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 2006) in its eighth 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. General information about the International System of Units, definitions of quantities, and details of unit symbols are to be found in ISO 80000-1:2009 Quantities and units - Part 1: General.
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. The unit represents a specific example or prototype of the quantity concerned, which is used as a point of reference. The number represents the ratio of the value of the quantity to the unit. In the case of the kilogram, the prototype is a platinum-iridium cylinder held under tightly controlled conditions in a vault at the BIPM, although there are a number of identical copies kept under identical conditions located throughout the world. A quantity of two kilograms (2 kg) would have exactly twice the mass of the prototype or one of its copies. There are seven base quantities used in the International System of Units. The seven base quantities and their corresponding units are:
- length (metre)
- mass (kilogram)
- time (second)
- 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).
|length||l||metre||m||The length of the path travelled by light in 1/299 792 458 of a second|
|mass||m||kilogram||kg||The mass of the International Prototype Kilogram|
|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, at rest, at a temperature of 0 K|
|I||ampere||A||The constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in a vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per metre of length|
|T||kelvin||K||The fraction 1/273.16 of the thermodynamic temperature of the triple point of water|
|n||mole||mol||The amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12 (elementary entities, which must be specified, may be atoms, molecules, ions, electrons, other particles or specified groups of such particles)|
|Iv||candela||cd||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 watts per steridian|
Coherent derived units expressed in terms of SI base 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 table below gives some examples of coherent derived units.
|speed, velocity||v||metre per second||m/s|
|acceleration||a||metre per second squared||m/s2|
|density, mass density||ρ||kilogram per cubic metre||kg/m3|
|surface density||ρA||kilogram per square metre||kg/m2|
|specific volume||v||cubic metre per kilogram||m3/kg|
|current density||j||ampere per square metre||A/m2|
|magnetic field strength||H||ampere per metre||A/m|
|amount concentration, concentration||c||mole per cubic metre||mol/m3|
|mass concentration||ρ||kilogram per cubic metre||kg/m3|
|luminance||Lv||candela per square metre||cd/m2|
Coherent derived units with special names and symbols
There are a number of derived SI units that 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. Each 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.
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 will be described shortly). 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.
|force||newton||N||-||m kg s-2|
|pascal||Pa||N/m2||m-1 kg s-2|
amount of heat
|joule||J||N m||m2 kg s-2|
|watt||W||J/s||m2 kg s-3|
amount of electricity
|electric potential difference,|
|volt||V||W/A||m2 kg s-3 A-1|
|capacitance||farad||F||C/V||m-2 kg-1 s4 A2|
|electric resistance||ohm||Ω||V/A||m2 kg s-3 A-2|
|electric conductance||siemens||S||A/V||m-2 kg-1 s3 A2|
|magnetic flux||weber||Wb||V s||m2 kg s-2 A-1|
|magnetic flux density||tesla||T||Wb/m2||kg s-2 A-1|
|inductance||henry||H||Wb/A||m2 kg s-2 A-2|
|Celsius temperature||degree Celsius||°C||-||K|
|luminous flux||lumen||lm||cd sr||cd|
|activity referred to a radio nuclide||becquerel||Bq||-||s-1|
specific energy (imparted),
ambient dose equivalent,
directional dose equivalent,
personal dose equivalent
|catalytic activity||katal||kat||-||s-1 mol|
Coherent derived units with hybrid names and symbols
The 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.
|dynamic viscosity||pascal second||Pa s||m-1 kg s-1|
|moment of force||newton metre||N m||m2 kg s-2|
|surface tension||newton per metre||N/m||kg s-2|
|angular velocity||radian per second||rad/s||m m-1 s-1 = s-1|
|angular acceleration||radian per second squared||rad/s2||m m-1 s-2 = s-2|
|heat flux density,|
|watt per square metre||W/m2||kg s-3|
|joule per kelvin||J/K||m2 kg s-2 K-1|
|Specific heat capacity,|
|joule per kilogram kelvin||J/(kg K)||m2 s-2 K-1|
|specific energy||joule per kilogram||J/kg||m2 s-2|
|thermal conductivity||watt per metre kelvin||W/(m K)||m kg s-3 K-1|
|energy density||joule per cubic metre||J/m3||m-1 kg s-2|
|electric field strength||volt per metre||V/m||m kg s-3 A-1|
|electric charge density||coulomb per cubic metre||C/m3||m-3 s A|
|surface charge density||coulomb per square metre||C/m2||m-2 s A|
|electric flux density,|
|coulomb per square metre||C/m2||m-2 s A|
|permittivity||farad per metre||F/m||m-3 kg-1 s4 A2|
|permeability||henry per metre||H/m||m kg s-2 A-2|
|molar energy||joule per mole||J/mol||m2 kg s-2 mol-1|
molar heat capacity
|joule per mole kelvin||J/(mol K)||m2 kg s-2 K-1 mol-1|
|exposure (x- and γ-rays)||coulomb per kilogram||C/kg||kg-1 s A|
|absorbed dose rate||gray per second||Gy/s||m2 s-3|
|radiant intensity||watt per steradian||W/sr||m4 m-2 kg s-3 = m2 kg s-3|
|radiance||watt per square metre steradian||W/(m2 sr)||m2 m-2 kg s-3 = kg s-3|
|catalytic activity concentration||katal per cubic metre||kat/m3||m-3 s-1 mol|
Non-SI units accepted for use with the International System of Units
The units detailed in the remaining four tables 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 tables, the equivalent definition in terms of SI units is also shown. The first of these tables lists units that are still 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.
