Bureau International des Poids et Mesures 2019 The International System of Units (SI)

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Bureau International des Poids et Mesures (2019) The International System of Units (SI). 9th edition:117-216 ISBN 978-92-822-2272-0.

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Bureau International des Poids et Mesures (2019) (SI)

Abstract: The International System of Units, the SI, has been used around the world as the preferred system of units, the basic language for science, technology, industry and trade since it was established in 1960 by a resolution at the 11th meeting of the Conférence Générale des Poids et Mesures, the CGPM (known in English as the General Conference on Weights and Measures).

Bioblast editor: Gnaiger E

Canonical reviewer's comments on: Bureau International des Poids et Mesures (2019) The International System of Units (SI) 9th ed.

Communicated by Gnaiger Erich (2020-07-05) last update 2020-07-13
in: Anastrophe XX Entity X and the elementary unit x of A X-mass Carol

General comment

This SI publication is one of the most significant scientific, interdisciplinary and transdisciplinary publications of the century. Too many teachers and editors resist to making it a highly influential publication with actual impact on scientific publication, and thus contribute to the current communication crisis. This does not reflect any faults of the SI authors but a lack of governance to improve scientific communication, and the limitations of the non-SI readers trapped into the International System of Impact Factors rather than guided towards implementing the International System of Units. What is the numerical value and the practical meaning of the Impact Factor of this SI publication?
In Resolution 1 (p. 189) a consensus is reached on the importance of "a redefinition of a number of units of the International System of Units (SI)". A unit should not be a number. Therefore, the redefinition of the unit of count remains an open issue.

Technical comments


Since 'small spelling variations occur in the language of the English speaking countries (for instance, "metre" and "meter", "litre" and "liter")' (p. 124), a decision should be taken for consistent spelling in a document. The English text of the SI brochure follows the style "metre" and "litre". It is found that in the scientific literature the spelling style "meter" and "liter" prevails even in European journals. Below are direct quotes from the SI brochure (with reference to the page number in the 9th edition), implementing corresponding changes in spelling style.

Quantity calculus (p. 147)

Symbols for units are treated as mathematical entities. In expressing the value of a quantity as the product of a numerical value and a unit, both the numerical value and the unit may be treated by the ordinary rules of algebra. This procedure is described as the use of quantity calculus, or the algebra of quantities. For example, the equation p = 48 kPa may equally be written as p/kPa = 48. It is common practice to write the quotient of a quantity and a unit in this way for a column heading in a table, so that the entries in the table are simply numbers.
Suggestion: pO2/[kPa] = 18.6; SO2/[µmol·kPa-1] = 9.72

Quantity symbols and unit symbols (p. 149)

Unit symbols must not be used to provide specific information about the quantity and should never be the sole source of information on the quantity. Units are never qualified by further information about the nature of the quantity; any extra information on the nature of the quantity should be attached to the quantity symbol and not to the unit symbol.
Comment: Quantity calculus can be extended by providing specific information about the entity (e.g. entity O2, this is not related to information on the quantity 'amount') together with the unit symbol, e.g., [mol O2] or [kJ·mol-1 O2], where the entity-type is not presented in the place occupied by the unit symbol for division or multiplication, respectively.

No comment

Defining the unit of a quantity (p. 127)

The value of a quantity is generally expressed as the product of a number and a unit. The unit is simply a particular example of the quantity concerned which is used as a reference, and the number is the ratio of the value of the quantity to the unit.
For example, the speed of light in vacuum is a constant of nature, denoted by c, whose value in SI units is given by the relation c = 299 792 458 m/s where the numerical value is 299 792 458 and the unit is m/s.
For a particular quantity different units may be used. For example, the value of the speed v of a particle may be expressed as v = 25 m/s or v = 90 km/h, where meter per second and kilometer per hour are alternative units for the same value of the quantity speed.
Before stating the result of a measurement, it is essential that the quantity being presented is adequately described. This may be simple, as in the case of the length of a particular steel rod, but can become more complex when higher accuracy is required and where additional parameters, such as temperature, need to be specified.
When a measurement result of a quantity is reported, the estimated value of the measurand (the quantity to be measured), and the uncertainty associated with that value, are necessary. Both are expressed in the same unit.

