Chapter One The tools of the trade

# Check your existing knowledge

# 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.

In a similar fashion, an answer of ‘17’ is even more unhelpful. Seventeen what? Miles? Kilometres? Leagues? Furlongs? So numbers, when used for measuring things, need units to quantify them.

## PARSEC

An astronomical measure equivalent to the distance travelled by light in 3.26 years (30 700 000 000 000 km), a distance that would take you three-quarters of the way to alpha centauri proxima, the nearest star (other than the sun) to earth.

## LEAGUE

An archaic measure of distance equal to about three miles, now only remembered from the fairy tale of the *Seven league boots* and Tennyson’s ‘Half a league, half a league, half a league onward’ from *The charge of the light brigade*.

## 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:

(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 10^{80} (estimates actually range from 10^{72} to 10^{87})!

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 (10^{9}) and, on the other, it has traditionally been a *million* million (10^{12}), 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 (10^{18}) 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 (*x*^{1} is *x*; 10^{1} is 10) and any number raised to the power of zero is 1 (*x*^{0} is 1; 10^{0} is also 1). It is also worth pointing out at this stage that multiplying powers is quite simple – you just add them together:

or

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:

or

so

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: 10^{6} is not *twice* as big as 10^{3}, 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!

## 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.

### THE MILE

The Roman foot measured 11.68 modern inches, divided, as is its modern day successor, into 12 parts (uniciae). Five feet made for one passus (from pace, meaning ‘double step’) and the mille passus (Roman mile) was 1000 paces or 5000 feet long – for the record, the cubit was one and a half feet long.

The statute mile of 5280 feet is so-called as it was formalized in a Parliamentary statute in 1592 by Elizabeth I, having been in usage since the 13^{th} century. Different countries (including the US) had slight variations on this distance (the US Survey mile is longer by inch) and it was not until 1959 that a standard length was agreed.

The nautical mile of 2025 yards bears only a coincidental relationship to the statute mile, being a measure of 1′ (one minute = one-sixtieth of a degree) of latitude.

### The Imperial system

The British Empire, being predominantly a vehicle for trade, had a little more success at imposing its weights and measures upon the world … indeed, that is why the system of feet and inches; ounces and pounds; pints and gallons is called the Imperial system (of, or pertaining to, an empire). Even then, different towns often had differing ideas as to what constituted a pound and what qualified as short measure and, even today, there is a difference between the amount – and spelling – of units: a US gallon is 3.79 *liters* and a UK gallon, 4.55 *litres*.

If you think this haphazard, bear in mind that, until the need for conformity was driven by railway timetabling in the mid-19^{th} century, different towns not only could have different weights and measures but even their clocks were set differently, not just by seconds but by minutes!

### The MKS system

Whilst science was still limited to the leisurely pursuit of gentleman amateurs, such variations mattered little as long as there was internal consistency; however, as the pace of scientific and technical advance started to snowball in the early 19th century, scientists began to seek a means of developing international conformity. The first international system devised was called the metre-kilogram-second (MKS) system whereby scientists agreed to use metres for measuring all distances, kilograms for mass and seconds for time. There was, of course, a schism almost straight away with some scientists championing centimetres, grammes and seconds (CGS system), which, as you will see later, explains some of the derived units with which we have been historically endowed.

Importantly though, there was also, for the first time, international agreement as to what the exact definition of these quantities should be.

A metre had already been defined by the French Academy of Science in the post-revolution fervour for change as ^{1}/_{10 000 000} of the quadrant of the Earth’s circumference running from the North Pole through Paris to the equator. The kilogram was defined as the mass of 1000 cubic centimetres of water, the second as ^{1}/_{86 400} of the average period of rotation of the Earth on its axis relative to the Sun. Unfortunately, as measurements became ever more precise, the definitions were no longer accurate enough: the Earth’s crust is a dynamic, moving structure subject to alteration at short notice; its rotation is also (very) gradually slowing. The density of water changes according to temperature and pressure (not to mention to regional fluctuations in the Earth’s gravitational field).

The problem was temporarily solved by relating the definitions of mass and length to the parameters of two lumps of platinum and iridium (and thus highly inert and resistant to oxidation) that were kept locked in a Parisian vault … which was fine if you were a Parisian scientist but was a bit tough on anyone else who wanted to calibrate their instruments.

Today, wherever possible, we use highly precise definitions based on constant phenomena that are observable by any scientist working anywhere in the world or, indeed, off it. A metre is now defined as the distance travelled by light in a vacuum in ^{1}/_{299 792 458} of a second; a second is 9 192 631 770 cycles of radiation associated with the transition between the two hyperfine levels of the ground state of the caesium-133 atom (a statement that will be understandable by the end of Ch. 8). A kilogram, however, remains as the mass of a cylinder of French platinum-iridium until somebody can think of anything better.

### The SI system

By this time, three new units had been added, all named after eminent scientists: a unit of force (the **newton**, N), defined as that force which gives to a mass of one kilogram an acceleration of one metre per second per second; a unit of energy (the **joule**, J), defined as the work done when the point of application of a newton is displaced one metre in the direction of the force; and a unit of power (the **watt**, W), which is the power that, in one second, gives rise to energy of one joule.

These additional units, which built upon the MKS system, were called the *Système International D’unités*, known as the **SI system**. Since its formal adoption in 1960, many more units have been added. There are now seven basic units (Table 1.1): in addition to the metre, the kilogram and the second, we now have the **ampere**, A, for electric current; for luminous intensity, the **candela**, cd; for temperature, **kelvin**, K; and for quantity of substance, the **mole**, mol.

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 × 10^{12} 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.

Quantity | Unit | Symbol |
---|---|---|

Acceleration (a) |
Metres per second per second | m/s^{2} = ms^{−2} |

Area (A) |
Square metre | m^{2} |

Current density (j) |
Ampere per metre | A/m^{2} = Am^{−2} |

Density (ρ) |
Kilogram per cubic metre | kg/m^{3} = kgm^{−3} |

Luminance (L)_{v} |
Candela per square metre | cd/m^{2} = cdm^{−2} |

Magnetic field strength (H) |
Ampere per metre | A/m = Am^{−1} |

Substance concentration (c) |
Mole per cubic metre | mol/m^{3} = mol.m^{−3} |

Volume (V) |
Cubic metre | m^{3} |

Velocity (v) |
Metres per second | m/s = ms^{−1} |

Wave number (σ) | Reciprocal metre | m^{−1} |

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^{−1}K^{−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 *kilo*metres’, ‘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 (10^{3}), 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 10^{24}, 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:

10^{24} |
Yotta | Y |

10^{21} |
Zetta | Z |

10^{18} |
Exa | E |

10^{15} |
Peta | P |

10^{12} |
Tera | T |

10^{9} |
Giga | G |

10^{6} |
Mega | M |

10^{3} |
Kilo | k |

10^{2} |
Hecto | h |

10^{1} |
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 |