Physical Quantities and SI Units
Overview
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 10^{th} 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 11^{th} CGPM) introduced the new system to the world.
A seventh base unit, the mole, was added following the 14^{th} 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 m^{2} s^{−1}, where the metre and the second are defined in terms of c and Δν_{Cs }."
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:
- The unperturbed ground state hyperfine transition frequency of the caesium 133 atom, Δν_{Cs }, is 9 192 631 770 Hz
- The speed of light in vacuum, c, is 299 792 458 m/s
- The Planck constant h is 6.626 070 15 × 10^{−34} J s
- The elementary charge e is 1.602 176 634 × 10^{−19} C
- The Boltzmann constant k is 1.380 649 × 10^{−23} J/K
- The Avogadro constant N_{A } is 6.022 140 76 × 10^{23} mol^{−1}
- The luminous efficacy of monochromatic radiation of frequency 540 × 10^{12} Hz, K_{cd }, is 683 lm/W
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 m^{2} s^{–2}, C = A s, lm = cd m^{2} m^{–2} = cd sr, and W = kg m^{2} s^{–3}.
There are seven base quantities used in the International System of Units. The seven base quantities and their corresponding units are:
- time (second)
- length (metre)
- mass (kilogram)
- electric current (ampere)
- thermodynamic temperature (kelvin)
- amount of substance (mole)
- luminous intensity (candela)
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).
Qty | Sym. | Unit | Unit Sym. | Unit Definition |
---|---|---|---|---|
time | t | second | s | 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 caesium 133 atom |
length | l | metre | m | The 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 |
mass | m | kilogram | kg | The 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 Δν_{Cs }. 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 × 10^{50}] hertz. |
electric current | I | ampere | A | The electric current corresponding to the flow of 1/(1.602 176 634 × 10^{−19}) elementary charges per second |
thermodynamic temperature | T | kelvin | K | The change of thermodynamic temperature that results in a change of thermal energy kT by 1.380 649 × 10^{−23} J |
amount of substance | n | mole | mol | The amount of substance of a system that contains 6.022 140 76 × 10^{23} specified elementary entities (elementary entities may be atoms, molecules, ions, electrons, other particles or specified groups of such particles) |
luminous intensity | I_{v } | candela | cd | The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 10^{12} 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 = L^{α} M^{β} T^{λ} I^{δ} Θ^{ε} N^{ζ} J^{η}
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.
Qty | Unit | Unit Symbol | Base Units | Other Units |
---|---|---|---|---|
plane angle | radian | rad | m/m | - |
solid angle | steradian | sr | m^{2}/m^{2} | - |
frequency | hertz | Hz | s^{-1} | - |
force | newton | N | kg m s^{-2} | - |
pressure, stress | pascal | Pa | kg m^{-1} s^{-2} | - |
energy, work, amount of heat | joule | J | kg m^{2} s^{-2} | N m |
power, radiant flux | watt | W | kg m^{2} s^{-3} | J/s |
electric charge, amount of electricity | coulomb | C | A s | - |
electric potential difference, electromotive force | volt | V | kg m^{2} s^{-3} A^{-1} | W/A |
capacitance | farad | F | kg^{-1} m^{-2} s^{4} A^{2} | C/V |
electric resistance | ohm | Ω | kg m^{2} s^{-3} A^{-2} | V/A |
electric conductance | siemens | S | kg^{-1} m^{-2} s^{3} A^{2} | A/V |
magnetic flux | weber | Wb | kg m^{2} s^{-2} A^{-1} | V s |
magnetic flux density | tesla | T | kg s^{-2} A^{-1} | Wb/m^{2} |
inductance | henry | H | kg m^{2} s^{-2} A^{-2} | Wb/A |
Celsius temperature | degree Celsius | °C | K | - |
luminous flux | lumen | lm | cd sr | cd sr |
illuminance | lux | lx | cd sr m^{-2} | lm/m^{2} |
activity referred to a radio nuclide | becquerel | Bq | s^{-1} | - |
absorbed dose, specific energy (imparted), kerma | gray | Gy | m^{2} s^{-2} | J/kg |
dose equivalent, ambient dose equivalent, directional dose equivalent, personal dose equivalent | sievert | Sv | m^{2} s^{-2} | J/kg |
catalytic activity | katal | kat | mol 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.
