Chapter One The tools of the trade

Check your existing knowledge

Section 1

1 If 4x + 4y − 12z = 32, derive an expression for x.

2 Expand (x − y)(2x + y)

In the diagram below:

10 What is 3 mm + 4 μm?

11 What is 10⁰ ×10¹ × 10² ×10³?

12 What is 9^½?

13 What is 9⁻¹?

14 What is the SI unit of temperature?

15 What is the SI unit of electrical resistance?

16 What is a scalar?

17 An object is acted upon by two forces. One, from the south, acts with a force of 4 newtons; the other, of equal magnitude, acts from the west.

a In which direction does the object move?

b What is the magnitude of the resultant force?

18 The YZ orthogonal plane is the equivalent to which anatomical plane?

19 If a person’s body undergoes −θ_y rotation, what action are they performing?

Weights and measures

In order to measure something, you need two elements: a quantity and a unit. If you enquire as to the distance to the nearest town and got the reply, ‘miles’ (as can often be the case in certain rural parts), it is of limited use. Of course, you can infer that the distance is one that you would be inclined to measure using miles rather than, say, inches or parsecs and that it is, by the use of the plural, more than a single mile; however, knowing whether it was 2 miles or 200 miles would be useful. Units require numbers to quantify them.

Numbers

The problem with numbers is that there are an awful lot of them. In fact, there are an infinite number of them – they continue, quite literally, forever: take the largest number you can imagine … and then add one! With smaller numbers, it is often easier to write the numeral than the number itself: ‘8’ is quicker to write (and spell) than ‘eight’; ‘1327’ requires seven pen-strokes, ‘one-thousand, three-hundred-and-twenty-seven’ needs 78.

Numbers are also much more useful for mathematics than words. Organizing them into representations of units, tens, hundreds, thousands and having a ‘zero’ allows us to manipulate them arithmetically – see how easy you find the following sums:

a Twenty-six plus forty-eight plus nineteen equals?

b XXVI + XLVIII + XIX =

(The answers are: a) ninety-three; b) XCIII and c) 93.)

The decimal (base ten) numeral system that we use is certainly more convenient on an everyday basis, and having a ‘zero’ enables us to perform mental calculations that were unavailable to the Romans – which is why the Greeks and the Arabs, from whom we obtained our digits (including the zero), were far more advanced in mathematics than the speakers of Latin, including the mediaeval scholars of Western Europe.

However, left to themselves, numbers become cumbersome for physicists, who must be able to measure everything from the size of a sub-atomic particle to the number of such particles in the universe. There are three ways of doing this: you can put up with writing down very long numbers, you can adapt the units you are using or you can find a short-hand way of recording very large (and very small) numbers.

The first of these options is impractical. Whereas the diameter of an atomic nucleus is approximately 0.000 000 000 000 01 metres – a number that is tedious to write on a regular basis – the number of particles in the universe is estimated as being somewhere in the region of 1 followed by eighty zeros, a number so big that it would fill several lines, take several minutes to write and even longer to read, painstakingly counting out the zeros in groups of three without losing track of one’s place.

It wasn’t long before people began adapting units to suit their needs, as has always been the case (an inch is the width of a thumb, a foot the length of a foot, a pace is a yard, and a fathom the arm span of a man). So, carpenters, in this metric age, use millimetres; engineers, microns (a thousandth of a millimetre); molecular scientists, Angstrøms (a ten-millionth of a millimetre). Although this is convenient if all your measurements are conducted in the same way and for discussions or recording of data, it does not allow for easy manipulation of data. It is important to know and remember that you can only add, subtract, multiply and divide quantities if they are in the same units. You can’t add millimetres directly to Angstrøms any more than you can directly add a distance measured in miles to one measured in kilometres.

Scientific notation

There is, however, a way of recording all numbers in a convenient way using scientific notation, a universally agreed system for expressing any number in terms of its power of ten. You will already be familiar with certain powers of ten:

The digit to the top right is the number of times the main figure is multiplied together:

The number ten is useful in that multiplying by ten simply involves adding a nought to the end of the number you are multiplying (e.g. 10 × 10 000 = 100 000); therefore, the power to which ten is raised is also equal to the number of noughts:

Suddenly, the number of particles in the universe becomes a much more manageable 10⁸⁰ (estimates actually range from 10⁷² to 10⁸⁷)!

