Physical Quantities and SI Units


The International System of Units (abbreviated as SI Units from its French name, Système International d'unités) is an internationally agreed metric system of units of measurement that has been in existence since 1960. The history of the metre and the kilogram, two of the fundamental units on which the system is based, goes back to the French Revolution. The system itself is based on the concept of seven fundamental base units of quantity, from which all other units of quantity can be derived. Following the end of the Second World War, it became increasingly apparent that a worldwide system of measurement was needed to replace the numerous and diverse systems of measurements in use at that time. In 1954, the 10th General Conference on Weights and Measures, acting on the findings of an earlier study, proposed a system based on six base quantities. The quantities recommended were the metre, kilogram, second, ampere, kelvin and candela.

The General Conference on Weights and Measures (abbreviated as CGPM from its French title, Conférence Générale des Poids et Mesures), the first of which took place in 1889, has taken place every few years since 1897 in Sèvres, near Paris. Following the 1954 proposals, the conference of 1960 (the 11th CGPM) introduced the new system to the world.

A seventh base unit, the mole, was added following the 14th CGPM, which took place in 1971. An official description of the system called the SI Brochure, first published in 1970 and currently (as of 2019) in its ninth edition, can be downloaded free of charge from the website of the Bureau International des Poids et Mesures (BIPM). The brochure is written and maintained by a subcommittee of the International Committee for Weights and Measures (abbreviated as CIPM from its French name - Comité International des Poids et Mesures). The relevant international standard is ISO/IEC 80000.

The role of the BIPM includes the establishment of standards for the principal physical quantities, and the maintenance of international prototypes. Its work includes metrological research (metrology is the science of measurement), making comparisons of international prototypes for verification purposes, and the calibration of standards. The work of the BIPM is supervised by the CIPM, which in turn is responsible to the CGPM. The General Conference currently meets every four years to confirm new standards and resolutions, and to agree on financial, organisational and developmental issues.

SI base quantities and units

The value of a physical quantity is usually expressed as the product of a number and a unit. In the past (and in some cases up until very recently) the unit represented a specific example or prototype of the quantity concerned, which was used as a point of reference. The number represents the ratio of the value of the quantity to the unit.

As of 2019, all of the base units are now defined with reference to seven "defining" physical constants that include fundamental constants of nature such as the Planck constant and the speed of light. The most recent changes occurred with the publication of the ninth edition of the SI brochure in 2019. Four base units - the kilogram, ampere, kelvin and mole - were redefined using physical constants. The second, metre, and candela, already defined using physical constants, were subject to corrections.

As a case in point, the kilogram was previously defined with reference to a prototype. The prototype in question was a platinum-iridium cylinder held under tightly controlled conditions in a vault at the BIPM, identical copies of which are kept under identical conditions located throughout the world. A quantity of two kilograms (2 kg) would have been defined as exactly twice the mass of the prototype or one of its copies. Now, however, according to the 2019 version of the SI Brochure:

"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 6.626 070 15 × 10−34 when expressed in the unit J s, which is equal to kg m2 s−1, where the metre and the second are defined in terms of c and ΔνCs ."

Also according to the 2019 edition of the SI Brochure, the seven defining physical constants used to define the SI units:

" . . . are chosen in such a way that any unit of the SI can be written either through a defining constant itself or through products or quotients of defining constants."

The seven defining constants used to define the SI units are:

where, according to the SI Brochure, the hertz, joule, coulomb, lumen, and watt, with unit symbols Hz, J, C, lm, and W, respectively, are related to the units second, metre, kilogram, ampere, kelvin, mole, and candela, with unit symbols s, m, kg, A, K, mol, and cd, respectively, according to Hz = s–1, J = kg m2 s–2, C = A s, lm = cd m2 m–2 = cd sr, and W = kg m2 s–3.

There are seven base quantities used in the International System of Units. The seven base quantities and their corresponding units are:

These base quantities are assumed to be independent of one another. In other words, no base quantity needs to be defined in terms of any other base quantity (or quantities). Note however that although the base quantities themselves are considered to be independent, their respective base units are in some cases dependent on one another. The metre, for example, is defined as the length of the path travelled by light in a vacuum in a time interval of 1/299 792 458 of a second.