Canonical comments

Definition of the SI (p. 127)

As for any quantity, the value of a fundamental constant can be expressed as the product of a number and a unit.
The definitions below specify the exact numerical value of each constant when its value is expressed in the corresponding SI unit. By fixing the exact numerical value the unit becomes defined, since the product of the numerical value and the unit has to equal the value of the constant, which is postulated to be invariant.
Comment: The terms 'numerical value' and 'number' are used as being equivalent: value = "product of a number and a unit"; value = "product of the numerical value and the unit". It should be considered to define: quantity Q = product of the numerical value of a number N and a unit uQ. Symbols for specific quantities Q are, e.g., m and V for mass and volume, respectively. Do they represent merely the quantity type? Interpret a formula such as m = 60 kg: The symbol m represents the quantity 'mass', the numerical value of the number N is 60 (N = 60), the unit is um = kg, and the value of the quantity m is 60 kg. Just in case that these definitions appear to be acceptable, then it follows: quantity m = value of the quantity m.
The seven constants 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 International System of Units, the SI, is the system of units in which
  • the unperturbed ground state hyperfine transition frequency of the caesium 133 atom ∆νCs 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 x-1,
  • the Boltzmann constant k is 1.380 649 × 10−23 J x-1 K-1,
  • the Avogadro constant NA is 6.022 140 76 × 1023 x mol−1,
  • the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz, Kcd, is 683 lm/W,
where the hertz, joule, coulomb, lumen, and watt, with unit symbols Hz, J, C, lm, and W, respectively, are related to the units second, meter, 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.
The numerical values of the seven defining constants have no uncertainty.

The nature of the seven defining constants (p. 129)

The Avogadro constant NA is a proportionality constant between the quantity amount of substance (with unit mole) and the quantity for counting entities (with unit one, symbol 1).
Comment: Comparing the terms 'quantity amount of substance (with unit mole)' and 'quantity for counting entities (with unit one, symbol 1)' raises several questions that lead to the following comments: 
(1) The authors should replace 'quantity for counting entities' by the proper name for this quantity, comparable to amount. 
(2) The term 'quantity for counting entities' is ambiguous: Is it a quantity for counting? Is it a quantity for entities that are counting (entities such as counting machines, cell counters, ticket counters)? 
(3) Unit one, symbol 1: This is a profound mixup of a numerical value or number 'one' (symbol 1) with a unit. There is an incosistency compared to all other units in the SI, since the 'one' (symbol 1) could be added to any other SI unit, e.g. volume per mass V/m with unit [m3]/[1 kg] instead of [m3]/[kg], compared to volume per count V/NX with unit [m3]/[1 x] instead of [m3]/[x].

Writing and printing of unit symbols and of numbers — Resolution 7 (p. 162)

Roman (upright) type, in general lower-case, is used for symbols of units; if, however, the symbols are derived from proper names, capital roman type is used. These symbols are not followed by a full stop.
Comment: The SI symbol '1' suggested for the unit of the quantity count, does not follow this SI procedure. A symbol for the unit of count is required, that is consistent with SI Resolution 7.
In numbers, the comma (French practice) or the dot (British practice) is used only to separate the integral part of numbers from the decimal part. Numbers may be divided in groups of three in order to facilitate reading; neither dots nor commas are ever inserted in the spaces between groups.
Comment: In the last sentence above, there is a confusion between numbers, numerals (representing numbers, such as 4, 12, 5093.78, 6, in a specific numeral system), and symbols (or characters, such as the ten characters in the decimal numeral system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ). When a large number is expressed by a numeral as a string of several symbols, the symbols may be divided in groups of three ..  
For comprison, IUPAC presents the same message in the following form: "To facilitate the reading of long numbers the digits may be grouped in threes about the decimal sign but no point or comma should be used except for the decimal sign." 

