# SI Units

SI Units are the most widely used system of units. They are the most common system for everyday commerce in the world, and are almost universally used in the realm of science. The name derives from the French phrase, Système International d'Unités, or in English International System of Units. The system consists of a set of seven base units together with a set of prefixes from which all other units are derived.

The emergence of an internationally recognized system of units at a time of increasing international cooperation and trade is highly significant. It has provided a necessary common base for the scientific, technical, and industrial exchange that has fostered a growing consciousness of the need to approach issues from a global perspective.

## History

The metric system was officially adopted in France after the French Revolution. During the history of the metric system, a number of variations have evolved and their use spread around the world replacing many traditional measurement systems.

By the end of World War II a number of different systems of measurement were still in use throughout the world. Some of these systems were metric system variations while others were based on the Imperial and American systems. It was recognized that additional steps were needed to promote a worldwide measurement system. As a result, the 9th General Conference on Weights and Measures (CGPM) in 1948, asked the International Committee for Weights and Measures (CIPM) to conduct an international study of the measurement needs of the scientific, technical, and educational communities.

Based on the findings of this study, the 10th CGPM in 1954 decided that an international system should be derived from six base units to provide for the measurement of temperature and optical radiation in addition to mechanical and electromagnetic quantities. The six base units recommended were the meter, kilogram, second, ampere, Kelvin (later renamed kelvin), and the candela. In 1960, the 11th CGPM named the system the International System of Units. The seventh base unit, the mole, was added in 1970 by the 14th CGPM.

SI units are still sometimes referred to as the metric system, especially in the United States, whose population has not widely adopted it, and in the United Kingdom, where conversion is only partial. SI units are a specific canon of measurements derived and extended from the Metric system; however, not all metric units of measurement are accepted as SI units. This international system of units is now either obligatory or permissible throughout the world. It is administered by the standards organization: the Bureau International des Poids et Mesures (International Bureau of Weights and Measures, BIPM).

## 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.

 Name Symbol Measure Definition kilogram kg Mass The 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). second s Time The 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). meter (or metre) m Length The 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). ampere A Electrical current The 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 meter apart in a vacuum, would produce a force between these conductors equal to 2×10 −7 newtons per meter of length (9th CGPM (1948) Resolution 7, CR 70). kelvin K Thermodynamic temperature The 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). mole mol Quantity of matter (mass/mass) A mole is the quantity of substance that contains the same number of elementary entities (atoms, molecules, ions, electrons or particles, depending on the substance) as there are atoms in 0.012 kilograms of pure carbon-12 (14th CGPM (1971) Resolution 3, CR 78). This number (NA) is approximately equal to 6.02214199×1023. candela cd Luminous intensity The 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).

### Length

The most important unit is that of length: one meter 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 percent 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 meter's length. Due to the difficulty of actually measuring the length of a meridian quadrant in the eighteenth century, the first platinum prototype was short by 0.2 millimeters. More recently, the meter was redefined as a certain multiple of a specific radiation wavelength, and currently it is defined as the distance traveled by light in a vacuum in a specific period of time. Attempts to relate an integer multiple of the meter to any meridian have been abandoned.

### Mass

The original base unit of mass in the metric system was the gram, chosen to match the mass of one cubic centimeter of water. For practical reasons, the reference standard that was deposited at the Archives de la république on June 22, 1799, was a kilogram (a cylinder of platinum). One kilogram is about 2.2 pounds. In 1889, the first General Conference on Weights and Measures (CGPM) sanctioned a replacement prototype, a cylinder of a 90 percent platinum, 10 percent iridium alloy; this has served as the standard ever since and is stored in a Paris vault. The kilogram became the base unit in 1901.

Also in 1901, a kilogram of distilled pure water at its densest (+3.98° C) under a standard atmosphere of pressure was used to define the liter, a more convenient unit than the very large cubic meter. Because this liter turned out to be different from the cubic decimeter by about 28 millionths, this definition was abandoned in 1964 in favor of the cubic decimeter.

The kilogram is the only base unit not to have been redefined in terms of an unchanging natural phenomenon. Such a definition, said to be in terms of an artefact (the cylinder in Paris), is particularly inconvenient, because, in principle, it can be used only by traveling to Paris and, with permission, comparing one's own candidate standard to the reference one. For this reason, as well as the effort required to protect the standard from absorption or dispersion of gases and vapors, at a meeting of the Royal Society in London on February 15, 2005, scientists called for the mass of the standard kilogramme in Paris to be replaced by a standard based on "an invariable property of nature"; but no decision on redefinition can be taken before 2007.

