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Physical Quantities and Units

Ian Mills

Department of Chemistry

University of Reading, RG6 6AD

England

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 [1], 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 [2]. 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 [2] 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’ [3] which is revised every few years, and the

International Union of Pure and Applied Physics (IUPAP) has a similar but less

extensive publication [4].

The Joint Committee on Guides in Metrology is another international committee

with publications in this area. Their International Vocabulary of Metrology [5] is an

important guide to the specialized language of metrology.

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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 [6],

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 [7]

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.

2. Preamble

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)

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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.

(2.2a)

(2.2b)

(2.2c)

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.

3. Quantities

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

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italic or a roman font depending on whether they represent either other quantities or

mnemonic labels.

Examples:

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

methane

Table 1

quantity symbol

------------- --------

length l, x

volume V

mass m

time t

velocity

v

energy E

power P

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.

4. Units

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).

quantity

-------------

concentration

pressure

temperature

molar mass

amount of substance n

mole fraction

surface tension

electric dipole

moment

symbol

--------

c

p, or P

T

M

x

?, or ?

p, or ?

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Units are actually just particular examples of quantities, which have been chosen

for easy reference for the purpose of comparing the values of quantities of the same

kind. They are standard quantities, which should ideally be readily available, easily

realized in any laboratory, and internationally agreed and recognized.

Table 2

quantity unit

---------- -------------

length metre

centimetre

kilometre

ångström

mass gram

kilogram

milligram

atomic mass unit

pressure pascal

bar

millibar

torr

5. The equations of physics, and definitional constants

The equations of physics play an important role in the way we define quantities and

units. The area A of a rectangle is related to its length l and its height h according to

the equation

A = hl

The area of a triangle of base length l and height h is

A = ½ hl

and the area of an ellipse of length (major axis) l and height (minor axis) h is

A = (?/4) hl

In principle it would be possible to define area in a different way, so that the

numerical factor became 1 instead of ½ in the second equation, or 1 instead of (?/4) in

the third equation. But once the definition of area is settled, the numerical factor in

each formula is uniquely determined, and whatever definition is chosen the numerical

factor can be unity in at most one of these three formulae. It is useful to refer to such

numerical factors as definitional constants.

symbol for unit

-----------------------------------------

m

cm = 10–2 m

km = 103 m

Å = 10–10 m

g

kg = 103 g

mg = 10–3 g

u = (1/6.022 137 × 1023) g

Pa

bar = 105 Pa

mbar = 100 Pa

Torr = 133.3 Pa

(5.1)

(5.2)

(5.3)

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A more instructive example of a definitional constant is the factor of ½ in the

formula

T = ½ mv2 (5.4)

relating the kinetic energy T to mass m and velocity v. This formula is related to the

formula expressing the change in potential energy U as the product of a force F acting

over a distance x

U = – ? F dx

It would in principle be possible to re-define energy so that the numerical factor in

(5.4) became 1, but then the factor 2 would appear in (5.5), so that the two formulae

would take the form T? = mv2 and U? = – ? 2 F dx. I have added a prime to the

symbols for kinetic and potential energy here in order to emphasize that they are not

the same quantities that appear on the left of (5.4) and (5.5), because we have defined

energy in a different way, so that T ? = 2T and U ? = 2U. Changing the value of a

definitional constant in any equation corresponds to changing the definition of one of

the quantities in the equation, and to avoid confusion we should always then adopt a

new symbol for the re-defined quantity.

As a third example of a definitional constant we may consider the commutator of

the operators for coordinate q and momentum p in quantum mechanics, given by the

relation

qp – pq = i h / 2?

where h is the Planck constant. The factor 1/2? in this equation may be regarded as a

definitional constant. Dirac chose to re-define the constant h in this equation so that

the definitional constant became 1, by writing (5.6) in the form

qp – pq = i

(5.5)

(5.6)

? (5.7)

where it is clear that

in common use today, but there is no confusion because the different symbols h and

are used for the two forms of the Planck constant. The change in the definitional

constant between (5.6) and (5.7) is sometimes called rationalization. A further

example of rationalization occurs in the equations of electromagnetic theory, as we

shall discuss below in Section 11.

The equations of physics play a key role in defining the quantities that we use, and

hence in defining units. These equations are for the most part well established today,

so that definitional constants such as the factor ½ in (5.4) are taken for granted, and

we never even think of defining things another way. The example illustrated by

equations (5.6) and (5.7) above, and the further example of rationalization in the

equations of electromagnetic theory to be discussed in Section 11 below, are the only

examples that I know of where two alternative choices for a definitional constant are

both still in common use.