The historical development of the realization of SI units (p. 204)

.. the masses of a silicon atom (averaged over the three isotopes used for the sphere) mSi, and the electron me ..
Comment: Defining entity X as Si, the SI symbol for the mass of a sample of Si is mSi. From a count NSi of SI in the sample, the mass per silicon atom is mSi·NSi-1. The same holds for the mass per any elementary entity of entity-type X. For consistency, the general term 'mass of an entity' (masses of a silicon atom .. and the electron) has to be replaced by the general term 'mass per elementary entity' (masses per silicon atom .. and per electron). The symbol mSi cannot be used for the expression mSi·NSi-1. This elementary formal inconsistency must be resolved in the SI.

Unit of amount of substance, mole (p. 209)

The quantity used by chemists to specify the amount of chemical elements or compounds is called “amount of substance”. Amount of substance, symbol n, is defined to be proportional to the number of specified elementary entities N in a sample, the proportionality constant being a universal constant which is the same for all entities. The proportionality constant is the reciprocal of the Avogadro constant NA, so that n = N/NA. The unit of amount of substance is called the mole, symbol mol. Following proposals by the IUPAP, IUPAC and ISO, the CIPM developed a definition of the mole in 1967 and confirmed it in 1969, by specifying that the molar mass of carbon 12 should be exactly 0.012 kg/mol. This allowed the amount of substance nS(X) of any pure sample S of entity X to be determined directly from the mass of the sample mS and the molar mass M(X) of entity X, the molar mass being determined from its relative atomic mass Ar (atomic or molecular weight) without the need for a precise knowledge of the Avogadro constant, by using the relations
nS(X) = mS/M(X), and M(X) = Ar(X) g/mol
Thus, this definition of the mole was dependent on the artefact definition of the kilogram. The numerical value of the Avogadro constant defined in this way was equal to the number of atoms in 12 grams of carbon 12. However, because of recent technological advances, this number is now known with such precision that a simpler and more universal definition of the mole has become possible, namely, by specifying exactly the number of entities in one mole of any substance, thus fixing the numerical value of the Avogadro constant. This has the effect that the new definition of the mole and the value of the Avogadro constant are no longer dependent on the definition of the kilogram.
Comment (1): Clarification is required to explain the difference between a number and a "number of specified elementary entities N in a sample". If symbol N is used to represent a 'number of specified elementary entities', then the same symbol N cannot be used to represent a 'number'. The term 'number' is used in defining the unit of a quantity (Section 2.1): "The value of a quantity is generally expressed as the product of a number and a unit." Distinguishable symbols should be used for "number of entities" NX with elementary unit [x] and dimensionless "number" N (without subscript). It might be appropriate, to use the lower case n for dimensionless number, but this is not practical, since it might be confused with the symbol n for amount, as in nX = NX/NA.
Comment (2): The terms entity and entities are used with two different meanings. Whereas these meanings can be more or less perfectly understood by decoding in context in practical language, this ambiguity should be avoided in a formal system of terminology: (i) In the context "amount of substance nS(X) of any pure sample S of entity X", and "molar mass M(X) of entity X", the term entity and symbol X are used with the meaning entity-type. (ii) If this interpretation in terms of specification of entity-type is taken rigorously, then the term "by specifying exactly the number of entities in one mole of any substance, thus fixing the numerical value of the Avogadro constant" must be understood as indicating, that in one mole of any such substance there are NA different entity-types. The intention of this comment is not to suggest, that anybody should make such a non-sensical interpretation, but rather to point to the formal inconsistency of the terminological system. The SI lacks a distinction between the term entity-tpye X (which does not express any quantity) and the elementary entity UX (which is an elementary quantity which needs to be defined before counting can start with a count NX = 1 x). With this clarification in mind, it then makes sense to use practical language: (i) "sample S of entity X"; (ii) "number of elementary entities" = count, NX = N·UX with elementary unit [x].
