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A prospectus of some Books in the Series ‘Basic Books in Science’ (Last updated 23 October 2008)

# Book 1. Number and symbols—from counting to abstract algebras

## Roy McWeeny

Professore Emerito di Chimica Teorica, Università di Pisa, Pisa (Italy)
Book 1 in the Series ‘Basic Books in Science’

This book, like the others in the Series, is written in simple English – chosen only because it is the language most widely used in science and technology, industry, commerce, and international business and travel. Its subject “Number and symbols” is basic to the whole of science; but it introduces a new language, nothing like the one we use in everyday writing and speaking. The spoken word came ﬁrst in the evolution of language; then the written word (starting about four thousand years ago). Mathematics began to develop somewhat later – in China, India, and the countries around the Mediterranean. But the symbols of mathematics, though still just marks on paper, are not connected in any way with speech, or sounds, or the written word: usually they stand for operations such as counting or moving things through space, operations which are often performed only in the mind. It is this abstract and symbolic nature of mathematics that makes it seem diﬃcult to so many people and shuts them oﬀ from an increasingly large part of science.

The aim of this ﬁrst book in the Series is to open the door into Mathematics, ready for going on into Physics, Chemistry, and the other Sciences.
Note: The Chapter and Section titles listed on p.3 serve only to indicate the overall structure of the Book. Chapters vary in length between 5 and 15 pages; the whole text extends to 54 pages.

Your’re starting on a long journey – a journey of discovery that takes you from ancient times, when people ﬁrst invented languages and were able to share their ideas with each other by talking and writing, to the present day.

Science began to develop only a few thousand years ago, with the study of the stars in the sky (leading to Astronomy) and the measurement of distances in dividing out the land and sailing the seas (leading to Mathematics). With what we know now, the journey can be made in a very short time. But it’s still the same journey – ﬁlled with surprises – and the further you go the more you will understand the world around you and the way it works. Along the way there are many important ‘milestones’:

• After the ﬁrst two Chapters in WorkBook 1, you will know how to work with numbers, going from the operation of counting, to various ways of combining numbers – like adding and multiplying. And you’ll have learnt that other symbols (such as the letters of the alphabet) can be used to stand for any numbers, so that $a \times b = b \times a\,$ is a way of saying that one number $(a)\,$ times another number $(b)\,$ gives exactly the same answer as $b\,$ times $a\,$ – whatever numbers $a\,$ and $b\,$ may stand for. Mathematics just uses its own special language.

• In Chapter 3 you’ll ﬁnd how questions that seem to have no answer can be given one by inventing new kinds of number – negative numbers and fractions.

• After Chapter 4 you will be able to use the decimal system and understand its meaning for all the rational numbers.

• In Chapter 5 you pass two more milestones: after the ﬁrst you go from the rational numbers to the ‘ﬁeld’ of all real numbers, including those that lie between the rational numbers and are called ‘irrational’. The second breakthrough takes you into the ﬁeld of complex numbers which can only be described if you deﬁne one completely new number, represented by the symbol $i\,$, with the strange property that $i \times i=-1\,$. There are no more numbers to ﬁnd, as long as we stick to the rules set up so far.

• But we’re not ﬁnished: human beings are very creative animals! The last Chapter shows how we can extend the use of symbols to include operations quite diﬀerent from the ones we’ve used so far.

### CONTENTS

1.1 Why do we need numbers?
1.2 Counting: the natural numbers
1.3 The naming of numbers

Chapter 2 Combining numbers
2.2 Combining by multiplication

Chapter 3 Inventing new numbers – equations
3.1 Negative numbers and simple equations
3.2 Representing numbers in a picture – vectors
3.3 More new numbers – fractions

Chapter 4 The decimal system
4.1 Rational fractions
4.2 Powers and their properties
4.3 Decimal numbers that never end

Chapter 5 Real and complex numbers
5.1 Real numbers and series
5.2 The new ﬁeld of complex numbers
5.3 Equations with complex solutions

Chapter 6 Beyond numbers – operators
6.1 Symmetry and groups
6.2 Sorting things into categories
6.3 Arguing with symbols – logic

### Looking back –

We started from almost nothing – just a few ideas about counting – and we’ve come quite a long way. Let’s take stock:

• Chapters 1 and 2 must have reminded you of when you ﬁrst learnt about numbers – as a small child, and the endless chanting of multiplication tables – learning without understanding. But now you know what it all means : the rules of arithmetic are made by us, to help us to understand and use what we see around us. You know that you can use any symbols in place of numbers and write down the rules of arithmetic using only the symbols – and then you are doing algebra.

• In Chapter 3 you met equations, containing a number you don’t know (call it ‘$x\,$’), along with the integers $(1,2,3,...)\,$ , and were able to solve the equations, ﬁnding for $x\,$ the new numbers $(0,\text{ zero; and } -1,-2,\text{ ..., negative numbers })\,$. You found how useful pictures could be in talking about numbers and their properties; and how a number could be given to any point on a line – bringing with it the idea of fractions as the numbers labelling points in between the integers.