? = h / 2?. In this case both of the expressions (5.6) and (5.7) are

?

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6. Coherent systems of units

Equation (2.1) expresses the value of a quantity as a product of a number and a

unit. A useful general way of expressing such equations for any quantity X,

introduced by James Clerk Maxwell in the 19th century, is to write

X = {X}[X]

where {X} denotes the numerical value and [X] denotes the unit of the quantity X.

Now consider once again the equation for kinetic energy

T = ½ mv2

(6.1)

(6.2)

If we write each of T, m and v as the product of a numerical value and a unit we have

{T}[T] = ½{m}{v}2 [m][v]2 (6.3)

This expression is valid for any choice of the units. Suppose we now choose the units

such that

[T] = [m][v]2 (6.4)

Then it follows that

{T} = ½{m}{v}2 (6.5)

a relation of precisely the same form as (6.2). It is important to appreciate the

difference between the equations (6.2) and (6.5). Equation (6.2) is an equation

between the quantities T, m, and v, and it is true whatever units may be used (provided

only that we adopt the equations of physics as currently accepted). Equation (6.5) is

an equation between the numerical values {T}, {m} and {v}, and it is true only for a

choice of units that satisfies equation (6.4).

A set of units that satisfies equations like (6.4), without numerical factors, is called

a coherent set of units. Coherent units can only have meaning in relation to a set of

equations relating the quantities involved. The use of coherent units is not essential,

but it is a matter of convenience, because it leads to relations between the numerical

values that are exactly similar to the relations between the corresponding quantities.

7. Dimensions

I shall use the concept of dimensions only in the following sense. We shall

describe a pure number as having no dimensions, or as being dimensionless. If the

ratio of two quantities is a pure number, then the two quantities will be said to have

the same dimension. Otherwise I shall not use the word dimension, and I shall not

attempt to define it further.

According to this terminology any quantity X has the same dimension as its unit

[X]. Again, in any equation between physical quantities the two sides of the equation

must have the same dimension. The subject of dimensional analysis consists of using

this fact to verify that some formula is at least a possible relation between the

quantities concerned. Since the dimension of any quantity is the same as the

dimension of its unit, a dimensional check may be made simply by comparing the

units on each side of an equation, which must be consistent with each other.

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8. Base quantities and units; the equations of physics; derived quantities and

units; and the International System

It is clear that units play a key role in international science, technology, and trade.

To give a few examples, we need to be able to construct the components of an

airplane, or the components of a complex electronic device, in different parts of the

world, and when we bring them together they must fit. The international trade in wine

is based on our knowledge that a litre in Australia or South Africa is the same as a

litre in France or Chile. We need to be able to measure the concentration of ozone in

the arctic atmosphere, at a known location and altitude, and to compare the result

today with the result ten years ago with confidence. We need to be able to measure

and compare the concentrations of drugs, or of microtoxins in food, with confidence.

We therefore need a system of units that is universally accepted, readily available

to all, and for which the units are easily realized. To realize a unit is to use it to

measure the value of a quantity. The International System of units, the Système

International, or the SI, is our attempt to fulfill these requirements.

To establish a system of quantities and units we proceed in the following steps.

(i) We choose a set of base quantities

(ii) We choose a set of equations to be used to define all other derived quantities in

terms of the base quantities

(iii) We choose a set of base units, one for each base quantity

(iv) Derived units are then defined in terms of the base units, using the chosen

equations. The derived units will form a coherent set provided that definitional

constants are omitted from all the defining equations for the derived units (as in

equation (6.4)).

We may illustrate this procedure with the example of the International System of

Units, the SI. The SI is founded on seven base quantities, for each of which we have a

carefully defined base unit. These are listed in Table 3. The actual definitions of the

base units must satisfy the requirements of being readily available, and easy to realize

with the highest possible precision. The definitions of the base units in the SI are

given in references [1], [2] and [3]. They are not important for the present discussion,

but for convenience they are reproduced in the Appendix.

The choice of base quantities is in principle arbitrary, and has been chosen in the

SI for historical reasons. All other quantities are called derived quantities, and are

defined in terms of the base quantities by the equations of physics. For the most part

the choice of these defining equations is obvious, and is taken for granted and not

often discussed.