The table below summarises the base quantities and their units. You may have noticed that an anomaly arises with respect to the kilogram (the unit of mass). The kilogram is the only SI base unit whose name and symbol include a prefix. You should be aware that multiples and submultiples of this unit are formed by attaching the appropriate prefix name to the unit name gram, and the appropriate prefix symbol to the unit symbol g. For example, one millionth of a kilogram is one milligram (1 mg), and not one microkilogram (1 μkg).

SI Base Units
Unit Definition
timetsecondsThe 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 caesium 133 atom
lengthlmetremThe length of the path travelled by light in a vacuum during a time interval with a duration of 1/299 792 458 of a second
massmkilogramkgThe kilogram is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 15 × 10−34 when expressed in the unit J s, which is equal to kg m2 s−1, where the metre and the second are defined in terms of c and ΔνCs .

An earlier proposed definition, equivalent to the above, describes the kilogram as the mass of a body at rest whose equivalent energy equals the energy of a collection of photons whose frequencies sum to [1.356392489652 × 1050] hertz.
IampereAThe electric current corresponding to the flow of 1/(1.602 176 634 × 10−19) elementary charges per second
TkelvinKThe change of thermodynamic temperature that results in a change of thermal energy kT by 1.380 649 × 10−23 J
nmolemolThe amount of substance of a system that contains 6.022 140 76 × 1023 specified elementary entities (elementary entities may be atoms, molecules, ions, electrons, other particles or specified groups of such particles)
Iv candelacdThe 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 watts per steridian

Dimensions of quantities

As stated earlier, each of the derived units of quantity identified by the International System of Units is defined as the product of powers of base units. Each base quantity is considered as having its own dimension, which is represented using an upper-case character printed in a sans serif roman font. Derived quantities are considered to have dimensions that can be expressed as products of powers of the dimensions of the base quantities from which they are derived. The dimension of any quantity Q is thus written as:

dim Q = Lα Mβ Tλ Iδ Θε Nζ Jη

The upper case characters L, M, T, I, Θ, N and J (Θ is the upper-case Greek character Theta) represent the dimensions of the base quantities length, mass, time, electric current, thermodynamic temperature, amount of substance and luminous intensity respectively. The superscripted characters are the first seven lower-case characters from the Greek alphabet (alpha, beta, lambda, delta, epsilon, zeta and eta), and represent integer values called the dimensional exponents. The dimensional exponents values can be positive, negative or zero. The dimension of a derived quantity essentially conveys the same information about the relationship between derived quantities and the base quantities from which they are derived as the SI unit symbol for the derived quantity.

In some cases, all of the dimensional exponents are zero (as is the case, for example, where a quantity is defined as the ratio of two quantities of the same kind). Such quantities are said to be dimensionless, or of dimension one. The coherent derived unit for such a quantity (as the ratio of two identical units) is the number one. The same principle applies to quantities that cannot be expressed in terms of base units, such as number of molecules, which is essentially simply the result of a count. These quantities are also regarded as being dimensionless, or of dimension one. Most dimensionless quantities are simply expressed as numbers. Exceptions include the radian and the steradian, used to express values of plane angles and solid angles respectively. Another notable exception is the decibel, which is described above.

Derived units

The derived units of quantity identified by the International System of Units are all defined as products of powers of base units. A derived quantity can therefore be expressed in terms of one or more base quantity in the form of an algebraic expression. Derived units that are products of powers of base units that include no numerical factor other than one are said to be coherent derived units. This means that they are derived purely using products or quotients of integer powers of base quantities, and that no numerical factor other than one is involved.

The seven base units and twenty-two coherent derived units of the SI form a coherent set of twenty-nine uinits which is referred to as the set of coherent SI units. All other SI units are combinations of some of these twenty-nine units.The word "coherent" in this context means that equations between the numerical values of quantities are in exactly the same form as the corresponding equations between the quantities themselves.

The twenty-two coherent derived units have special names and symbols. Often, the name chosen acknowledges the contribution of a particular scientist. The unit of force (the newton) is named after Sir Isaac Newton, one of the greatest contributors in the field of classical mechanics. The unit of pressure (the pascal) is named after Blaise Pascal for his work in the fields of hydrodynamics and hydrostatics. The table below lists the coherent derived units. Note that aach unit named in the table below has its own symbol, but can be defined in terms of other derived units or in terms of the SI base units, as shown in the last two columns.