### Temperature

The metric unit of temperature originally was the centigrade or Celsius scale. This was determined by divided into 100 equal-length parts the difference between a water-ice mixture at 0° C and the boiling point of pure distilled water at 100° C (under a standard atmosphere). This is still the metric unit of temperature in everyday use. With the discovery of absolute zero, a new temperature scale, the Kelvin Scale was developed, which relocates the zero point to absolute zero. The freezing point of water, 0° C, becomes 273.15 K. It is the Kelvin scale that is used as the base SI unit.

### Time

The metric unit of time is the second. It was originally defined as 1/86,400th of a mean solar day. The formal definition of the second has been changed several times as more accurate definitions became possible, based first on astronomic observations, then the tuning fork clock, quartz clock, and today the cesium atomic clock.

## SI prefixes

The SI system of units is a metric system. That is, the units are expressed in powers of 10 (i.e. 1×10 3 ). In order to simplify writing the powers of ten, they are expressed as prefixes and the symbol for the prefix put before the unit. Thus, 7.4×10 3 m is written as 7.4 km.

The following SI prefixes can be used to prefix any of the units to produce a multiple or submultiple of the original unit. This includes the degree Celsius (e.g. "1.2 m°C"); however, to avoid confusion, prefixes are not used with the time-related unit symbols min (minute), h (hour), d (day). They are not recommended for use with the angle-related symbols ° (degree), ' (minute of arc), and " (second of arc) [1], but for astronomical usage, they are sometimes used with seconds of arc.

10n Prefix Symbol Short scale* Long scale** Decimal equivalent
1024
yotta
Y
Septillion
1 000 000 000 000 000 000 000 000
1021
zeta
Z
Sextillion
Trilliard (thousand trillion)
1 000 000 000 000 000 000 000
1018
exa
E
Quintillion
Trillion
1 000 000 000 000 000 000
1015
peta
P
Billiard (thousand billion)
1 000 000 000 000 000
1012
tera
T
Trillion
Billion
1 000 000 000 000
109
giga
G
Billion
Milliard (thousand million)
1 000 000 000
106
mega
M
Million
1 000 000
103
kilo
k (K)
Thousand
1 000
102
hector
h (H)
Hundred
100
101
deca, deka
da (D)
Ten
10
100
none
none
One
1
10−1
deci
d
Tenth
0.1
10−2
centi
c
Hundredth
0.01
10−3
milli
m
Thousandth
0.001
10−6
micro
µ (u)
Millionth
0.000 001
10−9
nano
n
Billionth
Milliardth
0.000 000 001
10−12
pico
p
Trillionth
Billionth
0.000 000 000 001
10−15
femto
f
Billiardth
0.000 000 000 000 001
10−18
atto
a
Quintillionth
Trillionth
0.000 000 000 000 000 001
10−21
zepto
z
Sextillionth
Trilliardth
0.000 000 000 000 000 000 001
10−24
yocto
y
Septillionth
0.000 000 000 000 000 000 000 001

* Short scale is the English translation of the French term échelle courte, which designates a system of numeric names in which the word billion means a thousand millions.

** Long scale is the English translation of the French term échelle longue, which designates a system of numeric names in which the word billion means a million millions.

### 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, hectokilometers, millimicrons.

## Derived units

Units for the measurement of other quantities (such as pressure or electrical charge) are formed by combining base units. These types of units are thus called derived units. Some derived units have been given special names as shown in the table below.

 Name Symbol Quantity Expression in terms of other units Expression in terms of SI base units hertz Hz Frequency s−1 s−1 newton N Force, Weight m·kg·s−2 m·kg·s−2 joule J Energy, Work, Heat N•m m2·kg·s−2 watt W Power, Radiant flux J/s m2·kg·s−3 pascal Pa Pressure, Stress N/m2 m−1·kg·s−2 lumen lm Luminous flux cd·sr = m2·m−2·cd cd lux lx Illuminance lm/m2 = m2·m−4·cd m−2·cd coulomb C Electric charge or flux s·A s·A volt V Electrical potential difference, Electromotive force W/A = J/C m2·kg·s−3·A−1 ohm Ω Electric resistance, Impedance, Reactance V/A m2·kg·s−3·A−2 farad F Electric capacitance C/V m−2·kg−1·s4·A2 weber Wb Magnetic flux m2·kg·s−2·A−1 m2·kg·s−2·A−1 tesla T Magnetic flux density, Magnetic inductivity V•s•m−2 = Wb/m2 kg·s−2·A−1 henry H Inductance V•s•A−1 = Wb/A m2·kg·s−2·A−2 siemens S Electric conductance Ω−1 m−2·kg−1 s3·A2 becquerel Bq Radioactivity (decays per unit time) s−1 s−1 gray Gy Absorbed dose (of ionising radiation) J/kg m2·s−2 sievert Sv Equivalent dose (of ionising radiation) J/kg m2·s−2 katal kat Catalytic activity mol/s s−1·mol degree Celsius °C Thermodynamic temperature t°C = tK - 273.15 molarity M Concentration (moles of substance per liter of solution) mol/L 103•m−3•mol molality m Concentration (moles of substance per kilogram of solution) mol/kg kg−1•mol