• But between any two such numbers, ‘rational fractions’ of the form $a/b\,$, however close, there were still uncountable millions of numbers that you can’t represent in that simple way; you can get as close as you wish – but without ever really getting there. An ‘irrational’ number is deﬁned only by a recipe that tells you how to reach it – but that’s enough. The set of all the numbers deﬁned so far was called the Field of Real Numbers; and it’s enough for all everyday needs – like measuring.

• The last big step was made in Chapter 5, when we admitted the last ‘new’ number, denoted by $i\,$ and called the “imaginary unit”, with the property $i\times i=-1$ (not $1\,$). And when $i\,$ was included, and allowed to mix with all the real numbers, our number ﬁeld had to be extended to include both Real Numbers and Complex Numbers. All equations involving only such numbers could then be solved without inventing anything new: the ﬁeld was closed.

• In the last Chapter, however, you saw that symbols could be used to stand for other things – not only numbers. They can be used for operations, like moving things in space; or sorting a mixture of objects into objects ‘of the same kind’; or just for arguing about things!

# Book 2. Space — from Euclid to Einstein

## Roy McWeeny

Professore Emerito di Chimica Teorica, Universit`a di Pisa, Pisa (Italy)
Book 2 in the Series ‘Basic Books in Science’

This book takes the next big step beyond “Number and symbols” (the subject of Book 1), starting from our ﬁrst ideas about the measurement of distance and the relationships among objects in space. It goes back to the work of the philosophers and astronomers of two thousand years ago; and it extends to that of Einstein, whose work laid the foundations for our present-day ideas about the nature of space itself. This is only a small book; and it doesn’t follow the historical route, starting from geometry the way Euclid did it (as we learnt it in our schooldays); but it aims to give an easier and quicker way of getting to the higher levels needed in Physics and related sciences.

Note: The Chapter and Section titles listed on p.7 serve only to indicate the overall structure of the WorkBook. Chapters vary in length between 5 and 15 pages; the whole text extends to 70 pages.

Like the ﬁrst book in the Series, Book 2 spans more than two thousand years of discovery. It is about the science of space – geometry – starting with the Greek philosophers, Euclid and many others, and leading to the present – when space and space travel is written about even in the newspapers and almost everyone has heard of Einstein and his discoveries.

Euclid and his school didn’t trust the use of numbers in geometry (you saw why in Book 1): they used pictures instead. But now you’ve learnt things they didn’t know about – and will ﬁnd you can go further, and faster, by using numbers and algebra. And again, you’ll pass many ‘milestones’:

• In Chapter 1 you start from distance, expressed as a number of units, and see how Euclid’s ideas about straight lines, angles and triangles can be ‘translated’ into statements about distances and numbers.

• Most of Euclid’s work was on geometry of the plane; but in Chapter 2 you’ll see how any point in a plane is ﬁxed by giving two numbers and how lines can be described by equations.

• The ideas of area and angle come straight out of plane geometry (in Chapter 3): you ﬁnd how to get the area of a circle and how to measure angles.

• Chapter 4 is hard, because it ties together so many very diﬀerent ideas, mostly from Book 1 – operators, vectors, rotations, exponentials, and complex numbers – they are all connected!

• Points which are not all in the same plane, lie in 3-dimensional space – you need three numbers to say where each one is. In Chapter 5 you’ll ﬁnd the geometry of 3-space is just like that of 2-space; but it looks easier if you use vectors.

• Plane shapes, such as triangles, have properties like area, angle and side-lengths that don’t change if you move them around in space: they belong to the shape itself and are called invariants. Euclid used such ideas all the time. Now you’ll go from 2-space to 3-space, where objects also have volume; and you can still do everything without the pictures.

• After two thousand years people reached the last big milestone (Chapter 7): they found that Euclid’s geometry was very nearly, but not quite, perfect. And you’ll want to know how Einstein changed our ideas.

### CONTENTS

Chapter 1 Euclidean space
1.1 Distance
1.2 Foundations of Euclidean geometry

Chapter 2 Two-dimensional space
2.1 Parallel straight lines. Rectangles
2.2 Points and straight lines in 2-space
2.3 When and where will two straight lines meet?