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Table 3

Base quantities and units in the International System

base quantity

--------------------

length

mass

time

electric current

thermodynamic

temperature

amount of substance

luminous intensity

symbol for quantity

------------------------

l, x

m

t

I

T

base unit

-------------- -------------------

metre m

kilogram kg

second s

ampere A

kelvin K

symbol for unit

n

Iv

mole

candela

mol

cd

An example where a choice of alternative equations could be made arises for the

definition of force, F, in terms of the base quantities in Table 3. One might consider

achieving this using Newton’s first law involving the concept of inertia,

F = K ma , or F dt = K m dv

(8.1)

where K is a constant, m denotes mass, and a = dv/dt denotes acceleration.

Alternatively we might consider using Newton’s law of gravitation

F = G m1m2 / r2

where G is a constant, and F denotes the force of gravitational attraction between two

bodies of mass m1 and m2 at a distance r apart. In practice we use the equation (8.1) to

define force, and we take the constant K to be dimensionless and equal to 1. The

dimension of force is thus the same as that of mass times acceleration. The constant

G in (8.2) then has the same dimension as Fr2/m2, and the value of G must be

determined by experiment. Had we used equation (8.2) rather than (8.1) to define

force, and then taken G to be dimensionless and equal to 1, the dimension of force

would then have been the same as that of mass squared divided by length squared.

The constant K would then have had to be determined by experiment, and it would not

have been dimensionless. The decision to use (8.1) rather than (8.2) to define the

derived quantity force was taken by default long ago, presumably because the inertial

equations (8.1) play a greater role in daily life than the gravitational equation (8.2).

In any system of units the base units must be independently defined. The units for

all other quantities, which are derived quantities, are called derived units. They are

defined in terms of the base units by equations analogous to (6.4), obtained from the

corresponding equations between the quantities by omitting all definitional constants.

This procedure will give exactly one unit for each quantity, and the units will

necessarily be coherent.

Since the number of quantities in science is without limit, it is not possible to

provide a list of derived quantities and derived units, but a few selected examples are

shown in Table 4. In general the symbol for a derived unit is written simply as the

corresponding product of the base units, so that – for example – since the base unit for

length is the metre, m, and the base unit for time is the second, s, then the coherent

(8.2)

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derived unit for speed is the metre per second, m/s. However some derived units are

so frequently used that they are given special names. Examples are the newton, N, a

special name for the SI derived unit of force, and the joule, J, a special name for the

SI derived unit of energy. There are at present 22 derived units for which special

names have been approved, some of which appear in Table 4; the full list is given in

any of references [1] to [4]. These special names should be regarded as no more than

a shorthand for the corresponding product of base units, so that, for example, by

definition

N = m kg s–2, and J = m2 kg s–2

In Table 4, for those derived units that have a special name, this is given in the column

‘SI unit name’; otherwise this column is left blank. Derived units can frequently be

expressed in a number of different ways by making use of the special names, as shown

in the table; however these alternative expressions are of course all exactly equivalent.

In every case the final entry in each row of the table gives the expression for the

derived unit in terms of base units. More extensive lists of quantities, with their

recommended symbols (not included in Table 4) and their SI units, are given in

references [2] and [3].

(8.3)

This system will generally have the result that two or more different quantities may

have the same unit. An example in the SI is the quantity torque (or moment of force)

and energy (force times distance, or mass times velocity squared), for both of which

the SI unit is N m = m2 kg s–2. Another example is entropy and heat capacity, both of

which have the SI unit J/K = m2 kg s–2 K–1.

For five of the entries in Table 4 the SI unit is given as one, 1. These derived

quantities are described as dimensionless in the SI, and values of these quantities are

always pure numbers. For example, the quantity plane angle is defined as the ratio of

the arc distance over the radius, which is clearly dimensionless since it is the ratio of

two lengths. However for plane angle and also for solid angle it still proves

convenient to give the unit a name: the unit of plane angle is the radian, symbol rad,

and the unit of solid angle is the steradian, symbol sr. Nonetheless it is actually true

that the both the radian and the steradian are equal to one, rad = 1 and sr = 1.

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Table 4

Examples of derived quantities and units in the International System

derived quantity

-------------------

velocity, speed

linear momentum

angular momentum

plane angle

solid angle

frequency

angular velocity

force

pressure

energy, work, heat

power, radiant flux

heat capacity, entropy

thermal conductivity

electric charge

electric potential difference

SI unit

name

----------

radian

steradian

hertz

newton

pascal

joule

watt

coulomb

volt

SI unit symbol

--------------------------------------------------

m s–1

m kg s–1

m2 kg s–1

rad = 1

sr = 1

Hz = s–1

rad s–1

N = m kg s–2

Pa = N/m2

J = N m

W = J s–1

J/K

W m–1 K–1 = m kg s–3 K–1

C = A s

V = W/A

? = V/A

F = C/V

A/m2

mol/m3

kg/mol

N/m

1

1

1

= s–1

= m–1 kg s–2

= m2 kg s–2

= m2 kg s–3

= m2 kg s–2 K–1

= s A

= m2 kg s–3 A–1

= m2 kg s–3 A–2

= m–2 kg–1 s4 A2

= m–2 A

= m–3 mol

= kg mol–1

= kg s–2

electric resistance

electric capacitance

electric current density

concentration

molar mass

surface tension

number of molecules

mole fraction

refractive index

ohm

farad

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9. The SI prefixes

As observed in the previous section, for any specified quantity there is only one

