Dear Marcus, and All,
I have cobbled together some thoughts on coherence, but as yet I haven't
edited this closely. I understand you have some tight deadlines.
Coherence
A coherent system of units uses a small group of base units, and these can
be multiplied or divided to produce all of the necessary practical units for
all of our activities, without the need to introduce any numerical
constants. Examples are best to demonstrate coherence.
This first example is a story about floor covering for a kitchen floor.
A few months ago, a friend of mine, in the USA, went to buy some vinyl floor
covering. He was shown some imported material that was about 6 ft 6 3/4 in
wide (My friend reckoned that they were actually 2 metres wide, but the shop
assistant didn't know that).
As my friend needed a little less than 2.5 metres, he asked for 8 feet 2 1/2
inches.
When my friend asked the price, he found that the assistant had to work it
out to figure a price in square yards. Now let's see: 8' 2 1/2" x 6' 6 3/4"
÷ 9 = ?
My friend says the assistant was there, battling with his calculator, for a
long time!
In the meantime, my friend multiplied 2 metres x 2.5 metres to get 5 square
metres (m2).
>From this story, the first obvious point is that the calculations are easier
using metres than they are using feet and inches, but the reason for this is
not immediately clear it is due, in large part, to the coherence of the SI
(metric) units.
In SI, a coherent system, there is only one unit for measuring area a
square metre, and to calculate a quantity of area you multiply a length in
metres by another length in metres. In this example:
2 metres x 2.5 metres = 5 square metres
Compare this with the calculation that the sales assistant was attempting on
his calculator. If he were wise, he would begin from the end he required
(square yards) and convert all his original measures into yards. As feet and
inches are pre-decimal, they have traditionally been used with vulgar
fractions rather than decimal fractions.
Length: 8 feet 2 1/2 inches = 2 yards 2 feet 2 1/2 inches
= 2 yards + 2/3 yard + 2/36 yard + 1/72 yard
= 2 yards + 48/72 yard + 4/72 yard + 1/72 yard
= 2 53/72 yards
Width: 6 feet 6 3/4 inches = 2 yards + 6/36 yards + 3/144 yards
= 2 9/144 yards
Notice that with these non-coherent units we still haven't actually
multiplied the length by the width to find the area. All we have done is to
adjust the various units to make our final calculation possible.
Area = Length x Width
= 2 53/72 yards x 2 9/144 yards
= 197/72 yards x 297/144 yards
We still haven't done the calculation.
Area = 197/72 yards x 297/144 yards
= 58509/10368
= 5 square yards 5 square feet 113 5/8 square inches
It is important to note that with a coherent system, such as SI, there is
only one SI unit for each physical quantity; in the case of area the unit is
always the square metre.
The sales assistant, however, had a choice of doing the calculations in
square yards, square feet, square inches, square half-inches, or square
quarter-inches;. In a non-coherent measuring regime it does not matter which
units you choose they are all more difficult to use than SI.
Coherence in SI also works for any other quantity that we need to measure.
International Standard ISO 1000 uses velocity as an example. 'The unit of
velocity is the metre per second equal to one metre ÷ one second exactly.'
There are no conversions to be done.
A slightly more complex example comes from ISO 31-0. This refers to the
physical formula: 'force = mass x acceleration'. As you can see, this
formula is written using physical quantities, rather than units. In a
coherent system, the physical quantities and the units are interchangeable
in a formula.
When we apply units to the same formula, it reads:
force (in newtons) = mass (in kilograms) x acceleration (in metres per
second per second)
or with only the units:
newtons = kilograms x metre per second per second.
Again, there is not a conversion factor in sight.
However, if we use old units we can still use the same formula:
force = mass x acceleration
but we have to be very wary of our choice of units. If we want the force in
pounds (that are really pounds force) then we have to change the mass in
pounds (that are really pounds mass) into slugs of mass (32.2 pounds of mass
approximately) and the acceleration to feet per second per second (say from
miles per hour per second).
Alternatively, we could have the force in poundals if we keep the mass in
pounds (that are really pounds mass) with the same unit for acceleration. As
you can see, this is not easy; and this is largely because these old units
are not coherent.
In short, SI is a coherent system; old units, such as imperial, are not.
These examples all show that one of the advantages of SI is its coherence,
where all derived units are formed from the base units using the numerical
factor of one. Historically, examples of coherent sets of units are rare;
the cgs system of units was used from about 1872 until 1875; the mksA system
of units was used from about 1901 until 1960, when it was replaced by Le
Système Internationale d'Unités (SI). Currently SI is the only coherent
system of units available in the world.
The principle reference supporting the International System of Units is 'Le
Système Internationale d'Unités'. This document refers to coherence in
several places but the two main items are in sections 1.2 and 1.3 of the
Introduction. These are on page 92 of the 7th Edition (1998). I quote them
here.
Note: CGPM means Conférence Générale des Poids et Mesures General
Conference on Weights and Measures and CIPM means Comité International des
Poids et Mesures International Committee on Weights and Measures.
1.2 Two classes of SI units
SI units are divided into two classes:
� base units;
� derived units.
>From the scientific point of view, the division of SI units into these two
classes is to a certain extent arbitrary, because it is not essential to the
physics of the subject. Nevertheless, the CGPM, considering the advantages
of a single, practical, world-wide system of units for international
relations, for teaching and for scientific work, decided to base the
International System on a choice of seven well-defined units which by
convention are regarded as dimensionally independent: the metre, the
kilogram, the second, the ampere, the kelvin, the mole and the candela (see
2.1, p. 94). These SI units are called base units.
The second class of SI units is that of derived units. These are units that
are formed as products of powers of the base units according to the
algebraic relations linking the quantities concerned. The names and symbols
of some units thus formed in terms of base units may be replaced by special
names and symbols which can themselves be used to form expressions and
symbols for other derived units (see 2.2, p. 98).
The SI units of these two classes form a coherent set of units, where
coherent is used in the specialist sense of a system whose units are
mutually related by rules of multiplication and division with no numerical
factor other than 1. Following CIPM Recommendation 1 (1969; PV,37,30-31 and
Metrologia, 1970,6,66), the units of this coherent set of units are
designated by the name SI units.
It is important to emphasize that each physical quantity has only one SI
unit, even if this unit can be expressed in different forms. The inverse,
however, is not true; in some cases the same SI unit can be used to express
the values of several different quantities (see p. 101).
1.3. The SI prefixes
The CGPM adopted a series of prefixes for use in forming the decimal
multiples and submultiples of SI units (see 3.1 and 3.2, page 103).
Following CIPM Recommendation 1 (1969) mentioned above, these are designated
by the name SI prefixes.
The SI units, that is to say the base and derived units of the SI, form a
coherent set, the set of SI units. The multiples and submultiples of the SI
units formed by using the SI units combined with SI prefixes are designated
by their complete name, multiples and submultiples of SI units. These
decimal multiples and submultiples of SI units are not coherent with the SI
units themselves.
As an exception, the multiples and submultiples of the kilogram are formed
by attaching prefix names to the unit name "gram", and prefix symbols to the
prefix symbol "g".
