46 MATHEMATICIAN'S DELIGHT

with a dictionary should clear up that difficulty. * The next thing is to find out what knowledge you are expected to have before you attempt to understand the proof of a new result. It is possible to make a diagram showing how a book hangs together, how one section depends on previous sections. One should learn a book both backwards and forwards: one should know that the result on page 50 is proved by the result on page 29, and that it is used to prove the result on page 144. (Of course no sane person will learn the numbers of the actual pages on which results occur. But it may be worth while to write in the margin of page 50, 'See p. 29; used, p. 144') Many people learn separate results, but never link them together in this way.

In this book it has not been possible, in every single sentence, to give references to all the remarks, made earlier in the book, that may help towards understanding. If you cannot understand some sentence, underline it. The chances are that somewhere earlier in the chapter, or in the book, a remark has been made that was especially intended to prepare for the difficult sentence. At the first reading you may not have noticed this remark at all. It seemed pointless. Look back for such remarks. If you succeed in finding them, put a note in the margin, 'This explains sentence underlined on page ... '

It may seem to you that this advice does not amount to much, that it is obvious. Tt may be obvious - but people need a lot of persuading before they do it. As a rule, someone who has difficulty with calculus or trigonometry is not prepared to believe that the real trouble is ignorance of algebra or arithmetic. There is always an examination coming in six weeks, or a year, or whatever it is, and this examination is on calculus or trigonometry - not on algebra and arithmetic. Trying to learn higher mathe­matics without a firm grasp of the earlier part is like trying to invent an aeroplane without knowing anything about motor-car engines. Until the motor industry had been developed, all attempts at aeroplanes were complete failures.

*1 have tried to keep words in this book as short as possible. One or two \Yords may not be known to everyone. It is only fair that readers should take the trouble to look these up.

THE STRATEGY AND TACTICS OF STUDY 47

COMMON SENSE & EVERYDAY EXPERIENCE

THE GENERAL PLAN OF THIS BOOK In this diagram each block represents a chapler. Chapters 1,3, and 4

are of a general nature, and are not included in the diag.ram. Each block depends upon the blocks below. TLl~s, it is impossible to

understand Chapter 11 without first having read Chapters 6,9, and 10, and Chapters 9 and 10 in turn cannot be understood without Chapter 8, etc.

In some cases, the upper block depends only on a small part of the lower one. For instance, Chapter 8 can be understood without under­standing the whole of Chapter 6. In fact, it is only the part of Chapter 6 explaining the meaning of the signs 4'. lOS, etc., that is needed for Chapter 8. It is not possible to show this on the diagram.

Chapter 13 is split into two parts. 13a represents the greater part of the chapter, which is quite elementary. 13b represents the end of the chapter, which is more advanced.

If a reader finds difficulty, say in Chapter 10, he may find it worth while to leave Chapters 10, 11, and 12, for the time being, and to read the easier part of Chapter 13.

To revise elementary mathematics takes much less time than people imagine. How many text-books has a student of eighteen used? One on arithmetic, one on algebra, several perhaps on geometry, an elementary trigonometry, perhaps a book on cal­culus. Geometry we may leave on one side for the moment. How long does it take to look through an arithmetic book and one on algebra, and find out if there is aJ?Y important result which you missed at school? How long does it take to write down on a

48 MATHEMATICIAN'S DELIGHT

sheet of paper a list of the contents of these books, and to put a tick against the results which you thoroughly understand? Not very long. The advantage of doing this is that you begin to see how much (or how little) you have to learn. One tends to think of algebra as a vast jungle of confusion, in the midst of which one wanders without map or compass. It is much better to think of algebra (or that part of algebra which you need to know) as being half a dozen methods, and twenty or so results, of which you probably already know 60%. Nor need you revise the whole of this at once. Suppose for instance you are finding difficulty with calculus because you do not properly know the Binomial Theorem. Get down your book of algebra, and look up Binomial Theorem. Never mind about the proof for the moment. First get quite clear what the Binomial Theorem is. It is full of signs such

as nCr or (;) - different signs are used in different books. These

signs are explained in the chapter on Permutations and Com­binations. Again, do not bother about proof. See what these signs mean. Work out a few examples - 4C1 4C2 and 4C3, for instance. Work these right out, as numbers. Come back to the Binomial Theorem, and take particular examples of it. Put n = 4, for instance. * The binomial theorem deals with the ex­pression (x + at. Put x = 10 and a = 1. Work out 112, 113, 114. What is the connexion between 114 and the numbers worked out above? Work out 101 X 101 and 101 X 101 X 101. What do you notice about 11 X 11 and 101 X 101? What do you notice apout 11 X 11 X 11 and 101 X 101 X 101? The same numbers turn up both times? Do you think the same numbers will tum up in 1001 X 1001 as in 11 X 11? In 1001 X 1001 X 1001 as in 11 X 11 X II? If so, you are not far from discovering the Bino­mial Theorem for yourself. (If you are not clear what 114 stands for, the remarks above will be meaningless to you. What 114 means is explained in Chapter 6.)

*if you are in the fortunate position of having never been taught algebra, and therefore having no mistaken ideas about it, take no notice of this paragraph. The meaning of algebraic signs is explained in Chapter 7.