coherent SI unit. For particular applications, however, the SI unit may be

inconveniently large or inconveniently small. For example, the SI unit of length is the

metre, symbol m. However for a designer and builder of wristwatches the metre is

inconveniently large, and he may well prefer to work in centimetres or millimetres.

For a spectroscopist studying the wavelength of light, the micrometre or the

nanometre is more useful, and for a mapmaker the kilometre is more useful. Similar

comments apply to any of the other units.

For this reason a set of SI multiple and sub-multiple prefixes has been adopted,

many of which are familiar to us in everyday life. They represent multiples in integral

powers of ten, and they may be applied to any SI unit. They are shown in Table 5, and

a complete list is also given in any of the references [1], [2] or [3]. Thus a milligram,

symbol mg, is 10–3 g; a micrometre, ?m, is 10–6 m; a gigahertz, GHz, is 109 Hz.

These prefixes are useful, and are widely used.

The kilogram is a base unit whose name and symbol incorporate the prefix kilo.

This breaks the rule that base units should involve no prefixes, but in this case the

name and symbol were in use before the SI was established. Thus the kilogram is the

base unit of mass despite the fact that its name incorporates a prefix. Multiples and

sub-multiples of the kilogram are symbolized by adding prefixes to the symbol g for

the gram (e.g. milligram, mg, not microkilogram, ?kg, for 10–6 kg).

There is one point to be noted about using the prefixes: it is only the SI units

without prefixes that form a coherent set of units. Thus if one wishes to make use of

coherence, one must use SI units without any prefix, replacing each prefix by the

appropriate power of ten. However coherence in a set of units is not essential, and the

convenience of using units of a magnitude comparable to the quantity being measured

is a strong argument for the use of prefixes. Provided that one always uses quantity

calculus, in which quantities are expressed as the product of a number and a unit, all

manipulations (such as the substitution of values for quantities in equations and the

conversion of units) is straightforward whether or not the units are coherent.

Examples are shown in Chapter 7 of the IUPAC Green Book [3]. Difficulties arise

only when symbols are used to represent the numerical values of quantities in

particular units specified elsewhere in the text.

There is a further point to note concerning the naming of units in the SI. The

name ‘SI unit’ should be used only for the base units and derived units that do not

include any multiple or sub-multiple prefixes. This ensures that SI units form a

coherent set, and it is the recommendation of the CIPM on the advice of the CCU that

the name ‘SI unit’ should be used only in this restricted sense. However SI units to

which a multiple or sub-multiple prefix has been added are clearly a part of the

International System, and there is thus a need for a name to cover the complete set of

units in the International System, including those with multiple or sub-multiple

prefixes in addition to the base units and derived units without prefixes. It is the

present recommendation that this complete set of units should be referred to as ‘units

of the SI’, but this is not a good name, because to make a distinction between ‘SI

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units’ and ‘units of the SI’ is artificial in the normal meaning of words in the English

language. This is a problem that needs further attention from the CCU and the CIPM,

which are the international committees responsible for the SI. In the meantime my

personal inclination is to retain the name ‘SI units’ to span only the coherent set that

do not include any prefixes (as currently recommended by the CCU), and to refer to

the complete set as ‘the SI units including prefixes’. This is perhaps somewhat

cumbersome, but it is at least clear.

Table 5

SI multiple and submultiple prefixes

prefix

symbol

---------

d

c

m

?

n

p

f

a

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submultiple prefix

name

-------

deci

centi

milli

multiple

----------

10

102

103

106

109

1012

1015

1018

1021

1024

prefix

name

--------

deca

hecto

kilo

prefix

symbol

---------

da

h

k

---------------

10–1

10–2

10–3

10–6

10–9

10–12

10–15

10–18

10–21

10–24

micro

nano

pico

femto

atto

zepto

yocto

mega

giga

tera

peta

exa

zetta

yotta

M

G

T

P

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Y

varenna-a-1aug00

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10. On the number of independent base quantities and units

It is appropriate to consider, at this stage, what is the correct number of independent

base quantities in a system of quantities and units. In the SI this number is taken to be

seven, as given in Table 3 in Section 8. However some people might question whether

it is essential that this number should be seven. Why, for example, should we not

regard the quantity luminous intensity , which is taken to be a base quantity in the SI, as

being defined to be equal to the derived quantity power per solid angle, with the derived

SI unit watt per steradian, W/sr, rather than the base unit candela, cd?

