Physical Quantities and Units
Department of Chemistry
University of Reading, RG6 6AD
1. Reference material
Establishing a system of quantities and units for use in science, technology, and
commerce is a subject of vital importance to the world community. There are many
choices and conventions involved in achieving such a system, and these choices have
grown up over the years with the history of science and technology. Today these
decisions are made by several different international committees. These committees
and their publications are an important source of information.
The Bureau International des Poids et Mesures, the BIPM (in English: the
International Bureau of Weights and Measures) is established at the Pavillon de
Breteuil at Sèvres, near Paris, and this is the international home of metrology. It is
also the home of the annual meeting of the Comité International des Poids et Mesures,
the CIPM, and the meeting once every four years of the Conférence Générale des
Poids et Mesures, the CGPM. These bodies are responsible for establishing the
Système International des Unités, the SI (in English: the International System of
Units). The most important publication from the BIPM is the SI Brochure , which
is prepared by the Comité Consultatif des Unités, one of the consultative committees
of the CIPM. The SI Brochure provides the official definition of the SI. It is revised
every few years; the current 7th edition appeared in 1998. Thus the BIPM is
responsible for the world’s units of measurement.
The system of quantities (or physical quantities), and the equations relating these
quantities, is also an important part of the SI. This is sometimes referred to as the
International System of Quantities, or the ISQ. It has developed and grown as science
has grown, and it is still growing and changing. The most commonly used quantities,
with their names and symbols, are reviewed by the Technical Committee 12 of the
International Organization for Standardization, ISO/TC 12, who publish a review of
quantities and their symbols every few years . They also collaborate with the
International Electrotechnical Commission, Technical Committee 25, IEC/TC 25, and
they are together at present preparing a revised agreed list of quantities and symbols
used in science and technology which will displace reference  when it is published.
The International Unions in the various specialized fields also publish valuable
guides to quantities and units. In particular the International Union of Pure and
Applied Chemistry, IUPAC, publishes the so-called Green Book, ‘Quantities, Units
and Symbols in Physical Chemistry’  which is revised every few years, and the
International Union of Pure and Applied Physics (IUPAP) has a similar but less
extensive publication .
The Joint Committee on Guides in Metrology is another international committee
with publications in this area. Their International Vocabulary of Metrology  is an
important guide to the specialized language of metrology.
Finally there are of course very many individual papers in this field. I shall
mention only two. The first is a paper by my old teacher Edward Guggenheim ,
published in 1942, which is a review of how a system of quantities and units has to be
established, with reference to the system we all use today. Although it is now dated,
this paper has always been an inspiration to me, and is the model for many of the ideas
that I shall express in these lectures. The second is a recent paper by Jan de Boer 
published in a special issue of Metrologia in 1995 devoted to the subject of physical
units. There is interest in many papers in this issue, but de Boer’s paper – which is the
first in the issue – is an important current review of the subject.
A leading physicist of the 19th century is quoted as having said that one can
multiply together only numbers, and that the idea of multiplying a length by a mass is
nonsense. I take the opposite view, that we are entitled to multiply together – or to
divide one by the other – any two quantities, provided that our definition of
multiplication and division is self-consistent and obeys the associative and distributive
laws. (In general the commutative law is also obeyed in multiplication of quantities,
although operator quantities in quantum mechanics are an exception.) Thus if a reader
asks me what is the product of a kilogram and a metre, I would say the answer is a
kilogram metre. And if the reader suggests that this answer is somehow
unsatisfactory, I would point out that when a quarter is multiplied by three the answer
is three quarters, and when ? is multiplied by ?3 the answer is ?3?, and no simpler
answer is possible.
Similarly we may say that the product of three metres and four metres is twelve
metres squared, and the ratio of eight metres and two seconds is four metres per
second. In these lectures I shall consistently make use of the concept of multiplying or
dividing physical quantities by one another.
Having said that, I should add that multiplying quantities like a kilogram and a
metre to get a kilogram metre is not quite the same as multiplying 3 by 4 to get 12.
We should recognize that we are extending the meaning of ‘multiply’ to cover the
multiplication of quantities, and even to cover the multiplication of operators in
quantum mechanics. But this extension of the meaning of multiplication leads to no
inconsistencies, and proves to be a valuable addition to the language of science.
In science and technology we frequently write equations of the form
m = 150 g , T = 273 K ,
In each case the symbol on the left of the equation represents a quantity (mass,
temperature, or electric current in the examples above), and the symbols on the right
of the equation give the value of the quantity expressed as the product of a number
(the numerical value) and a unit (gram, kelvin or ampere in these examples). There
are conventions regarding the symbols used to denote quantities and units which we
review below, but one that we note here is that the symbols for quantities are generally
single letters which are always printed in an italic (sloping) font, whereas the symbols
for units are always in a roman (upright) font, as in these examples.
