## Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

### Part I. ALGEBRA AND ANALYTIC GEOMETRY

### Chapter 1. Prologue

*1.1 THE IMPORTANCE OF MATHEMATICS*

You live in an age that is dominated by science and engineering. Whether you like it or not, they have significant effects on your life. And it seems probable that in the near future their effects will become even greater than they are now. Thus if you wish to be effective in this world and to achieve the things you want, it is necessary to understand both science and engineering (and these require mathematics).

Long ago Pythagoras (died about 492 B.C.) said:

Number is the measure of all things.

(I am told that the strict translation is simply “everything is number.”) Galileo (1564–1642) similarly said:

Mathematics is the language of science.

Mathematics clearly plays a fundamental role in the older sciences, such as astronomy, physics, and chemistry, and is of increasing importance in the other “hard” sciences. But mathematics is also rapidly invading all the biological sciences, especially such fields as genetics and molecular biology. Even in the humanities we find that questions of authorship and style are being decided by applying statistical tests to the written material. For a long time business administration has been using more and more mathematics as people have tried to understand both the workings of the large, complex organizations they have to manage and the competition between such organizations. Mathematics is similarly needed in modem government administration. The social sciences are also heavily dependent on the statistical approach to many of their problems. Indeed, one may say

that Science is a habit of the mind as well as a way of life, and that mathematics is an aspect of culture as well as a collection of algorithms. (C. B. Boyer, 1906–1976)

It appears, therefore, that mathematics, in one form or another, will invade most fields of knowledge as we try to make them more reliable. There is an inevitability of this happening. But it has long been observed that the mathematics that is not learned in school is very seldom learned later, no matter how valuable it would be to the learner. Any unwillingness to learn mathematics today can greatly restrict your possibilities tomorrow.

This is not an assertion that all of mathematics will be useful; all that can be done is to look at both the past and the present, and then make educated guesses as to future needs for mathematics. This book covers what is believed to be the most useful applications (on the average). The three fields, calculus, probability, and statistics are all in constant use. Mathematicians in the past have tended to avoid the latter two, but probability and statistics are now so obviously necessary tools for understanding many diverse things that we must not ignore them even for the average student.

Calculus is the mathematics of change. The mathematics you have learned up to this point has served mainly to describe *static* (unchanging) situations; the calculus handles *dynamic* (changing) situations. Change is characteristic of the world. As Heraclitus (sixth to fifth century B.C.) said,

You cannot step in the same river twice.

and

Everything is in a state of becoming and flux.

Probability is the mathematics of uncertainty. Not only do we constantly face situations in which there is neither adequate data nor an adequate theory, but many modem theories have uncertainty built into their foundations. Thus learning to think in terms of probability is essential.

Statistics is the reverse of probability (glibly speaking). In probability you go from the model of the situation to what you expect to see; in statistics you have the observations and you wish to estimate features of the underlying model. There is, of course, much more to statistics than this.

This book is *not* mainly about the results obtained in mathematics; rather it is concerned with mathematics itself. There is simply too much mathematics in current use, let alone what will be in use in the near future, to try to cover all the applications of mathematics. Instead of concentrating on the *results*, we will concentrate more on the *methods* from which the results follow. Thus this book is fundamentally different from the other books on mathematics you have studied. Most mathematics books are filled with finished theorems and polished proofs, and to a surprising extent they ignore the methods used to create mathematics. It is as if you were merely walked through a picture gallery and never told how to mix paints, how to compose pictures, or all the other “tricks of the trade.” Of course it would be simpler if I could tell you all the things you need to know about mathematics, but this approach seems to be hopeless for the coming years. Showing you the methods for doing mathematics covers a wider range of applications, but it does leave more creativity to you when you need some specific result. You will be left more able to do mathematics, to create mathematics as you need it, but less able to recall some specific result you happen to need. In short, in the face of almost infinite useful knowledge, we have adopted the strategy of “information regeneration rather than information retrieval.” This means, most importantly, you should be able to generate the result you need even if no one has ever done it before you—you will not be dependent on the past to have done everything you will ever need in mathematics.