Comment (3): The symbols nS(X) and mS are well defined, such that the meaning of the message can be understood. There remains, however, the difficulty to understand the logic of selecting these symbols. The amount nS(X) [mol] of X can be calculated from the mass mS [g] of pure sample S and the molar mass M(X) [g·mol-1] of entity X only, if there is a pure sample S of entity X. Why then is this essential information not added to the symbol mS(X), comparable to nS(X)?
Comment (4): If mS or mS(X) [g] is the symbol for the mass of a sample of entity X, and nS(X) [mol] is the symbol for amount of substance of any pure sample S of entity X, are these then equivalent to mX [g] and nX [mol], respectively, as the symbols for the mass and amount of entity X? Can a mass or amount be determined from entity X that is not sampled? It may be clarified, that a quantity by definition can be measured (mass) or counted (amount) only on a defined sample of a defined entity X. It should be noted that "defined entity X" may be something rather undefined: A number of trees can be sampled; the mass of these trees can be measured; the number of trees (converted from count to amount) that are contained in the sample may or may not be known; the species of trees may be a pine or any other mixture of trees; the number of pinecones and needles may or may not be determined. In contrast to this potentially but not necessarily vague definition of entity X, the sample must be well defined in the sense of separated from the rest of the world for measuring or counting, otherwise a quantity does not make sense. Therefore, the symbols nS(X) and mS(X) do not make logical sense and should be replaced by nX and mX.
Comment (5): The symbol M(X) is well defined to indicate the molar mass M(X) [g·mol-1] of entity X, but there remains a problem with an extension to obtain a consistent system of symbols for other derived quantities, when mass is normalized not only for amount [g·mol-1] but count [g·x-1] and volume [g·L-1] (or for the SI [kg·m-3]). In different contexts there are ad hoc practical symbols in use, but a formally consistent system of symbols does not exist, as shown in the following table, comparing canonical and practical symbols for harmonization.
Quantity Unit Normalized for Unit Canonical symbol Unit Practical symbol Quantity
count NX x count NX x 1 1 -- --
amount nX mol count NX x nUX mol·x-1 NA-1 1/Avogadro constant
volume VX m3 count NX x VUX m3·x-1 elementary volume
mass mX kg count NX x MUX kg·x-1 elementary mass
count NX x amount nX mol NnX x·mol-1 NA Avogadro constant
amount nX mol amount nX mol 1 1 -- --
volume VX m3 amount nX mol VnX m3·mol-1 Vm(X) molar volume (IUPAC)
mass mX kg amount nX mol MnX kg·mol-1 M(X) molar mass
count NX x volume VX m3 NVX x·m-3 CX number concentration (IUPAC)
amount nX mol volume VX m3 nVX mol·m-3 Vm(X)-1 1/molar volume
volume VX m3 volume VX m3 1 1 -- --
mass mX kg volume VX m3 mVX kg·m-3 ρX density
count NX x mass mX kg NmX x·kg-1
amount nX mol mass mX kg nmX mol·kg-1 M(X)-1 1/molar mass
volume VX m3 mass mX kg VmX m3·kg-1 vX specific volume (IUPAC)
mass mX kg mass mX kg 1 1 -- --
Dualistic interpretation of body mass, if entity X is interpreted as indicating per elementary entity (left) or meaning of entity-type (right).
The Figure compares elementary quantities that are stoichiometrically linked to counting of elementary entities (left) with extensive quantities based on measurements of a pure sample. Count, amount, and charge depend on the definition of elementary entities, whereas measurements of extensive quantities are independent of information on discrete units. Red-shaded terms and units are not implemented in the SI. Entity X is used strictly in the sens of entity-type X. In practical language, however, the entity-type X = body in the context of human bodies is understood as indicating the elementary entity in terms of a single individual body, linked to the perception, that body mass is measured routinely on a sample of a single individual.
Let a biomedical scientist measure the body mass of patients on the third floor. Here the meaning of X is definitely "body mass per elementary entity". Let the same biomedical scientist take blood samples from each patient, and move to the fourth floor for isolation of PBMC and platelets, taking a cell count, and measuring the cell mass. The meaning of X=body as perceived on the third floor (body mass MUX = 60 kg·x-1) switches uncounciously upon arrival in the fourth floor for studying respiration of living cells and X=ce, when cell mass is the mass measured in a sample containing a large number of cells (cell mass mX = 6 mg). All this works practically and automatically very well even in a single person, as long as the quantities obtained on the patient's blood cells (metabolic oxygen consumption per mass of cells) are not related to quantities obtained on the individual patient (VO2max).