This convention also skirts around another problematical area, that of nomenclature. Although people are generally agreed as to the meanings of ‘hundred’, ‘thousand’ and ‘million’, thereafter American English diverges from its mother tongue and works in increments of a thousand rather than a million so, on one side of the Atlantic, a billion is a thousand million (10⁹) and, on the other, it has traditionally been a million million (10¹²), except in some European countries, such as France, who have adopted the American convention. As the numbers get larger, so does the confusion: a British trillion (10¹⁸) is an American quintillion whilst an American trillion is a British billion. Because international finance uses the American convention, the trend is increasingly to follow this, even in the UK, but the use of scientific convention removes all ambiguity … as well as doing away with the need to remember how many noughts there are in a septillion! (For the record, in the US, 24; in the UK, 42.)

For numbers that aren’t exact multiples of ten, scientific notation uses multiplication so:

and

For numbers between 0 and 1000, it is usual to write them in full but, just so you know, any number raised to the power of one is itself (x¹ is x; 10¹ is 10) and any number raised to the power of zero is 1 (x⁰ is 1; 10⁰ is also 1). It is also worth pointing out at this stage that multiplying powers is quite simple – you just add them together:

It is also, at this stage, worth quickly dealing with fractional powers: if a number is raised to the power of a ½, it is the number’s square root; raised to ⅓, it is the number’s cube root and so on:

Having dealt with the very big, it is a relatively easy matter to deal with the very small. If a number is raised to a negative power, it is the inverse of the positive power:

Now we have a way of representing any number, large or small, in a concise, consistent and comprehensible manner. This can take us from the unimaginably small to the incomprehensibly large (Fig. 1.1); however, bear in mind that the scale in this figure is not linear but logarithmic; that is, it rises in powers of ten: 10⁶ is not twice as big as 10³, it is one thousand times bigger. If we used a normal, linear scale, it wouldn’t just be a case being hard to fit Figure 1.1 on the page – it would be hard to fit into the known universe!

Figure 1.1 • From the very small to the very large.

Units

Since the dawn of time, humans have been measuring things. Thousands of years ago, the ancient Egyptians and Chinese had discovered sophisticated ways of measuring distances (for building purposes – walls, pyramids and the like) and time (the cyclical passage of astronomical bodies; clocks would come later). Thousands – possibly tens of thousands – of years before that, early man had almost certainly found ways of describing to their friends how far it was to the mammoth hunting grounds, and to their womenfolk the enormous size of the mammoth they had so nearly managed to kill.

The problem with ancient measuring systems was that they were either comparative (my club is bigger than your club) or local to one tribe or one area. On a day-to-day basis, this was not a problem but, as civilization became more international, the need for uniformity became more pronounced. One only has to look at the story of the cubit to get an idea of size of the problem.

Originally, a cubit was the distance from one’s fingertips to one’s elbow. Quite obviously, this varies from one individual to the next but will suffice if the measurement is a personal one (I want to cut a piece of wood as long as the measurement I just made) or if there’s only one carpenter in town and he’s willing to make house calls in order to measure up for jobs.

As soon as commerce was invented, this definition lost its usefulness – as anyone familiar with the story of The King’s New Bed will know (a king kept having problems ordering a bed that he had measured as 6 feet long by 4 feet wide until he found a carpenter with feet the same size as his own; thereafter, he made models of his foot for his citizens to use for measurements throughout his realm and so everyone lived happily ever after … except for a couple of carpenters who had had their heads removed for failing to make a bed that came up to royal requirements).

A number of non-fictional rulers tried to standardize the cubit; the trouble was – much like the eponymous king – they presumably used their own personal cubit as the standard. This means that the biblical cubit (as used to lay down Noah’s Ark, God presumably having Noah’s arm measurements down pat when issuing his omnipotent blueprints) at 56 cm was different from that of the Egyptians (53 cm), which, in turn, was different from that of the Romans (44 cm), which was slightly shorter than that of the British (46 cm).

Where trade failed to agree communal weights and measures, empires enforced them. The Romans at one time ruled almost half the population of the known world and, even if they failed in the long-term standardization of the cubit, from them, we get the mille or mile, a thousand (double) paces … even if the statute mile is approximately 140 yards longer than the Roman one … and, for various technical reasons, 265 yards less than a nautical mile.