I have also chosen to raise many questions about the relationship between the mathematical models developed in the text and the physical world in which you live. I have not attempted to supply you with all the answers; rather it is up to you to think about them and come to your own opinions of how far you can trust the results of applying mathematics in the real world. If you are to go very far in your chosen field, it is doubtful if you can long avoid some new applications of mathematics; yet, as you will see, not all applications give sensible answers!

The assumptions and definitions of mathematics and science come from our intuition, which is based ultimately on experience. They then get shaped by further experience in using them and are occasionally revised. They are not fixed for all eternity. In many applications it is essential that you be able to trace the effects of various assumptions and definitions on any conclusions you draw—perhaps the particular mathematics you used was inappropriate for your case! New applications of mathematics will, from time to time, require new assumptions and altered definitions, and it is the intent of the text to prepare you to make them when needed, but naturally we cannot tell what they will be.

It is not claimed that one course will make you a great mathematician able to create all of mathematics for yourself; all that can be done is to start you down the path of learning to create mathematics. In a very real sense, all we can do is coach you; you must have both some talent and the willingness to practice what is being taught. If you expect to continue learning all your life, you will be teaching yourself much of the time. You must learn to learn, especially the difficult topic of mathematics.

*1.2 THE UNIQUENESS OF MATHEMATICS*

At first glance much of mathematics seems arbitrary, but, at least for most useful mathematics, this is not so. To study the essential uniqueness of mathematics, I asked many expert scientists and engineers the following question: “If we ever find ourselves in two-way communication with a distant world, will they have essentially the same mathematics as we do?” The answers were all a definite “Yes.” They generally reasoned along the following lines:

1.The physical phenomena we see in space resemble those we see on Earth.

2.From this we infer that the same laws of physics apply everywhere.

3.Since mathematics is the language of science, it too must be essentially the same.

Their arguments were much more detailed and complete than this, but at present we are in no position to explain the technical details they used, nor the breadth of their arguments.

The meaning of the word “essentially” needs some explanation. In Euclidean geometry (approximately 300 B.C.), for example, the Greeks apparently chose to ignore *orientation*. They said that a left-handed and a right-handed triangle could be congruent (__Figure 1.2-1__). To carry out the proof, they allowed the flopping over in three dimensions of a triangle, even though a triangle is a two-dimensional figure. As a result of this choice, when they came to three-dimensional geometry they could not get the important theorem that in three dimensions there are only two orientations, the left-handed and the right-handed spirals. The distant intelligent beings we are imagining might have chosen to include orientation in their geometry. But it is claimed that this is not an *essential* difference. Again, Ptolemy (second century A.D.) used the chords of the double angle where we now use sines (__Figure 1.2-2__), but this is hardly an essential difference; it is merely a notational difference, although an important one in practice.

**Figure 1.2-1** Congruent triangles

**Figure 1.2-2** Ptolemy’s chord = 2 sin *θ*/2

Although the major users of mathematics are almost all in agreement that mathematics is essentially unique, we need to consider the idea often expressed by pure mathematicians that

Mathematics is the free creation of the human mind.

In a sense the users of mathematics are saying that mathematics, like science itself, is *discovered* rather than *created*. The pure mathematicians are saying that mathematics resembles the other arts in the sense that creation is a personal thing. In science creativity is comparatively impersonal; if one scientist does not discover something then another one will. But if Shakespeare had not written the play *King Lear*, we would probably have no closely similar play.

There may be less conflict between these two extreme opinions than appears; perhaps they are talking about different things. The users are talking about the mathematics they have found to be useful, while the pure mathematician may be talking about the mathematics now being created.

Another explanation of the difference is that those who daily work closely with the real world tend to believe that our senses, while occasionally deceived, report fairly accurately what is out there. On the other hand, those who work more with their imaginations tend to believe that our senses are rather unreliable, and the world out there could be very different from what we think it is. All are agreed that we cannot know with absolute certainty.

The mathematical results in this book are user oriented; they are the kinds that have been found to be useful in helping us to understand the universe in which we live. Textbooks in the past, especially at the calculus level, have concentrated on physical science applications. This was appropriate both because they are the historical source of much of the mathematics and because they are usually easier to understand. But the applications of mathematics of interest to the typical student are of far wider range than this. We will do a little in this direction of selecting suitable applications, but we are forced to minimize them lest they get in the way of the essential part of the book—how *you do mathematics*. The applications are often illustrated in the case histories.

