinks the proper pronunciation is. A shift occurred in The US sometime in the mid 1990s (at about the time the Internet took off in popular culture) when "gig" replaced "jig" as the preferred pronunciation. BIPM guidelines don't care how any SI terms are pronounced as long as they're always represented with the proper symbol.

Divisions of the International System  unofficial* also norwegian
factor prefix symbol linguistic origin
10−1 deci d latin: ten (decem)
10−2 centi c latin: hundred (centum)
10−3 milli m latin: thousand (mille) 1000−1
10−6 micro µ greek: small (μικρος, mikros) 1000−2
10−9 nano n greek: dwarf (νανος, nanos) 1000−3
10−12 pico p italian: small (piccolo) 1000−4
10−15 femto f danish*: fifteen (femten) 1000−5
10−18 atto a danish*: eighteen (atten) 1000−6
10−21 zepto z greek: seven (επτα, epta) 1000−7
10−24 yocto y greek: eight (οκτω, octo) 1000−8
10−27 xenno x greek: nine (εννεα, ennea) 1000−9
10−30 weko w greek: ten (δεκα, deka) 1000−10
10−33 vendeko v greek: eleven (ενδεκα, endeka) 1000−11
10−36 udodeko u greek: twelve (δωδεκα, dodeka) 1000−12
Multiples of the International System  unofficial
factor prefix symbol linguistic origin
101 deca da greek: ten (δεκα, deka)
102 hecto h greek: hundred (εκατο, ekato)
103 kilo k greek: thousand (χιλια, khilia) 10001
106 mega M greek: big (μεγαλος, megalos) 10002
109 giga G greek: giant (γιγας, gigas) 10003
1012 tera T greek: four (τετρατος, tetratos) 10004
1015 peta P greek: five (πεντε, pente) 10005
1018 exa E greek: six (εξι, exi) 10006
1021 zetta Z greek: seven (επτα, epta) 10007
1024 yotta Y greek: eight (οκτω, octo) 10008
1027 xenna X greek: nine (εννεα, ennea) 10009
1030 weka W greek: ten (δεκα, deka) 100010
1033 vendeka V greek: eleven (ενδεκα, endeka) 100011
1036 udodeka u greek: twelve (δωδεκα, dodeka) 100012

Other scientific, traditional, and practical units and unit systems are still in use and are still useful.

Acceptable non-SI units defined in terms of SI units
quantity unit definition
time minute min 60 s
time hour h 3,600 s
time day d 86,400 s
plane angle degree ° (π/180) rad
plane angle minute ' (π/10,800) rad
plane angle second " (π/648,000) rad
volume liter l 0.001 m3
mass metric ton (tonne) t 1,000 kg
power level difference neper Np 1 W/W
power level difference bel B (0.5 ln 10) Np
Acceptable non-SI units defined experimentally
quantity unit definition
energy electron volt eV 1.602176462(63) × 10−19 J
mass atomic mass unit u 1.66053873(13) × 10−27 kg
distance astronomical unit ua 1.49597870691(6) × 1011 m
Acceptable non-SI units subject to review
quantity unit definition
distance nautical mile   1,852 m
velocity knot kn (1,852/3,600) m/s
area are a 100 m2
pressure bar bar 100 kPa
distance ångström Å 10−10 m
area barn b 100 fm2
exposure roentgen R 258 μC/kg
equivalent dose rem rem 0.01 Sv

units named after people

Derived units of the International System named after scientists * also known as Jhone Neper also spelled roentgen
unit   scientist   quantity
ångström Å Anders Ångström (1814-1874) Sweden distance
becquerel Bq Henri Becquerel (1852-1908) France radioactivity
bel B Alexander Graham Bell (1847-1922) Scotland power level
coulomb C Charles-Augustin Coulomb (1736-1806) France electric charge
curie Ci Marie Curie (1867-1934) Poland radioactivity
degree celsius Anders Celsius (1701-1744) Sweden celsius temperature
gray Gy Louis Gray (1905-1965) England absorbed dose
henry H Joseph Henry (1797-1878) USA inductance
hertz Hz Heinrich Hertz (1857-1894) Germany frequency
joule J James Joule (1818-1889) England energy, work, heat
kelvin K William Thomson, Lord Kelvin (1824-1907) Ireland temperature
neper Np John Napier* (1550-1617) Scotland power level
newton N Isaac Newton (1642-1727) England force
ohm Ω Georg Ohm (1787-1854) Germany resistance
pascal Pa Blaise Pascal (1623-1662) France pressure, stress
röntgen R Wilhelm Röntgen (1845-1923) Germany exposure
siemens S Werner von Siemens (1816-1892) Germany conductance
sievert Sv Rolf Sievert (1896-1966) Sweden equivalent dose
tesla T Nikola Tesla (1856-1943) Croatia magnetic field
volt V Alessandro Volta (1745-1827) Italy electric potential
watt W James Watt (1736-1819) Scotland power, heat flow
weber Wb Wilhelm Weber (1804-1891) Germany magnetic flux