The meaning of the word dimension has been deliberately restricted in this paper to

the usage specified in Section 7. However most authors extend the meaning of the

word dimension to include such statements as: ‘the dimensions of velocity can be

expressed in terms of the dimensions of length and time’. If we temporarily adopt this

usage, then the question in the previous paragraph can be re-phrased as: What is the

least number of fundamental quantities such that the dimensions of all other quantities

can be expressed in terms of the dimensions of the fundamental quantities?

The following discussion is based on that presented by Guggenheim in 1942 [6].

Before the introduction into the SI in the 1960s of the base quantities amount of

substance and luminous intensity, with the mole and the candela as the corresponding

base units, it was argued that the number of base dimensions was five. This was

justified by arguing that the study of mechanics and dynamics requires three base

quantities: length, mass and time. Extending to the study of electrical phenomena

requires the addition of an electrical quantity, such as electric current, and then the

extension to thermodynamics requires the further introduction of a thermal quantity,

which we may take to be thermodynamic temperature. This argument seems reasonable.

Of course there still remains the choice of which particular quantities are regarded as

the independent base quantities. For example it might seem more logical to take energy

rather than mass as a base quantity in mechanics and dynamics, or electric charge rather

electric current as the extra base quantity in electrical phenomena, or entropy rather

than temperature as the extra base quantity in thermodynamics. However that does not

seem to change the choice of five as the number of base quantities.

Guggenheim then argues as follows. Let us initially take the five base quantities to

be length, time, energy, electric potential, and temperature. Now it is widely accepted

that the speed of light in vacuum, c0, is a fundamental constant of nature, and this can

be used to relate any distance to an equivalent time interval through the equation

l = c0 t

Thus one might argue that time and length have the same dimension, and are not

independent, so that we can dispense with length as an independent dimension. (It is

interesting that in 1983 the base unit of length, the metre, was re-defined as the distance

traveled by light in a vacuum in a specified length of time – although we still regard

length and time as independent dimensions. But this was long after Guggenheim was

writing his paper.)

(10.1)

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In a similar way energy can be related to time through the equation

E = h?

using the Planck constant h, another of the fundamental constants, noting that frequency

? is the reciprocal of the characteristic period or time interval T of an oscillation. Thus

energy may be regarded as having the same dimension as (time)–1. Then the relation

E = kT

can be used to relate energy to temperature through the Boltzmann constant k, and

finally the relation

E = eU

can be used to relate energy to electric potential difference U through the elementary

charge e (the charge on a proton). Thus by using the fundamental constants c0, h, k, and

e we may argue that there is only one independent dimension where originally we had

considered there to be five. Guggenheim comments that this conclusion is, perhaps, not

absurd, but if it is accepted then there remains nothing of dimensional analysis, and we

might as well give up altogether the concept of dimensions. It is clear that reducing the

number of dimensions below five reduces the power of dimensional analysis.

In the following section we discuss the CGS-ESU and CGS-EMU systems, which

differ from the SI in having one less base quantity and one less independent dimension

than the SI, in exactly the manner described above.

Consider now whether the number of independent base quantities may not be greater

than five. We have tacitly assumed in the discussion so far that lengths in different

directions have the same dimension. Although this seems reasonable in general, in a

particular field of study it may be that vertical and horizontal lengths never enter any

formula in the same way. Ships navigating the sea used to use the fathom as a unit of

length to measure depths and the nautical mile to measure horizontal distances, and no

conversion between these units was ever required. Similarly airplanes measure height

in thousands of feet, but horizontal distances in nautical miles. Thus in these fields it

may be useful to regard vertical and horizontal distances as dimensionally different.

In the study of liquid flow through a tube or the flow of electricity along a wire the

important characteristics of the tube or the wire are its length and its cross sectional

area. The fact that the cross section can, if desired, be considered to have the same

dimensions as a length squared is here largely irrelevant, and the power of dimensional

analysis is actually increased by regarding cross sectional area and length as

independent fundamental quantities.