However, we do not neglect the beauty of mathematics; it often makes the subject matter much more attractive and hence easier to master. The student should often find beauty in this book and gradually learn to do mathematics in attractive ways. As Edna St. Vincent Millay (1892–1950) wrote,

Euclid alone has looked on beauty bare.

*1.3 THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS*

There is a universality about mathematics; what was created to explain one phenomenon is very often later found to be useful in explaining other, apparently unrelated, phenomena. Theories that were developed to explain some poorly measured effects are often found to fit later, much more accurate measurements. Furthermore, from measurements over a limited range the theory is often found to fit a far wider range. Finally, and perhaps most unreasonably, quite regularly from the mathematics alone new pehnomena, previously unknown and unsuspected, are successfully predicted. This universality of mathematics could, of course, be a reflection of the way the human mind works and not of the external world, but most people believe it reflects reality.

This remarkable effectiveness is difficult to explain unless one makes some kind of assumption, such as that there exists, in some sense, both a physical and a logical universe to be discovered, and that these two universes are somehow intimately related. Mathematics is then seen to be a description of the logical structure of the universe.

A second explanation is that the useful mathematics, being based on long experience, follows the scientific approach. Only after being well established were the postulates (or axioms) carefully abstracted. Currently, mathematicians make no distinction between axiom and postulate; even Archimedes (287?–212 B.C.) did not bother to distinguish between them. Mathematics based on arbitrary postulates, or possibly picked either for their elegance or else for the elegance of what follows from them, would seem to have little chance of being successfully used in explaining the universe.

The postulational approach is widely used in mathematics and is very valuable when used properly. There is an innate elegance about the postulational approach. But it has been well said, and repeated by several well-known mathematicians, that

A book which starts off with axioms should be preceded by another volume explaining how and why these axioms have been chosen, and with what end in view.

We will carefully observe this rule and examine closely any assumptions we lay down.

Yet it must be acknowledged that an eminent physicist, the Nobel laureate P. A. M. Dirac (1902–), has said

I want to emphasize the necessity for a sound mathematical basis for any fundamental physical theory. Any philosophical ideas that one may have play only a subordinate role. Unless such ideas have a mathematical basis they will be ineffective.

Apparently, the elegance of mathematics should not be dismissed lightly.

When a theory is sufficiently general to cover many fields of application, it acquires some “truth” from each of them. Thus there is a positive value for generalization in mathematics that may not be apparent to the beginner. This is one of the many reasons this book emphasizes the processes of extension, generalization, and abstraction. They often bring increased confidence in the results of a specific application, as well as new viewpoints. Notice that we are mainly interested in the *processes*, and we are not interested in presenting mathematics in its most abstract form. On the contrary, we will often begin with concrete forms and then exhibit the *process* of abstraction.

It is necessary to note that science and mathematics do not explain everything. In more than 2000 years they have added little to our understanding of such things as Truth, Beauty, and Justice. There may be definite limits to the applicability of the scientific method.

*1.4 MATHEMATICS AS A LANGUAGE*

As Galileo said,

Mathematics is the language of science.

Not only must you learn to think in the language of mathematics, you also need to read it. To do this, you must learn the alphabet, the vocabulary, and the grammar. In this language there are no syllables, and the individual characters tend to be the words, while the equations tend to be the sentences. Much of mathematics consists of rewriting a sentence in another logically equivalent form.

This book is filled with strange symbols that you need to recognize easily. For example, you have already met in school the Greek lowercase letter *π* (pi). Because the individual letters often play the role of words, we need a rich alphabet of symbols in mathematics, and for this reason mathematicians have been driven to the use of the Greek alphabet, both upper- and lowercase. It would be foolish to try to avoid the Greek alphabet because it is in constant use in mathematics and its applications. Thus you should learn to recognize these symbols, and for this purpose they are given in __Appendix C__. Learn to recognize them rapidly so that when they occur in the middle of some difficult passage you need not be distracted by, “What symbol is that?” There are also other strange symbols such as the elongated S (∫), which is an operation called “integration.” Thus the language has both processes (verbs) and things (nouns).