Qty | Sym. | Unit | Unit Sym. |
---|---|---|---|
area | A | square metre | m^{2} |
volume | V | cubic metre | m^{3} |
speed, velocity | v | metre per second | m s^{-1} |
acceleration | a | metre per second squared | m s^{-2} |
wavenumber | σ | reciprocal metre | m^{-1} |
density, mass density | ρ | kilogram per cubic metre | kg m^{-3} |
surface density | ρ_{A } | kilogram per square metre | kg m^{-2} |
specific volume | v | cubic metre per kilogram | m^{3} kg^{-1} |
current density | j | ampere per square metre | A m^{-2} |
magnetic field strength | H | ampere per metre | A m^{-1} |
amount of substance concentration | c | mole per cubic metre | mol m^{-3} |
mass concentration | ρ, γ | kilogram per cubic metre | kg m^{-3} |
luminance | L_{v } | candela per square metre | cd 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.
Qty | Unit | Unit Symbol | Base Units |
---|---|---|---|
dynamic viscosity | pascal second | Pa s | kg m^{-1} s^{-1} |
moment of force | newton metre | N m | kg m^{2} s^{-2} |
surface tension | newton per metre | N m^{-1} | kg s^{-2} |
angular velocity, angular frequency | radian per second | rad s^{-1} | s^{-1} |
angular acceleration | radian per second squared | rad/s^{2} | s^{-2} |
heat flux density, irradiance | watt per square metre | W/m^{2} | kg s^{-3} |
heat capacity, entropy | joule per kelvin | J K^{-1} | kg m^{2} s^{-2} K^{-1} |
Specific heat capacity, specific entropy | joule per kilogram kelvin | J K^{-1} kg^{-1} | m^{2} s^{-2} K^{-1} |
specific energy | joule per kilogram | J kg^{-1} | m^{2} s^{-2} |
thermal conductivity | watt per metre kelvin | W m^{-1} K^{-1} | kg m s^{-3} K^{-1} |
energy density | joule per cubic metre | J m^{-3} | kg m^{-1} s^{-2} |
electric field strength | volt per metre | V m^{-1} | kg m s^{-3} A^{-1} |
electric charge density | coulomb per cubic metre | C m^{-3} | A s m^{-3} |
surface charge density | coulomb per square metre | C m^{-2} | A s m^{-2} |
electric flux density, electric displacement | coulomb per square metre | C m^{-2} | A s m^{-2} |
permittivity | farad per metre | F m^{-1} | kg^{-1} m^{-3} s^{4} A^{2} |
permeability | henry per metre | H m^{-1} | kg m s^{-2} A^{-2} |
molar energy | joule per mole | J mol^{-1} | kg m^{2} s^{-2} mol^{-1} |
molar entropy, molar heat capacity | joule per mole kelvin | J K^{-1} mol^{-1} | kg m^{2} s^{-2} mol^{-1} K^{-1} |
exposure (x- and γ-rays) | coulomb per kilogram | C kg^{-1} | A s kg^{-1} |
absorbed dose rate | gray per second | Gy s^{-1} | m^{2} s^{-3} |
radiant intensity | watt per steradian | W sr^{-1} | kg m^{2} s^{-3} |
radiance | watt per square metre steradian | W sr^{-1} m^{-2} | kg s^{-3} |
catalytic activity concentration | katal per cubic metre | kat m^{-3} | mol 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.
Qty | Unit | Unit Symbol | SI Units |
---|---|---|---|
time | minute | min | 1 min = 60 s |
time | hour | h | 1 h = 60 min = 3600 s |
time | day | d | 1 d = 24 h = 86 400 s |
length | astronomical unit | ua | 1 ua = 1.495 978 706 91 (6) × 10^{11} m |
plane and phase angle | degree | ° | 1° = (π/180) rad |
plane and phase angle | minute | ′ | 1′ = (1/60)° = (π/10 800) rad |
plane and phase angle | second | ″ | 1″ = (1/60)′ = (π/648 000) rad |
area | hectare | ha | 1 ha = 1 hm^{2} = 10^{4} m^{2} |
volume | litre | L or l | 1L = 1 dm^{3} = 10^{3} cm^{3} = 10^{-3} m^{3} |
mass | tonne | t | 1 t = 10^{3} kg |
mass | dalton | Da | 1 Da = 1.660 539 040 (20) × 10^{-27} kg |
energy | electronvolt | eV | 1 eV = 1.602 176 634 × 10^{-19} J |
logarithmic ratio | neper | Np | - |
logarithmic ratio | bel | B | - |
logarithmic ratio | decibel | dB | - |
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.