SI Units with Special Names and Symbols
QtyUnitUnit SymbolBase UnitsOther Units
plane angleradianradm/m-
solid anglesteradiansrm2/m2-
forcenewtonNkg m s-2-
pascalPakg m-1 s-2-
amount of heat
jouleJkg m2 s-2N m
radiant flux
wattWkg m2 s-3J/s
electric charge,
amount of electricity
coulombCA s-
electric potential difference,
electromotive force
voltVkg m2 s-3 A-1W/A
capacitancefaradFkg-1 m-2 s4 A2C/V
electric resistanceohmΩkg m2 s-3 A-2V/A
electric conductancesiemensSkg-1 m-2 s3 A2A/V
magnetic fluxweberWbkg m2 s-2 A-1V s
magnetic flux densityteslaTkg s-2 A-1Wb/m2
inductancehenryHkg m2 s-2 A-2Wb/A
Celsius temperaturedegree Celsius°CK-
luminous fluxlumenlmcd srcd sr
illuminanceluxlxcd sr m-2lm/m2
activity referred to a radio nuclidebecquerelBqs-1-
absorbed dose,
specific energy (imparted),
grayGym2 s-2J/kg
dose equivalent,
ambient dose equivalent,
directional dose equivalent,
personal dose equivalent
sievertSvm2 s-2J/kg
catalytic activitykatalkatmol s-1-

Note that the units for the plane angle and the solid angle (the radian and steradian respectively) are both derived as the quotient of two identical SI base units. They are thus said to have the unit one (1). They are described as dimensionless units or units of dimension one (the concept of dimension was described above).

Note that a temperature difference of one degree Celsius has exactly the same value as a temperature difference of one kelvin. The Celsius temperature scale tends to be used for day-to-day non-scientific purposes such as reporting the weather, or for specifying the temperature at which foodstuffs and medicines should be stored. In this kind of context it is somewhat more meaningful to a member of the public than the Kelvin temperature scale.

The units in the coherent set can be combined to express the units of other derived quantities. Since this allows a potentially unlimited number of combinations, it is not possible to list them all here. The table below lists some examples of derived quantities, together with the corresponding coherent derived units expressed in terms of base units.

Coherent Derived Units expressed in terms of Base Units
areaAsquare metrem2
volumeVcubic metrem3
speed, velocityvmetre per secondm s-1
accelerationametre per second squaredm s-2
wavenumberσreciprocal metrem-1
density, mass densityρkilogram per cubic metrekg m-3
surface densityρA kilogram per square metrekg m-2
specific volumevcubic metre per kilogramm3 kg-1
current densityjampere per square metreA m-2
magnetic field strengthHampere per metreA m-1
amount of substance concentrationcmole per cubic metremol m-3
mass concentrationρ, γkilogram per cubic metrekg m-3
luminanceLv candela per square metrecd m-2

The example coherent SI derived units shown in the table below are based on a combination of derived units with special names and the SI base units. The names and symbols for these units reflects the hybrid nature of these units. As with the units in the previous table, each unit has its own symbol but can be defined in terms of the SI base units, as shown in the final column. The value of being able to use both special and hybrid symbols in equations can be appreciated when we look at the length of some of the base unit expressions.

SI Derived Units with hybrid names
dynamic viscositypascal secondPa skg m-1 s-1
moment of forcenewton metreN mkg m2 s-2
surface tensionnewton per metreN m-1kg s-2
angular velocity, angular frequencyradian per secondrad s-1s-1
angular accelerationradian per second squaredrad/s2s-2
heat flux density,
watt per square metreW/m2kg s-3
heat capacity,
joule per kelvinJ K-1kg m2 s-2 K-1
Specific heat capacity,
specific entropy
joule per kilogram kelvinJ K-1 kg-1m2 s-2 K-1
specific energyjoule per kilogramJ kg-1m2 s-2
thermal conductivitywatt per metre kelvinW m-1 K-1kg m s-3 K-1
energy densityjoule per cubic metreJ m-3kg m-1 s-2
electric field strengthvolt per metreV m-1kg m s-3 A-1
electric charge densitycoulomb per cubic metreC m-3A s m-3
surface charge densitycoulomb per square metreC m-2A s m-2
electric flux density,
electric displacement
coulomb per square metreC m-2A s m-2
permittivityfarad per metreF m-1kg-1 m-3 s4 A2
permeabilityhenry per metreH m-1kg m s-2 A-2
molar energyjoule per moleJ mol-1kg m2 s-2 mol-1
molar entropy,
molar heat capacity
joule per mole kelvinJ K-1 mol-1kg m2 s-2 mol-1 K-1
exposure (x- and γ-rays)coulomb per kilogramC kg-1A s kg-1
absorbed dose rategray per secondGy s-1m2 s-3
radiant intensitywatt per steradianW sr-1kg m2 s-3
radiancewatt per square metre steradianW sr-1 m-2kg s-3
catalytic activity concentrationkatal per cubic metrekat m-3mol s-1 m-3