The conclusion of this discussion is that the number of independent base quantities,

or the number of independent dimensions in a system of quantities and units, is a matter

of choice. In the SI, we have made the choice of just seven base quantities as described

earlier, but it is entirely possible to make other choices. The choice clearly affects the

power of dimensional analysis, and Guggenheim remarks that “if in the same problem

or set of problems, two authors make a different choice, the one choosing the greater

number is likely to be the more competent physicist”.

(10.2)

(10.3)

(10.4)

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11. Electromagnetic quantities and units

Historically electromagnetic theory was first developed using two different but

related systems of quantities and units, the CGS-ESU system and the CGS-EMU

system, where CGS stands for centimetre-gram-second, and ESU and EMU for

electrostatic and electromagnetic units. For brevity I shall refer to these as the ESU and

EMU systems. These systems gave rise to base units of an inconvenient magnitude,

and in the years around 1900 a set of so-called ‘practical’ electrical units was developed

including the ohm, the volt, and the ampere. It was then shown by Giorgi in 1901 that a

coherent system of units based on the metre, kilogram, second, and ampere could be

developed in which the joule, the volt, the ohm, and the coulomb naturally appeared as

coherent derived units. This was called the MKSA (metre-kilogram-second-ampere)

system, and our present International System (the SI) is a straightforward development

of the MKSA system to include temperature, amount of substance, and luminous

intensity as base quantities with the base units kelvin, mole and candela. However

electromagnetic theory does not normally involve the latter three quantities, so that it

may be described in terms of only the first four base quantities. In this sense the SI is

equivalent to the MKSA system.

Relating the SI to the ESU and EMU systems is important, but it is not trivial,

because the systems involve slightly different defining equations with definitional

constants that differ by the factor 4?, a different number of base quantities, and

different definitions of the base units.

It may seem to the younger generation of scientists that there is no longer a reason to

use any system other than the SI, but this is not quite true. In the first place there is a

lot of valuable published literature in science in which the results are expressed in the

ESU and EMU systems. Secondly there are distinguished scientists who argue in

favour of the ESU and EMU systems over the SI. We should at least be prepared to

listen to their argument. Finally a strong scientist who expresses himself clearly and

unambiguously should surely be able to work in any internally consistent system of

quantities and units. It would seem to be an admission of failure to say that one is only

prepared to work in the SI. This section is therefore devoted to describing and building

a bridge between these different systems of quantities and units. The discussion which

follows is closely similar to that presented in Chapter 7 (Section 7.3) of the 1993

edition of the IUPAC Green Book [3].

The translation of the SI (or the MKSA system) into the ESU and EMU systems may

be considered in the following steps.

(i) We change the definitional constants in the equations of electromagnetic theory by

the factor 4?, and make corresponding changes in some of the quantities that appear

in the equations. This change is described as changing from rationalized to non-

rationalized equations.

(ii) We change the four base units from those of the MKSA system to four new units

which I shall call the CGSF units for the electrostatic system, or the CGSB units for

the electromagnetic system.

(iii) Finally we reduce the number of base quantities and units by one in each case, to

obtain the ESU or the EMU system respectively.

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I shall now describe these changes in greater detail.

We start with the formula for the electrostatic force of repulsion F between two

charges q1 and q2 separated by a distance r, and the electromagnetic force of attraction

between two current elements i1dl1 and i2dl2, in a vacuum. In the system of quantity

equations used in the SI these may be written

F = – q1q2 r / 4?? 0 r 3

d2F = (?0/4?) i1dl1×(i2dl2×r) / r3

The quantities ? 0 and ? 0, the permittivity and permeability of a vacuum respectively,

have the values

? 0 = (107/4?c02) kg–1 m–1 C2 = 8.854 188 ×10–12 C2 m–1 J–1

?0 = 4??×10–7 N A–2 = 1.256 637 ×10–6 N A–2

where c0 is the speed of light in a vacuum. The value of ?0 given in (11.4) results from

the definition of the ampere (given in the appendix); in fact one could say that the

choice of this value of ?0 may be taken as the definition of the ampere. The value of ?0

given in (11.3) then results from the Maxwell relation

?0?????????c???

(11.1)

(11.2)

(11.3)

(11.4)

???????

The factors of 4? in equations (11.1) and (11.2) are definitional constants. As

discussed earlier, in Section 5, the use of definitional constants of this kind is a matter

of choice. They are introduced in the SI equations (11.1) and (11.2) in order to avoid

their appearance in a number of other equations in electromagnetic theory. However in

the ESU and EMU systems of units and equations, which were developed before the SI,

these factors of 4? were not introduced, so that the equations equivalent to (11.1)

through (11.5) were written in a different way. When these factors of 4? were

introduced the new equations were described as ‘rationalized’. The earlier equations

are sometimes described as ‘unrationalized’, or ‘non-rationalized’, or ‘irrational’.