- Unit symbols - these appear in roman (upright) type. They are printed in lower case letters unless derived from a proper name, in which case the first letter is capitalized (e.g. "Pa" for pascal). The exception to the rule is the symbol for the litre, which may be written either as "l" or "L". The latter is allowed in order to distinguish the symbol used for the litre from the number one (1). Any multiple or sub-multiple prefix is considered to be part of the unit symbol, to which it is prepended without an intervening space (e.g. "km" for kilometer, "mm" for millimeter, or "μm" for micrometer).
- Unit names - these appear in roman (upright) type. All unit names are printed in lower case characters, including the first letter, regardless of whether or not they are named after a person, or whether or not the unit symbol begins with an upper-case character (i.e. "newton", not "Newton"). If a prefix is used with the unit name it becomes part of the unit name and is formed as a single word (e.g. "micropascal", not "micro pascal" or "micro-pascal"). If a derived unit is the product of two or more separate units, either a space or a hyphen can be used to separate the names (e.g. "newton metre" or "newton-metre"). For units raised to a power, the appropriate modifier may precede or follow the unit name (e.g. "square metre" or "metre cubed").
- Compound units - units expressed as the product or quotient of other units are written in the same way as standard algebraic expressions. Multiplication is represented either by a space, or by the use of the dot operator (also called a middle dot). For example the symbol for "newton metre" is written as "N m" or "N·m". Division is represented using a solidus (forward stroke) or using negative exponents. The symbol for "newton per meter" is written either as "N/m" or as "N m^{-1}").
- Variables - unknown quantities in equations are usually represented using a single character in an italic font, e.g. "m" for mass or "I" for electric current. The quantity symbol may be further qualified, typically by using a subscripted number or label, e.g. "R_{LOAD }" for an unknown load resistance, or "I_{1 }" for the unknown current in a specific branch of an electrical circuit (note, incidentally, that although serif fonts are often used for equations, this is not specifically mandated by the BIPM).
- Quantities - a quantity of known value is expressed as a number, followed by a space, and then the unit symbol. The space represents the multiplication operator. The exception to the rule is for a plane angle expressed in degrees, minutes and seconds. The degree, minute and second symbols always follow their respective numbers without an intervening space. For example, a value of thirty-five degrees is written as "35°". Numbers always appear as roman (upright) text.
- Combining units - different units should only be combined when expressing a quantity using non-SI units that is either a time or an angle. For example, time is commonly expressed in hours, minutes, and seconds. In fields such as navigation or astronomy, it is still customary to express plane angles in terms of degrees, minutes, and seconds. Note however that for other uses, angles given in degrees may alternatively be written as decimal fractions, e.g. "21.255°" rather than "21° 15′ 18″".
- Decimal markers - for any number that has a fractional part, the decimal marker (sometimes called the decimal point) is the symbol that separates the integral part of the number from its fractional part. This is usually either a period or a comma. For values of between minus one and one, the decimal marker is preceded by a zero, e.g. "0.123".
- The thousand separator - numbers consisting of long sequences of digits are often split into groups of three digits to make them easier to read. The preferred method of separating these groups is to use a space, since the use of dots or commas can be interpreted in different ways in different parts of the world. For example, the speed of light is expressed as "299 792 458 m/s". Note that if there are only four digits before or after the decimal marker, a separator is not generally considered to be necessary.
- Multiplication and division - various methods can be used to indicate multiplication. The names of the variables to be multiplied might be juxtaposed (placed next to one another), e.g. "xy". They may be placed within brackets, e.g. "(x)(y)". The multiplication sign can be used to indicate multiplication by placing it between the variables to be multiplied, e.g. "x × y". Note that the multiplication sign should always be used where numbers only are being multiplied together, but is best avoided if variables names are involved (in order to avoid confusion with the common variable name x). Use of the middle dot ("·") is discouraged. Division is indicated using a solidus, e.g. "x/y" or negative index, e.g. "x y^{-1}".
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.
Factor | Name | Symbol | Factor | Name | Symbol |
---|---|---|---|---|---|
10^{1} | deca | da | 10^{-1} | deci | d |
10^{2} | hecto | h | 10^{-2} | centi | c |
10^{3} | kilo | k | 10^{-3} | milli | m |
10^{6} | mega | M | 10^{-6} | micro | μ |
10^{9} | giga | G | 10^{-9} | nano | n |
10^{12} | tera | T | 10^{-12} | pico | p |
10^{15} | peta | P | 10^{-15} | femto | f |
10^{18} | exa | E | 10^{-18} | atto | a |
10^{21} | zetta | Z | 10^{-21} | zepto | z |
10^{24} | yotta | Y | 10^{-24} | yocto | y |