Fact File

Unit	Symbol	Definition
Metre	m	The distance travelled by light in a vacuum in ¹/_{299 792 458} second
Kilogram	kg	Defined by the international prototype kilogram of platinum-iridium in the keeping of the International Bureau of Weights and Measures in Sèvres, France
Second	s	The duration of 9 192 631 770 periods of radiation associated with a specified transition of the cesium-133 atom
Ampere	A	The current that, if maintained in two wires placed one metre apart in a vacuum, would produce a force of 2 × 10⁻⁷ newton per metre of length
Kelvin	K	¹/_273.16 of the triple point of pure water (corresponding to −273.15° on the Celsius scale and to −459.67° on the Fahrenheit scale). It is calculated by extrapolating the point at which an ideal gas and constant pressure would reach zero volume
Mole	mol	The amount of a given substance that contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12
Candela	cd	The intensity in a given direction of a source emitting radiation of frequency 540 × 10¹² hertz and that has a radiant intensity in that direction of ¹/₆₈₃ watt per steradian

The three original derived units have also been built upon considerably; however, all these other SI units can – as Tables 1.2, 1.3 and 1.4 show – be defined in terms of these seven basic units, either directly or indirectly. Many of these units have their own special names (such as the newton and joule). At this stage, there may appear to be an alarming number of these units with peculiar names that define quantities of which you may never have heard – and the lists given are by no means complete, these are merely the units that a health professional is likely to encounter! However, by the time you have finished the book, all these terms should be like regular acquaintances. Do not try to memorize them all at this stage, rather familiarize yourself with the names and use the table as a reference so that, when you encounter a new term or an unfamiliar symbol, you can refer back and learn the detail in its correct context.

Table 1.2 Some derived SI units

Quantity	Unit	Symbol
Acceleration (a)	Metres per second per second	m/s² = ms⁻²
Area (A)	Square metre	m²
Current density (j)	Ampere per metre	A/m² = Am⁻²
Density (ρ)	Kilogram per cubic metre	kg/m³ = kgm⁻³
Luminance (L_v)	Candela per square metre	cd/m² = cdm⁻²
Magnetic field strength (H)	Ampere per metre	A/m = Am⁻¹
Substance concentration (c)	Mole per cubic metre	mol/m³ = mol.m⁻³
Volume (V)	Cubic metre	m³
Velocity (v)	Metres per second	m/s = ms⁻¹
Wave number (σ)	Reciprocal metre	m⁻¹

Table 1.3 Some derived SI units with special names and symbols

Table 1.4 Some SI units with derived unit names

Quantity	Unit	Derivation
Absorbed dose rate	Gray per second	Gy.s⁻¹
Angular acceleration (α)	Radian per second per second	rad.s⁻²
Angular velocity (ω)	Radian per second	rad.s⁻¹
Electric charge density (P)	Coulomb per cubic metre	C.m⁻³
Electric field strength (E)	Volt per metre	V.m⁻¹
Exposure, X- & γ-ray	Coulomb per kilogram	C.kg⁻¹
Moment of force (F_m)	Newton metre	N.m
Thermal conductivity (λ)	Watts per metre per kelvin	Wm⁻¹K⁻¹
Viscosity (π)	Pascal second	Pa.s

There is one other trick to know about when it comes to units, which can – and often does – serve as an alternative to scientific nomenclature. In everyday conversation, it becomes a bit tedious to say ‘it’s six times ten to the three metres to the nearest town’ so, instead, we say ‘it’s six kilometres’, ‘kilo’ being a prefix meaning ‘one thousand’. Scientists do the same in their everyday conversation too: there are a stack of these prefixes, some of which – kilo (10³), centi (10⁻²), milli (10⁻³) – you will be familiar with as you will with their common abbreviations: k, c and m respectively. Table 1.5 shows the abbreviations and symbols from 10⁻²⁴ to 10²⁴, which, whilst not exhaustive, is enough for most eventualities that you are likely to encounter. So you are now able to express large and small quantities in two ways, by saying, for example:

Table 1.5 Commonly used prefixes for SI units. Note the use of capital letters for units of 10⁶ and above and lower case for 10³ and below

10²⁴	Yotta	Y
10²¹	Zetta	Z
10¹⁸	Exa	E
10¹⁵	Peta	P
10¹²	Tera	T
10⁹	Giga	G
10⁶	Mega	M
10³	Kilo	k
10²	Hecto	h
10¹	Deka	da
10⁻¹	Deci	d
10⁻²	Centi	c
10⁻³	Milli	m
10⁻⁶	Micro	μ
10⁻⁹	Nano	n
10⁻¹²	Pico	p
10⁻¹⁵	Femto	f
10⁻¹⁸	Atto	a
10⁻²¹	Zepto	z
10⁻²⁴	Yocto	y