It is fairly obvious that you need to both read and write mathematics. But it was a long time before I learned to listen to what a formula had to tell me. Formulas should be studied with this attitude in mind: what is it trying to say? Why is the formula the way it is? This attitude is much like that of the experimentalist who listens to what the physical (material) world says; the mathematician is similarly trying to learn what the world of abstract symbols has to say.

It is a common observation that in translating from one natural language, say Russian or Chinese to another, say English, the precise meaning cannot always be preserved. Partly this is because there may be no appropriate word, and partly because each language, to some extent, imposes its own patterns of thought. Mathematics, being very different from the natural languages, has its corresponding patterns of thought. Learning these patterns of thought is much more important than any particular result in the text. They are learned by the constant use of the language and cannot be easily taught in any other fashion. The mastery of the mathematical way of thinking is one of the main goals of this book.

*1.5 WHAT IS MATHEMATICS?*

In a conversation a friend of mine once said:

Mathematics is merely clear thinking.

There is a great deal of truth in his remark. Clear thinking can be done without the use of formal mathematical symbols, but in his view the clear thinker is doing mathematics. The use of special symbols is merely an economy in thinking and expressing these thoughts (a shorthand if you prefer). Algebra is not a random collection of arbitrary rules; rather it is based on a few simple rules that came from long experience in handling numbers plus drawing logical conclusions (see __Section 2.7__ for two examples). You probably remember many of the deductions from this simple base as rules for handling algebraic expressions with ease and rapidity. Theorems are further economies because they record more complex patterns of thinking that once shown to be valid need not be repeated every time they are needed. As George Temple (1901–) wrote,

there is some element of truth in maintaining that mathematics is not so much a subject as a way of studying any subject, not so much a science as a way of life.

When you began studying algebra the teacher probably said that the mysterious symbol *x* was some number that was as yet undetermined. Perhaps it was called “a general number” or some similar thing. But as you got further into the course, you began to see that often the *x* was never going to be a number. For example, in the equation

(*x* + 1)^{2} = *x*^{2} + 2*x* + 1

you really have an equality for any *x* (meaning for all *x*), and further thinking and watching what happens when you use algebra suggest that *x* is merely a symbol with certain rules of combination and need not always be thought of as a number. Thus the *x* is sometimes thought of as an abstract symbol having no particular meaning.

As you go through this book you will find more and more that mere numbers are left behind. Occasionally, specific numbers will be used to find a specific answer or to illustrate a point, but often the result will be left in a general form. For example, the area of a triangle is usually given by the general formula

without any specific numbers. At this stage in your education, substituting specific numbers into a general formula is one of the things we suppose you can do and is hardly to be considered an essential part of your further education. Thus, computing with numbers will have surprisingly little to do with the material in this book. As you will gradually see, calculus systematically evades a great deal of numerical calculation.

If mathematics is not just numbers, and *x* is not just a general number, then what is mathematics? There is no agreed upon definition of mathematics, but there is widespread agreement that the essence of doing mathematics is:

1.Extension

2.Generalization

3.Abstraction

These are all somewhat the same thing. They are important because faced with almost an infinity of details you cannot afford to deal constantly with the specific; you must learn to embrace more and more detail under the cover of generality.

We will not generalize for the sake of generalization, nor abstract for the fun of it. As Mark Kac (1914–) has written,

For unrestricted abstraction tends also to divert attention from whole areas of application whose very discovery depends on features that the abstract point of view rules out as being accidental.

We will deliberately give you practice in these arts by giving both examples and exercises in the form: “Extend the above results,” or “Generalize the above,” or “Give a reasonable abstraction of the above.” The difficulty facing you is that there is no unique, correct answer in most cases. It is a matter of taste, depending on the circumstances, your own personal traits and needs, and the particular age you live in. At first you will have to depend on the taste of this book and the professor. Gradually, you will develop your own taste, and along the way you may occasionally recognize that your taste may be the best one! It is the same as in an art course. During the course you must pay reasonable attention to the taste of the professor, but you should not neglect your own. The art is not simple, as the following remark shows:

As opposed to abstraction the art of doing mathematics consists in finding special cases which contain all the germs of generality. David Hilbert (1862–1943).