I = 2.5 A (2.1)
To give some further examples, suppose that v denotes the speed of a ship. Then
we may write
v = 4.00 knots
= 2.06 m/s
= 7.40 km/h
where knot, m/s, and km/h are units of speed. A knot is a nautical mile per hour,
and a nautical mile is 1852 m. The symbols m, km, s, and h are the symbols for a
metre, a kilometre, a second and an hour. Note that although the numerical values
and the units differ in the three equations (2.2), the product which expresses the value
of v is always the same.
It would however be wrong to write v = 4.00, or v = 2.06, or v = 7.40, or any
other number. v is a quantity, and the value of a quantity is in general the product of a
number and a unit. (The only exception is for a dimensionless quantity, whose value
is indeed given by a pure number, the unit being simply the number one, 1.)
We thus adopt a system in which we use symbols to denote quantities, and the
value of a quantity is always expressed as the product of a number and a unit.
Moreover we allow ourselves to multiply and divide quantities at will – and
correspondingly multiply and divide both the numbers and the units. This system is
known as quantity calculus, but it might be better known as the algebra of quantities,
because it is more to do with algebraic manipulation than calculus.
Quantity calculus is not universally used in science and technology. There are still
eminent scientists who use the alternative system in which symbols are used to
represent the numerical values of quantities, expressed in units which are specified in
the accompanying text. However I hope to convince you of the advantage of always
using quantity calculus, and its use is certainly becoming more widespread.
Examples of quantities are length, volume, mass, time, velocity, energy, power,
electric charge, and electric current. Further examples that are more specifically
concerned with chemistry are concentration, pressure, temperature, molar mass,
amount of substance (or chemical amount), mole fraction, surface tension, and electric
dipole moment. There are quantities associated with each specialized field of science,
and many examples span all fields. I shall use the word quantity to mean both a
quantity in the general sense (e.g. length, mass, time) and a quantity in the particular
sense (e.g. the length of my pencil, the mass of my car, the time interval between
sunrise and sunset at Stonehenge on the 21st of June 2000). This is common practice,
and the context is usually sufficient to distinguish these different meanings. I shall not
attempt to define the word quantity, other than by giving examples.
There is usually a recommended symbol (or sometimes two alternative symbols)
for each quantity, as illustrated in Table 1. Quantity symbols are generally single
letters of the Latin or Greek alphabet, but they may be further specified by subscripts
or superscripts, or information in brackets. They are always type-set in an italic
(sloping) font. Subscript indices and other qualifying information are set in either an
italic or a roman font depending on whether they represent either other quantities or
mi for the mass of the ith particle
Ti and Tf for the initial and final temperatures
M(NaCl) for the molar mass of NaCl
?fHo(CH4) for the standard enthalpy of formation of
length l, x
electric charge Q, or q
electric current I
Quantities are sometimes called physical quantities, but there seems little need for
the extra adjective, since they are concerned with all fields of science – chemistry and
biology just as much as physics.
Examples of units are the metre, centimetre, kilometre, and the ångström, all of
which are units of length; the gram, kilogram, milligram, and the unified atomic mass
unit, all of which are units of mass; the pascal, bar, millibar, and torr, all of which are
units of pressure. For each unit there is an internationally agreed symbol, as for the
examples in Table 2.
The use of the recommended symbols for units is mandatory. They are always
written in a roman (upright) font. They may consist of more than one letter, which is
never followed by a stop (as might be used for an abbreviation), and is never
pluralised by adding an s. However when writing products of units it is important to
always leave one space between the units for clarity (for example, m s denotes the
product of a metre and a second, but ms denotes a millisecond). A half-high dot
between the units may also be used to indicate multiplication. Another rule about the
symbols for units is that they begin with a capital letter when the unit is named for a
person (J for joule, K for kelvin), but with a lower case letter otherwise (s for second,
m for metre).
amount of substance n
p, or P
?, or ?
p, or ?
conversion back to the CGSF or CGSB system and hence to the SI. There seems to be
no easy answer to this question, so that to go from the SI to the ESU or EMU system is
much easier than to go in the reverse direction.
The Gaussian system: Many authors make use of the Gaussian system, which is a
mixture of the ESU and EMU systems in which those quantities and equations that
occur in electrostatics are expressed in the ESU system and those that occur in
electrodynamics are expressed in the EMU system. I personally find the mixture of the
two systems confusing, and there seems to be no authoritative definition of the way in
which the quantities and equations are to be divided between the two systems. For this
reason I shall make no further comment about the Gaussian system.
Quantities should for preference always be handled by the rules of quantity calculus,
according to which the symbol for a quantity should be used to denote its value
represented as the product of a number and a unit. This is to be contrasted with the
alternative system in which the symbol for a quantity is used to represent the numerical
value of that quantity, in units that are specified in the accompanying text.
Any system of units is always intimately connected with some corresponding
system of quantities, and a set of equations relating these quantities. A system of units
is constructed from a small number of base units that are independently defined, and
then a much larger number of derived units constructed as products of powers of the
base units according to a set of equations between the corresponding quantities. There
is choice both concerning the number of base units, and concerning which units are
taken to be the base units. There is also choice concerning the quantity equations used
to define the derived quantities, and hence the derived units. The choice of these
equations is a step which is often taken for granted, but it plays an important role, and
when there is the possibility of adopting alternative equations, alternative base
quantities, and even alternative numbers of base quantities, care is required to avoid
confusion. All these differences occur in choosing the equations and units of
electromagnetic theory, as between the SI, the ESU system and the EMU system.