Canonical recommendation

The BIPM and the Metre Convention (p. 118)

The 59 SI member states and 42 states and economies that were associates of the General Conference should consider to extend the present number of 10 Consultative Committees, which cover
  1. electricity and magnetism,
  2. photometry and radiometry,
  3. thermometry,
  4. length,
  5. time and frequency,
  6. ionizing radiation,
  7. units,
  8. mass and related quantities,
  9. amount of substance: metrology in chemistry and biology,
  10. acoustics, ultrasound and vibration.
Insufficient care is taken to emphasize the fundamental role of the quantity count, define a consistent symbol for the quantity count, and implement a unit and appropriate symbol for the unit of the quantity count, such that a unit is not confused with a number, and a numeral is not used as a symbol for a unit. To take these deficiencies into account, it is recommended to implement an 11th Consultative Committee on:
11. elemental entities and count.

Quantity Symbol for quantity Q Symbol for dimension Name of SI unit Symbol for SI unit uQ [*]
length l L meter m
mass m M kilogram kg
time t T second s
electric current I I ampere A
thermodynamic temperature T Θ kelvin K
amount of substance *,§ nX = NX·NA-1 N mole mol
count *,$ NX X elementary unit x
elementary entity *,$ UX U elementary unit x
charge *,€ QX = NX·zX·e I·T coulomb C = A·s
luminous intensity Iv J candela cd
[*] »SI base units, except for the canonical 'elementary unit' [x]. The following footnotes are canonical comments.
* For the quantities n, N, U, and Q, the entity-type X of the elementary entity UX has to be specified in the text and indicated by a subscript: nO2; Nce; QX.
§ Amount nX is an elementary quantity, converting the elementary unit [x] into moles [mol] using the Avogadro constant, NA.
$ Count NX equals the number of elementary entities UX. In the SI, the quantity 'count' is explicitly considered as an exception: "Each of the seven base quantities used in the SI is regarded as having its own dimension. .. All other quantities, with the exception of counts, are derived quantities" (Bureau International des Poids et Mesures 2019 The International System of Units (SI)). An elementary entity UX is not a count (UX is not a number of UX). NX has the dimension X of a count and UX has the dimension U of an elementary entity, and both quantities have the same unit, the 'elementary unit' [x].
Charge is a derived SI quantity. Charge is an elementary quantity, converting the elementary unit [x] into coulombs [C] using the elementary charge, e, or converting moles [mol] into coulombs [C] using the Faraday constant, F. zX is the charge number of elementary entity UX, which is a constant for any defined elementary entity UX. QX = nX·zX·F



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Entity, count, and number, and SI base quantities / SI base units
Quantity name Symbol Unit name Symbol Comment
unit-entity UX elementary unit [x] UX, UB; [x] not in SI
count NX elementary unit [x] NX, NB; [x] not in SI
number N - dimensionless = NX·UX-1
amount of substance nB mole [mol] nX, nB
electric current I ampere [A] A = C·s-1
time t second [s]
length l meter [m] SI: metre
mass m kilogram [kg]
thermodynamic temperature T kelvin [K]
luminous intensity IV candela [cd]
Fundamental relationships
» Avogadro constant
» Boltzmann constant
» elementary charge
» Faraday constant
» gas constant
SI and related concepts
» International System of Units
» International Union of Pure and Applied Chemistry, IUPAC
» entity
» quantity
» dimension
» format
» motive unit


Amount of substance, Avogadro constant, Concentration, Count, Density, Dimension, International System of Units, MitoFit 2020.4, Volume, Quantity