Non-SI units accepted for use with the SI

The units detailed in the final table are accepted for use with the International System of Units for a variety of reasons. Many are still in use, some are required for the interpretation of scientific texts of historical importance, and some are used in specialised areas such as medicine. The hectare, for example, is still commonly used to express land area. The use of the equivalent SI units is preferred for modern scientific texts. Wherever reference is made to non-SI units, they should be cross referenced with their equivalent SI units. For the units shown in the following table, the equivalent definition in terms of SI units is also shown. Most of the units listed that are in widespread daily use, and likely to be so for the foreseeable future.

Note that for most purposes, it is recommended that fractional values for plane angles expressed in degrees should be expressed using decimal fractions rather than minutes and seconds. Exceptions include navigation and surveying (due to the fact that one minute of latitude on the Earth's surface corresponds to approximately one nautical mile), and astronomy. In the field of astronomy, very small angles are significant due to the enormous distances involved. It is therefore convenient for astronomers to use a unit of measurement that can represent very small differences in angle in a meaningful way. Very small angles can be represented in terms of arcseconds, microarcseconds and picoarcseconds.

Non-SI Units still in widespread use
timeminutemin1 min = 60 s
timehourh1 h = 60 min = 3600 s
timedayd1 d = 24 h = 86 400 s
lengthastronomical unitua1 ua = 1.495 978 706 91 (6) × 1011 m
plane and phase angledegree°1° = (π/180) rad
plane and phase angleminute1′ = (1/60)° = (π/10 800) rad
plane and phase anglesecond1″ = (1/60)′ = (π/648 000) rad
areahectareha1 ha = 1 hm2 = 104 m2
volumelitreL or l1L = 1 dm3 = 103 cm3 = 10-3 m3
masstonnet1 t = 103 kg
massdaltonDa1 Da = 1.660 539 040 (20) × 10-27 kg
energyelectronvolteV1 eV = 1.602 176 634 × 10-19 J
logarithmic rationeperNp-
logarithmic ratiobelB-
logarithmic ratiodecibeldB-

Presentational conventions

There are a number of widely accepted conventions for the expression of quantities in hand-written or printed documents and texts. These conventions have been in place with relatively little modification since the General Conference on Weights and Measures first introduced the System of International Units in 1960. They are primarily intended to ensure a uniform approach to the presentation of hand written or printed information, and to ensure the readability of scientific journals, textbooks, academic papers, data sheets, reports, and other related documents. The presentational requirements will vary to some extent according to the norms of the language in which the work is written. We are concerned here only with the conventions as they apply to the English language. The following list represents some of the more important requirements.

Multiples and submultiples of SI units

Multiples and submultiples of SI units are signified by attaching the appropriate prefix to the unit symbol. Prefixes are printed as roman (upright) characters prepended to the unit symbol with no intervening space. Most unit multiple prefixes are upper case characters (the exceptions are deca (da), hecto (h) and kilo (k). All unit submultiple prefixes are lower case characters. Prefix names are always printed in lower case characters, except where they appear at the beginning of a sentence, and prefixed units appear as single words (e.g. millimeter, micropascal and so on). All multiples and submultiples are integer powers of ten. Beyond one hundred (or one hundredth) multiples and submultiples are integer powers of one thousand, although they are still expressed as powers of ten. The following table lists the most commonly encountered multiple and submultiple prefixes.

SI Prefixes