The Physics
Hypertextbook
Opus in profectus

# International System of Units

## Discussion

### overview

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 four major components.

1. Seven defining constants (or reference constants) with exact values. These constants may be universal constants that arise from fundamental physical laws (Planck constant h, Boltzmann constant k, speed of light c), be connected to natural phenomena (speed of light c, hyperfine transition of a cesium atom ΔνCs, charge on a proton e), or have evolved from previous definitions of the base units (Avogadro constant NA, luminous efficacy of green light Kcd).
2. Seven dimensionally independent, base units (or fundamental units) defined in terms of the defining constants that are assumed irreducible by convention (second, meter, kilogram, coulomb, kelvin, mole, and candela).
3. 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 derived units are used for more than one physical quantity.)
4. Twenty currently agreed upon prefixes that can be attached to any of the base 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 these components are described in greater detail below. ### Defining constants

The defining constants listed below are used to build up the base units of the International System.

hyperfine transition
The unperturbed ground state hyperfine transition frequency of the cesium 133 atom (ΔνCs) is defined to be exactly 9,192,631,770 hertz. This definition can be used to to define the second [s] since the hertz [Hz] is an inverse second [Hz = 1/s]. The second is the base unit of time in the SI.
speed of light
The speed of electromagnetic radiation in vacuum (c) is defined to be exactly 299,792,458 meters per second [m/s]. This definition combined with the definition of the second can be used to define the meter [m], which is the base unit of length or distance in the SI.
Planck's constant
Planck's constant (h) is defined to be exactly 6.62607015 × 10−34 joule seconds, where the joule [J] is a unit of energy equivalent to a kilogram meter squared per second squared [kg m2/s2]. Planck's constant is therefore a universal constant that relates mass, length, and time. This definition combined with the definitions of the second and the meter can be used to define the kilogram [kg], which is the base unit of mass in the SI.
elementary charge
The elementary charge (e) is defined to be exactly 1.602176634 × 10−19 coulombs. The elementary charge is the magnitude of the charge on many subatomic particles. For example, the charge on a proton is +1 e and the charge on an electron is −1 e. This definition combined with the definition of a second can be used to define the ampère [A] since a coulomb [C] is the amount of charge transfered by one ampère of current in one second [C = A s]. The ampère is the base unit of electric current in the SI.
Boltzmann's constant
Boltzmann's constant (k) is defined to be exactly 1.380649 × 10−23 joule per kelvin. Boltzmann's constant is a universal constant that relates energy to temperature. Since the joule is related to the SI units of mass, length, and time [J = kg m2/s2] Boltzmann's constant can be used to define the kelvin [K], which is the base unit of temperature in the SI.
Avogadro's constant (NA) is defined to be exactly 6.02214076 × 1023 particles per mole. Avogadro's constant evolved over the history of chemistry and is not a universal constant. (I would call it a cultural constant.) It is a way to count the microscopic particles of chemistry and statistical thermodynamics (particles being things like molecules, atoms, ions, electrons, etc.). The unit that evolved out of this is the mole [mol], which is the base unit of number of particles in the SI.
luminous efficacy
The luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz (Kcd) is defined to be exactly 683 lumen per watt. The luminous efficacy is another example of a constant that evolved out of the cultural practice of humans doing science. It is a way to quantify how well a light source produces visible light — a concept similar to efficiency. Efficiency is for machines using energy to do work. Efficacy is for light sources using energy to make light. The watt [W] is a unit of power derived from the kilogram, meter, and second [W = kg m2/s3]. The lumen is a unit of luminous flux derived from the candela and the steradian [lm = cd sr]. A steradian is a unit of solid angle that is a unitless ratio of areas [m2/m2]. This simplified but still tortuous chain of logic can be used to define the candela [cd], which is the base unit of luminous intensity in the SI.
The seven defining constants of the International System
symbol value description
ΔνCs 9,192,631,770 Hz the unperturbed ground state hyperfine transition frequency of the cesium 133 atom
c 299,792,458 m/s the speed of light in vacuum
h 6.62607015 × 10−34 J s Planck's constant
e 1.602176634 × 10−19 C the elementary charge
k 1.380649 × 10−23 J/K Boltzmann's constant
NA 6.02214076 × 1023/mol Avogadro's constant
Kcd 683 lm/W the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz

### base 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 base units of the International System of Units.

#### time

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 not the world's best 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 186,400 of a mean solar day (24 × 60 × 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 tidal interactions between the Earth and the Moon.

Magnify

Magnify

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 is known as a tropical year. In 1960, the CGPM on advice from the International Astronomical Union declared the second to be 131,556,925.9747 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. 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 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.

Despite the changes made to the International System in 2018, this last definition is now worded differently, but is still practically true.

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. Accuracies of 1 part in 1014 are routinely achieved.

#### length

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. 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 at the poles and deviates from the geometric ideal by about one part in three hundred. A trip around the equator is 134 km longer that a trip around the poles. Second, a small mistake was made when measuring the equator to pole distance through Paris. Thus the length of the actual meter 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 travel to BIPM headquarters and copy the etched lines of the international prototype on to their own platinum-iridium bars to 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 practical 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 its experimental definition anywhere in the universe at any time. The original international prototype meter is still enshrined at the BIPM under conditions specified in 1889, but it is unlikely that it will ever have any official duties again.

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.

In 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 1299,792,458 of a second. The change meant that lengths could now be measured as accurately as times are measured — time being the most accurately measured of all physical quantities. This had the additional effect of fixing the speed of light in a vacuum at exactly 299,792,458 m/s.

Despite the changes made to the International System in 2018, this last definition is now worded differently, but is still practically true.