Coherence in a system of units is desirable but not essential. Coherence leads to the
result that the equations between the numerical values of quantities always mimic the
corresponding equations between the quantities themselves. Most of the commonly
used systems of quantities and units, and in particular the SI, are chosen to have this
property. The advantage of coherence is lost by making use of multiple and sub-
multiple prefixes on units, as may be used with the SI, but the convenience of using
prefixes with units so that the combined prefix-unit has a magnitude comparable with
the quantity being measured generally outweighs the disadvantage of losing coherence.
In any case coherence can easily be re-established, if desired, by replacing the prefixes
with appropriate powers of ten.
Finally I would like to comment on the question of whether we should all make use
of only one system of quantities and units, namely the SI, the International System.
There is much to be said for this policy. Non-SI units are today all defined in terms of
SI units, so that the SI becomes the meeting ground for all systems of units. The SI is
the only system that is widely recognized and internationally agreed, so that it has a
clear advantage for establishing a dialogue with the rest of the world. I also believe that
it would simplify the teaching of science and technology to the next generation if we all
used only this system. This is particularly true in relation to electromagnetic theory: in
my opinion the quantities and units in this field in particular are much easier to
understand and more logical in the SI than in either the ESU or the EMU system, just
because the SI involves the extra electrical base quantity and dimension that is missing
in the ESU and EMU systems. Thus I am personally a strong supporter of the SI.
However this is a subject in which I think dogmatism is to be avoided. Scientists
must discover for themselves the best way of doing things to achieve the results that
they desire. We should always be prepared to listen to a defender of other systems. If,
for example, workers in a particular specialist field find the use of a particular non-SI
unit convenient and helpful, as in the case of X-ray crystallographers who wish to
defend the use of the ångström, then that is their right. The convenience to be found in
other systems and other units should be considered and balanced against the advantages
of the International System, and that should be the basis of our actions. I must add,
however, that those who chose to use non-SI units should always define the units they
use in terms of the SI. Otherwise they run the risk of not being understood.
I wish to record my debt to Edward Guggenheim, who not only originally
stimulated my interest in this subject, but who wrote the 1942 paper , which is not
much read today, but on which the present paper is based. Many changes have occurred
in the 58 years since Guggenheim’s paper was written, but his basic ideas remain
unchanged, and in some cases I have used his words almost unchanged in preparing the
present text. I also wish to thank the following colleagues who made valuable
comments on the draft of this text that have significantly improved the final version:
Philip Bunker, Tom Cvitaš, Walter Emerson, Per Jensen, Roberto Marquardt, Terry
Quinn, and Anders Thor.
 Le Système International d’Unités (the SI Brochure), Bureau International des
Poids et Mesures, 7th Edn. 1998, ISBN 92-822-2154-7.
Quantities and Units (the ISO Standards Handbook), International Standards
Organization, 3rd Edn. 1993, ISBN 92-67-10185-4.
Quantities, Units and Symbols in Physical Chemistry (the IUPAC Green Book),
Blackwell Science, 2nd Edn. 1993, ISBN 0-632-03583-8.
E R Cohen and P Giacomo, Symbols, Units, Nomenclature and Fundamental
Constants in Physics, 1987 revision, IUPAP 25 (IUPAP-SUNAMCO 87-1); also
published in Physica 146A (1987) 1-68.
The International Vocabulary of Metrology (the VIM), International Standards
Organization 2nd Edn. 1993, ISBN
E A Guggenheim, Units and Dimensions, Phil. Mag. 33 (1942) 479-496.
 J de Boer, On the History of Quantity Calculus and the International System,
Metrologia 31 (1995) 405-429.
Definitions of the seven base units adopted for the seven base quantities of the
International System, the SI, as of the year 2000.
metre: The metre is the length of path travelled by light in vacuum during a time
interval of 1/299 792 458 of a second (17th CGPM, 1983).
kilogram: The kilogram is the unit of mass; it is equal to the mass of the international
prototype of the kilogram (3rd CGPM, 1901).
second: The second is 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 (13th CGPM, 1967).
ampere: The ampere is that constant current which, if maintained in two straight
parallel conductors of infinite length, of negligible circular cross section, and placed 1
metre apart in vacuum, would produce between these conductors a force equal to
2×10–7 newton per metre of length (9th CGPM 1948).
kelvin: The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of
the thermodynamic temperature of the triple point of water (13th CGPM 1967).
mole: The mole is the amount of substance of a system which contains as many
elementary entities as there are atoms in 0.012 kilograms of carbon-12. When the mole
is used, the elementary entities must be specified and may be atoms, molecules, ions,
electrons, other particles, or specified groups of such particles (14th CGPM 1971).
candela: The candela is 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) watt per steradian (16th CGPM, 1979).