Chapter 3 Area and angle
3.1 What is area?
3.2 How to measure angles
3.3 More on Euclid

Chapter 4 Rotations: bits and pieces

Chapter 5 Three-dimensional space
5.1 Planes and boxes in 3-space – coordinates
5.2 Describing simple objects in 3-space
5.3 Using vectors in 3-space
5.4 Scalar and vector products
5.5 Some examples

Chapter 6 Area and volume: invariance
6.1 Invariance of lengths and angles
6.2 Area and volume
6.3 Area in vector form
6.4 Volume in vector form

Chapter 7 Some other kinds of space
7.1 Many-dimensional space
7.2 Space-time and Relativity
7.3 Curved spaces: General Relativity

### Looking back –

You started this book knowing only about numbers and how to work with them, using the methods of algebra. Now you’ve learnt how to measure the quantities you meet in space (distances, area, volume), each one being a number of units. And you’ve seen that these ideas give you a new starting point for geometry, diﬀerent from the one used by Euclid, and lead you directly to modern forms of geometry. Again, you’ve passed many milestones on the way:

• Euclid started from a set of axioms, the most famous being that two parallel straight lines never meet, and used them to build up the whole of geometry: in Chapter 1 you started from diﬀerent axioms – a distance axiom and a metric axiom – which both follow from experiment.

• Two straight lines, with one point in common, deﬁne a plane; the metric axiom gave you a way of testing to see if the two lines are perpendicular; and then you were able to deﬁne two parallel straight lines – giving you a new way of looking at Euclid’s axiom. Using sets of perpendicular and parallel straight lines you could ﬁnd numbers $(x, y)\,$, the coordinates, that deﬁne any point in the plane. Any straight line in the plane was then described by a simple equation; and so was a circle.

• In Chapter 3 you learnt how to calculate the area of a triangle and of a circle and to evaluate $\pi\,$ (‘pi’) by the method of Archimedes. You studied angles and found some of the key results about the angles between straight lines that cross.

• Chapter 4 reminded you of some of the things you’d learnt in Book 1, all needed in the study of rotations. You learnt about the exponential function, $e^x\,$, deﬁned as a series, and its properties; and found its connection with angle and the ‘trigonometric’ functions.

• In talking about 3-space, the ﬁrst thing to do was to set up axes and decide how to label every point with three coordinates; after that everything looked much the same as in 2-space. But it’s not easy to picture things in 3-space and it’s better to use vector algebra. For any pair of vectors we found two new ‘products’ – a scalar product (just a number) and a vector product (a new vector), both depending on the lengths of the vectors and the angle between them. Examples and Exercises showed how useful they could be in 3-space geometry.

• Chapter 6 was quite hard! But the ideas underneath can be understood easily: lengths, areas and volumes are all unchanged if you move something through space – making a ‘transformation’. This fact was often used by Euclid (usually in 2-space) in proving theorems about areas; but by the end of the Chapter you have all the ‘tools’ for doing things much more generally, as we do them today.

• To end Book 2 (in Chapter 7) you took a look at the next big generalization – to spaces of $n\,$ dimensions, where $n\,$ is any integer. Of course, you couldn’t imagine them: but the algebra was the same, for any value of $n\,$. So you were able to invent new spaces, depending on what you wanted to use them for. One such space was invented by Einstein, just a hundred years ago, to bring time into the description of space – counting $t\,$ as a fourth coordinate, similar to $x,y,z\,$. And you got a glimpse of some of the amazing things that came out as a result, things that could be checked by experiment and were found to be true.

Before closing this book, stop for a minute and think about what you’ve done. Perhaps you started studying science with Book 1 (two years ago? three or four years ago?) and now you’re at the end of Book 2. You started from almost nothing; and after working through about 150 pages you can understand things that took people thousands of years to discover, some of the great creations of the human mind – of the Scientiﬁc Mind.

# Basic Books in Science: current developments

The Series, as envisaged at present (October 2008), will start with the following books:
Book 1. Number and symbols – from counting to abstract algebras
Book 2. Space – from Euclid to Einstein
Book 3. Relationships, change – and Mathematical Analysis
Book 4. Motion and mass – ﬁrst steps into Physics
Book 5. Atoms, molecules and matter – the stuﬀ of Chemistry
Book 6. The planet we live on – the beginnings of the Earth Sciences (Author Chris King, University of Keele, UK)
Book 7. The chemistry of life
Book 8. Living things, from one cell to many – the Life Sciences
Book 9. The evolution of living creatures – who were our ancestors?
Book 10. Electricity, elementary particles – and on into Modern Physics

**************************************

The ﬁrst three books are already published on the websites
<http://www.paricenter.com> (check out ‘Basic Books in Science’) or <http://www.learndev.org> (check out ‘For the Love of Science’)

Books 4 and 5 are also available in an advanced draft form, together with a supplementary module (Book 3A on Calculus and diﬀerential equations, by John Avery). Book 10 is nearing completion and Spanish translations of the ﬁrst few books are also available in near-ﬁnal form from the ‘learndev’ website. The intention is to seek translators for French and Arabic versions, along with authors for the other volumes.

It is important to maintain the momentum of the Series. To this end, proposals and suggestions are invited from prospective authors.

 Pisa, 24 October 2008 Roy McWeeny, Series Editor