### Dimensionless derived units

The following SI units are actually dimensionless ratios, formed by dividing two identical SI units. They are therefore considered by the BIPM to be derived. Formally, their SI unit is simply the number 1, but they are given these special names for use whenever the lack of a unit might be confusing.

 Name Symbol Quantity Definition radian rad Angle The unit of angle is the angle subtended at the center of a circle by an arc of the circumference equal in length to the radius of the circle. There are ${\displaystyle 2\pi }$ radians in a circle. steradian sr Solid angle The unit of solid angle is the solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere having an area r2. There are ${\displaystyle 4\pi }$ steradians on a sphere.

## Other Units

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

edit

Non-SI units accepted for use with SI

Name Symbol Quantity Equivalent SI unit
minute min time 1 min = 60 s
hour h time 1 h = 60 min = 3600 s
day d time 1 d = 24 h = 1440 min = 86400 s
degree of arc ° angle 1° = (π/180) rad
minute of arc angle 1′ = (1/60)° = (π/10800) rad
second of arc angle 1″ = (1/60)′ = (1/3600)° = (π/648000) rad
liter l or L volume 1dm3 = 0.001 m3
tonne t mass 1 t = 103 kg

Non-SI units not formally adopted by the CGPM

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

Non-SI units with values obtained only by experiment

electronvolt eV energy 1 eV = 1.60217733 (49) × 10−19 J
atomic mass unit u mass 1 u = 1.6605402 (10) × 10−27 kg
astronomical unit AU length 1 AU = 1.49597870691 (30) × 1011 m

Non-SI units whose use is not encouraged

nautical mile   length 1 nautical mile = 1852 m
knot   speed 1 knot = 1 nautical mile per hour = (1852/3600) m/s
are a area 1 a = 1 dam2 = 100 m2
hectare ha area 1 ha = 100 a = 10000 m2
bar bar pressure 1 bar = 105 Pa
ångström, angstrom Å length 1 Å = 0.1 nm = 10−10 m
barn b area 1 b = 10−28 m2

## 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 liter, whose original abbreviation "l" is dangerously similar to "1." The NIST recommends that "L" be used instead, a usage which is common in the U.S., Canada, and Australia, and has been accepted as an alternative by the CGPM. The cursive "ℓ" is occasionally seen, especially in Japan, but this is not currently recommended by any standards body.
• Symbols are written in singular form: i.e. "25 kg", not "25 kgs." Pluralization would be language dependent; "s" plurals (as in French and English) are particularly undesirable since "s" is the symbol of the second.
• Symbols do not have an appended period (.).
• It is preferable to write symbols in upright Roman type (m for meters, L for liters), 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," "22 °C" [2]. Exceptions are the symbols for plane angular degrees, minutes, and seconds (°, ′ and ″), 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).
• The 10th resolution of CGPM in 2003 declared that "the symbol for the decimal marker shall be either the point (period) on the line or the comma on the line." In practice, the period is used in [American] English, and the comma in most other European languages.
• Symbols for derived units formed from multiple units by multiplication are joined with a space or center 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 "meter per second" can be written "m/s," "m s-1," "m·s-1," or ${\displaystyle {\frac {\mbox{m}}{\mbox{s}}}}$. 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 is legally being used in every country in the world and many countries do not maintain definitions of other units. Those countries that still give recognition to non-SI units (e.g. the U.S. 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 meters. In the U.S., 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.)

## Cultural issues

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 the 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, which are fine for theoretical physics can be inconvenient for electrical engineering usage and are largely unfamiliar to householders using appliances rated 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 200 years, as experts have tried frequently to refine the metric system to fit the best scientific research (e.g. CGS to MKS to SI system changes or the invention of the 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.

## References

• Mills, I., Tomislav Cvitas, Klaus Homann, Nikola Kallay, IUPAC. Quantities, Units and Symbols in Physical Chemistry, 2nd ed. Blackwell Science, 1993. ISBN 0632035838.