|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|
|plane angle||degree||°||1° = (π/180) rad|
|plane angle||minute||′||1′ = (1/60)° = (π/10 800) rad|
|plane angle||second||″||1″ = (1/60)′ = (π/648 000) rad|
|area||hectare||ha||1 ha = 1 hm2 = 104 m2|
|volume||litre||L or l||1L = 1 dm3 = 103 cm3 = 10-3 m3|
|mass||tonne||t||1 t = 103 kg|
The table below lists a number of units related to fundamental constants (with the exception of the astronomical unit). These non-SI units are used to express quantities in certain specialised fields of scientific endeavor because it is simply more convenient to do so than using the equivalent SI unit, or because their use provides a more meaningful expression of quantities in a given context. The astronomical unit, for example, represents the approximate mean distance from the Earth to the Sun (149 597 870 700 metres). The value of these units must be determined experimentally due to the fact that, for one reason or another, they represent quantities that cannot be measured directly using conventional means. The abbreviations n.u. and a.u. stand for natural unit and atomic unit respectively.
The very nature of the quantities involved means that there will be a degree of uncertainty associated with each of them. This degree of uncertainty (known as the standard uncertainty) is expressed as a number in parentheses immediately following the value of the unit, and represents the amount by which the last two digits may vary. From the table below, for example, we can see that the electronvolt has a value (in Joules) of 1.602 176 53 × 10-19 plus or minus 0.000 000 14 × 10-19. This means its actual value may lie between 1.602 176 39 × 10-19 and 1.602 176 67 × 10-19 Joules.
|energy||electronvolt||eV||1 eV = 1.602 176 53 (14) × 10-19 J|
|mass||dalton||Da||1 Da = 1.660 538 86 (28) × 10-27 kg|
|mass||unified atomic mass unit||u||1 u = 1.660 538 86 (28) × 10-27 kg|
|length||astronomical unit||ua||1 ua = 1.495 978 706 91 (6) × 1011 m|
|speed||n.u. of speed (speed of light in a vacuum)||c0||299 792 458 m/s (exact)|
|action||n.u. of action (reduced Planck constant)||h||1.054 571 68 (18) × 10-34 J s|
|mass||n.u. of mass (electron mass)||me||9.109 3826 (16) × 10-31 kg|
|time||n.u. of time||h/(mec02)||1.288 088 6677 (86) × 10-21 s|
|charge||a.u. of charge||e||1.602 176 53 (14) × 10-19 C|
|mass||a.u. of mass (electron mass)||me||9.109 3826 (16) × 10-31 kg|
|action||a.u. of action (reduced Planck constant)||h||1.054 571 68 (18) × 10-34 J s|
|length||a.u. of length, bohr (Bohr radius)||a0||0.529 177 2108 (18) × 10-10 m|
|energy||a.u. of energy, hartree (Hartree energy)||Eh||4.359 744 17 (75) × 10-18 J|
|time||a.u. of time||h/Eh||2.418 884 326 505 (16) × 10-17 s|
The next table represents units that have precisely defined SI equivalents, but that are used by particular organisations of groups due to commercial or legal reasons, or because (again) they are simply more convenient or meaningful in a particular context. For example, the nautical mile is used to express the distances travelled by ships and aircraft, while the knot (or nautical mile per hour) is used to express their speeds. A nautical mile is equal to 1852 metres, and represents approximately one minute of latitude. Note in particular the three dimensionless units (the neper, bel and decibel) which represent logarithmic ratio quantities. The decibel (dB) is one tenth of a bel, and is probably the most frequently encountered of these units. In the field of telecommunications, the decibel is typically used to express signal strength relative to some specified reference level. For example, signal-to-noise ratio (SNR) is expressed using a decibel value of 10 log10 S/N. A signal-to-noise ratio of 10 would be expressed as 10 dB, a ratio of 100 as 20 dB, a ratio of 1000 as 30 dB and so on.
|pressure||bar||bar||1 bar = 0.1 MPa = 100 kPa = 105 Pa|
|pressure||millimeter of mercury||mmHg||1 mmHg ≈ 133.322 Pa|
|length||ångström||Å||1 Å = 0.1 nm = 100 pm = 10-10 m|
|distance||nautical mile||M||1 M = 1852 m|
|area||barn||b||1 b = 100 fm2 = (10-12 cm)2 = 10-28 m2|
|speed||knot||kn||1 kn = (1852/3600) m/s|
|energy||erg||erg||1 erg = 10-7 J|
|force||dyne||dyn||1 dyn = 10-5 N|
|dynamic viscosity||poise||P||1 P = 1 dyn s cm-2 = 0.1 Pa s|
|kinematic viscosity||stokes||St||1 St = 1 cm2 s-1 = 10-4 m2 s-1|
|luminance||stilb||sb||1 sb = 1 cd cm-2 = 104 cd m-2|
|illuminance||phot||ph||1 ph = 1 cd sr cm-2 = 104 lx|
|acceleration||gal||Gal||1 Gal = 1 cm s-2 = 10-2 m s-2|
|magnetic flux||maxwell||Mx||1 Mx = 1 G cm2 = 10-8 Wb|
|magnetic flux density||gaus||G||1 G = 1 Mx cm-2 = 10-4 T|
|magnetic field||œrsted||Oe||1 Oe ≙ (103/4π) A m-1|
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.
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.
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 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").
- 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).
- 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. "RLOAD" for an unknown load resistance, or "I1" 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".