Guggenheim describes them as ‘orthodox’. We need some name to label the two sets

of equations, and I shall use the name ‘rationalized’ to describe the equations (11.1)

through (11.5) and the related system of electromagnetic equations, and ‘irrational’ to

describe the equivalent equations without the factors of 4?.

As described in Section 5, when a definitional constant is changed in an equation

one of the other factors in the equation must also change. In (11.1) and (11.2) it is the

constants ?0 and ?0 which take different values in the irrational equations. In fact there

are a number of quantities in electromagnetic theory which take different values in the

two different sets of equations. We therefore need different symbols for the new

quantities.

Unfortunately almost all previous authors have used exactly the same symbols for

the electromagnetic quantities in the rationalized and the irrational sets of equations.

This makes it almost impossible to discuss the relation between the two sets of

quantities and equations. It is analogous to the situation that would exist in quantum

mechanics if we used the same symbol h for both the Planck constant and the Planck

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18 18

constant divided by 2?. Fortunately the different symbols h and ? are used in this case,

so that we are able to write equations like h = 2??.

In these notes I shall use the superscript (ir), as a mnemonic for irrational, to label

those quantities that take different values in the irrational set of equations. There is not

much precedent for this, and it is admittedly a clumsy notation, but I have not been able

to devise a simpler alternative. I would like to add that in describing the ESU and

EMU equations as irrational I am not trying to deprecate them: I believe that a good

scientist should be able to work in either system. I am simply searching for a simple

way of distinguishing between the different quantities. The SI equations have been

described as rationalized for many years, and irrational seems as good an adjective as I

can devise for the alternative equations.

With this lengthy preamble, we can now consider the irrational equations equivalent

to (11.1) through (11.5) above. They take the form:

F = – q1q2 r /? 0(ir) r 3

d2F = ?0(ir) i1dl1×(i2dl2×r) / r3

? 0(ir) = (107/c02) m–1 kg–1 C2 = 1.112 650 ×10–10 m–1 J–1 C2

?0(ir) = 10–7 N A–2

? 0(ir) ???ir????????c???

There are altogether eight quantities that are defined in a different way in the

rationalized from the irrational equations, which are as follows:

(11.6)

(11.7)

(11.8)

(11.9)

????????

?0(ir) = 4??0

? (ir) = 4??

?0(ir) = ?0/4?

? (ir) = ?/4??

D(ir) = 4?D

H(ir) = 4?H

?e(ir) = ?e/4?

?(ir) = ? /4?

(11.11)

(11.12)

(11.13)

????????

(11.15)

(11.16)

(11.17)

(11.18)

where D is the electric displacement, H is the magnetic field, ?e is the electric

susceptibility, and ? is the magnetic susceptibility. All of the equations of

electromagnetic theory can now be transformed from the rational to the corresponding

irrational form by using equations (11.11) through (11.18) to eliminate ?0, ?, ?0, ?, D,

H, ?e and ? in favour of the corresponding irrational quantities distinguished by a

superscript (ir). The irrational equations obtained in this way are the basis of the ESU

and EMU systems of quantities and equations.

Next we must change the base units from the four MKSA units to the four CGSF

units or the four CGSB units.

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19 19

The CGSF system: The CGSF system may be described in terms of the four base

quantities length, mass, time and electric charge, with the base units centimetre, gram,

second, and franklin. I shall use the name franklin, symbol Fr, for the quantity that is

customarily named the ‘esu of charge’, defined as that charge that repels a similar

charge with a force of 1 dyne at a distance of 1 centimetre. (The name ‘franklin’ for the

esu of charge was suggested by Guggenheim in 1942 [6], and although it has not been

much used I use it here for brevity and clarity.) Comparing with (11.6) we see that this

definition of the franklin gives ?0(ir) the value

?0(ir) = 1 Fr2 dyn–1 cm–2 = 1 Fr2 cm–1 erg–1

Comparing this with equations (11.11) and (11.3), and noting that cm = 10–2 m and

erg = 10–7 J, we find the relation between the franklin and the coulomb to be

1 Fr = (10/?) C = 3.335 641 ×10–10 C

Here I have used the symbol ? for the pure number 2.997 924 58 ×1010 = (c0/cm s–1). I

suggest that the system of quantities and units defined in this way might be best called

the CGSF system, i.e. the centimetre-gram-second-franklin system. All other derived

units in the CGSF system are obtained as products of powers of the base units (cm, g, s,

Fr) in the usual way. The CGSF system is a coherent system of units, just as is the SI,

for which the base units are the (m, kg, s, A).