#### mass

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 °C.

a mass equal to the mass of one liter of pure water under atmospheric pressure and at the temperature of its maximum density, which is approximately 4 °C.

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. 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 from normal use and to prevent its accidental or intentional destruction. 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.

Despite all the best efforts of the BIPM, the mass of the international prototype kilogram 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 had a mass of one kilogram only after it has been cleaned and washed by a specified method.

new def

#### charge

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

#### temperature

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

#### amount

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.

#### intensity

The SI unit of luminous intensity is the candela [cd]. The lumen [lm] is a unit of luminous flux, which is a measure of the total visible light output from some source over all directions. This is the hard part. How much that light is concentrated in any particular direction is called luminous intensity. The amount of spreading is measured using the solid angle unit called the steradian [sr], which is a measure of the fraction of a sphere cut out by a solid angle. A steradian is then a ratio of areas with units that cancel out [m2/m2]. One lumen spread out over one steradian is called a candela [cd = lm/sr], which is the base unit of luminous intensity in the SI.

Base units of the International System
quantity unit definition
time second s The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the cesium frequency νCs, the unperturbed ground-state hyperfine transition frequency of the cesium 133 atom, to be 9,192,631,770 when expressed in the unit Hz, which is equal to s−1.
length meter m The meter, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299,792,458 when expressed in the unit m/s, where the second is defined in terms of νCs.
mass kilogram kg The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015 × 10−34 when expressed in the unit J s, which is equal to kg m2s−1, where the meter and the second are defined in terms of c and νCs.
electric
current
ampère A The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602176634 × 10−19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of νCs.
thermodynamic
temperature
kelvin K The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380649 × 10−23 when expressed in the unit J K−1, which is equal to kg m2s−2K−1, where the kilogram, meter, and second are defined in terms of h, c, and νCs
amount of
substance
mole  mol  The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.022140769 × 1023 elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit mol−1 and is called the Avogadro number. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle, or specified group of particles.
luminous
intensity
candela cd The candela, symbol cd, is the SI unit of luminous intensity in a given direction. It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz, Kcd, to be 683 when expressed in the unit lm W−1, which is equal to cd sr W−1, or cd sr kg−1m−2s3, where the kilogram, meter, and second are defined in terms of h, c, and νCs.

### 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 (where it is called a newton meter)
• 1/s is used for frequency (cycles per second or hertz), angular frequency (radian per second), and 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
frequency hertz Hz   1/s
force newton N   kg m/s2
pressure, stress pascal Pa N/m2 kg/m s2
energy, work, heat joule J N m kg m2/s2
power, heat flow watt W J/s kg m2/s3
electric charge coulomb C   A s
electric potential volt V W/A kg m2/A s3
capacitance farad F C/V A2 s4/kg m2
resistance ohm Ω V/A kg m2/A2 s3
conductance siemens S A/V A2 s3/kg m2
magnetic flux weber Wb V s kg m2/A s2
magnetic flux density tesla T Wb/m2 kg/A s2
inductance henry H Wb/A kg m2/A2 s2
Celsius temperature degree Celsius °C   K
luminous flux lumen lm cd sr cd m2/m2
illuminance lux lx lm/m2 cd/m2
activity becquerel Bq   1/s
absorbed dose gray Gy J/kg m2/s2
equivalent dose sievert Sv J/kg m2/s2
catalytic activity katal kat   mol/s

### prefixes

Twenty currently agreed upon prefixes that can be attached to any of the base 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 * 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 spanish: small (pico) 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
Multiples of the International System
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

Other scientific, traditional, and practical units and unit systems are still in use and are still useful. The Anglo-American system of units still in official use in the United States today is actually just an extension of the International System. Many of the units that are unique to the system now have definitions that refer to their SI counterparts. An inch of distance is exactly 0.0245 m, for example, and a pound of mass is exactly 0.45359237 kg.

Acceptable non-SI units defined in terms of SI units * also known as a metric ton in the US
quantity unit definition
time minute min 60 s
time hour h 3,600 s
time day d 86,400 s
distance astronomical unit au 149,597,870,700 m
plane angle degree ° (π/180) rad
plane angle minute ' (π/10,800) rad
plane angle second " (π/648,000) rad
area hectare ha 10,000 m2
volume liter L 0.001 m3
mass tonne* t 1,000 kg
energy electron volt eV 1.602176634 × 10−19 J
Acceptable non-SI units defined experimentally * also known as the dalton [Da]
quantity unit definition
mass atomic mass unit* u 1.660539040(20) × 10−27 kg
Other acceptable non-SI units
quantity unit definition
natural logarithmic ratio bel B log(x/x0)
decadic logarithmic ratio neper Np ln(x/x0)

Units named after people…

Derived units of the International System named after scientists
unit scientist quantity
becquerel Bq Henri Becquerel activity
bel B Alexander Graham Bell level
coulomb C Charles-Augustin Coulomb electric charge
degree Celsius °C Anders Celsius celsius temperature
gray Gy Louis Gray absorbed dose
henry H Joseph Henry inductance
hertz Hz Heinrich Hertz frequency
joule J James Joule energy, work, heat
kelvin K William Thomson, Lord Kelvin temperature
neper Np John Napier level
newton N Isaac Newton force
ohm Ω Georg Ohm resistance
pascal Pa Blaise Pascal pressure, stress
siemens S Werner von Siemens conductance
sievert Sv Rolf Sievert equivalent dose
tesla T Nikola Tesla magnetic field
volt V Alessandro Volta electric potential
watt W James Watt power, heat flow
weber Wb Wilhelm Weber magnetic flux