(11.19)

(11.20)

The ESU system: In the ESU system as it is customarily presented the number of

independent dimensions is reduced by one, by defining the dimension of charge to be

the same as that of [(energy)×(length)]½, so that

1 Fr2 = 1 erg cm (11.21)

Although this seems an odd identity to adopt, it has the effect that in the ESU system

?0(ir) = 1 Fr2 erg–1 cm–1 = 1 (11.22)

so that ?0(ir) disappears entirely from all equations. Thus equation (11.6), for example,

may be written in the simpler form

F = – q1q2 r / r 3

The remaining equations of the ESU system are obtained from the CGSF irrational

equations in a similar way, simply by setting ? 0(ir) = 1 and ?0(ir) = 1/c02.

(11.23)

The CGSB system: The CGSB system is built on exactly the same (irrational)

equations as the CGSF system, but the four base quantities are length, mass, time, and

electric current, with the base units centimetre, gram, second, and biot. The name biot,

symbol Bi, is used for the quantity that is customarily named the ‘emu of current’,

defined as that electric current such that the force between two parallel wires, 1 cm

apart in a vacuum, each carrying a current of one biot, is 2 dyne per centimetre of wire.

(Like the franklin, the name ‘biot’ is not much used, but I use it here for brevity and

clarity.) Comparing with (11.7) we see that this definition of the biot gives ?0(ir) the

value

?0(ir) = 1 dyn Bi–2 = 1 erg cm–1 Bi–2

(11.24)

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Comparing this with equations (11.13) and (11.4) we find the relation between the biot

and the ampere to be

1 Bi = 10 A

The electrostatic unit of current is the franklin per second, Fr/s, which – from equation

(11.20) – is equal to (10/?) C/s = (10/?) A, and from this it follows that the relation

between the biot and the franklin per second is

1 Bi = ? Fr/s = 2.997 924 58 ×1010 Fr/s

I suggest that the system of quantities and units defined in this way might best be called

the CGSB system, i.e. the centimetre-gram-second-biot system. All other derived units

in the CGSB system are obtained as products of powers of the base units (cm, g, s, Bi)

in the usual way. The CGSB system of units is a coherent system of units, just as is the

SI for which the base units are the (m, kg, s, A). A number of the electromagnetic units

in the CGSB system are related to the corresponding units of the CGSF system by a

factor equal to the pure number ? = 2.997 924 58 ×1010.

(11.25)

(11.26)

The EMU system: In the EMU system, as it is customarily presented, the number of

independent dimensions is reduced by one, by defining the dimension of electric current

to be the same as that of (force)½, so that

1 Bi2 = 1 dyn = 1 g cm s–2

Again, as in the ESU system, it seems an odd identity to make (electric current)2

dimensionally the same as (force), but it has the simplifying effect that in the EMU

system

?0(ir) = 1 dyn Bi–2 = 1

so that ?0(ir) disappears entirely from all equations. Thus in the EMU system equation

(11.7), for example, takes the form

d2F = i1dl1×(i2dl2×r) / r3

The remaining equations of the EMU system are obtained from the CGSB irrational

equations in a similar way, by setting ?0(ir) = 1 and ?0(ir) = 1/c02.

(11.27)

(11.28)

(11.29)

Review: When comparing or converting the values of electromagnetic quantities

expressed in the different systems, the SI, ESU or EMU systems, it is my opinion that it

is always simplest not to reduce the number of dimensions by one as in the ESU and

EMU systems, but to use the CGSF rather than the ESU system, and the CGSB rather

than the EMU system. The conversion of the equations, and of the values of quantities,

between the SI and the CGSF and CGSB systems is a simple task using the relations

presented in this section, because all three systems have the same number of base

quantities and base units.

The final step of setting ?0(ir) = 1 in the ESU system, or ?0(ir) = 1 in the EMU

system, and then eliminating the franklin and the biot using equations (11.21) and

(11.27), is also an easy change to make. The difficulty arises when one is given an

equation, or the value of a quantity, in the ESU or the EMU system, because one has the

problem of knowing where to insert the missing factors of ?0(ir) or ?0(ir), or of knowing

where to insert the missing unit, the franklin or the biot, in order to make the

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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.

12. Summary

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

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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.

Acknowledgement

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 [6], 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.

References

[1] 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.

[2]

[3]

[4]

[5]

[6]

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[7] J de Boer, On the History of Quantity Calculus and the International System,

Metrologia 31 (1995) 405-429.

Appendix

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).

varenna-b-1aug00.doc