In mathematics, at first you are presented with specific problems. Gradually, you learn that there are many things you could do that have not been done. The taste to work on the right problem at the right time and in the right way is the secret of doing significant things. It is a matter of taste as to when and to what you apply the techniques you have mastered. A central problem in teaching mathematics is to communicate a reasonable sense of taste—meaning often when to, or not to, generalize, abstract, or extend something you have just done. Taste is the main difference between first- and second-rate mathematicians. Of necessity the exercises in this book must be fairly easy and of no great importance. When doing significant things, it often takes days, weeks, months, and at times even years to get the result, but such time is not available in a course like this. Thus you must develop your abilities by practicing on simple, often trivial, exercises. Occasionally, short *case studies* are included to illustrate how a number of different ideas can be combined into a larger piece of mathematics.

*1.6 MATHEMATICAL RIGOR*

When you yourself are responsible for some new application of mathematics in your chosen field, then your reputation, possibly millions of dollars and long delays in the work, and possibly even human lives, may depend on the results you predict. It is then that the *need* for mathematical rigor will become painfully obvious to you. Before this time, mathematical rigor will often seem to be needless pedantry (a pedant is one who unduly explains minutiae). Thus rigor requires some practice before the need first arises. It is not something that you can learn suddenly, but it requires the gradual development of your abilities.

Mathematical rigor is the clarification of the reasoning used in mathematics. Usually, mathematics first arises in some particular situation, and as the demand for rigor becomes apparent more careful definitions of what is being reasoned about are required, and a closer examination of the numerous “hidden assumptions” is made. The student should be aware that often the definitions are gradually changed so some desired results can be obtained, but *sometimes* the earlier definitions are retained and different results are found to follow once a closer examination is made of all the details involved.

Over the years there has been a gradually rising standard of rigor; proofs that satisfied the best mathematicians of one generation have been found to be inadequate by the next generation. Rigor is not a yes-no property of a proof, much as you might like it to be; rather it is a vague standard of careful treatment that is currently acceptable to a particular group. [__K–2__] and [__L__] (references are at the end of this chapter).

Ideally, when teaching a topic the degree of rigor should follow the student’s perceived need for it; but as just noted it is not something that you can suddenly master. It is necessary to require a gradually rising level of rigor so that when faced with a real need for it you are not left helpless. As a result, this book does not contain a uniform level of rigor, but rather a gradually rising level. Logically, this is indefensible, but psychologically there is little else that can be done.

The truth is that we do not always know what we are talking about. For example, the statement

All statements are false,

appears to contradict itself. First, it is surely a statement; therefore, accordingly, it must a be false statement. But if it is false, then the statement is true. There is a built-in contradiction in the grammatically correct statement.

Another example is the rule

Every rule has an exception.

If this rule is always true, then there are rules without an exception, and therefore this rule is false. Again, a contradiction!

Troubles like this can be made to arise whenever what is being said includes itself—a self-referral situation. Expressions like “the set of all sets” have this self-referral property and hence are dubious expressions; *you* are by no means sure that it does not contain a contradiction within itself.

The function of rigor is mainly critical and is seldom constructive. Rigor is the hygiene of mathematics, which is needed to protect us against careless thinking.

*1.7 ADVICE TO YOU*

The preceding discussion indicates that this book is unlike almost all other mathematics books, both in content and in aims. We intend to teach the doing of mathematics (of course at an elementary level). The applications of these methods produce the results of mathematics (which is usually only what is taught); the results are the worked out examples in the book and in the exercises to be solved. There is also a deliberate policy to force you to think abstractly. Not only is mathematics by its very nature abstract, but it is only through abstraction that any reasonable amount of the useful mathematics can be covered. There is simply too much known to continue the older approach of giving detailed results.

Consequently, you will find that these unconventional goals cause you a good deal of confusion. Yes, many of the results are important, but more important are the underlying methods. Things are scattered and repeated throughout the text; they are not organized in the conventional way, since the conventional goals are not the ones we have in mind. It is easy to measure your mastery of the results via a conventional examination; it is less easy to measure your mastery of doing mathematics, of creating new (to you) results, and of your ability to surmount the almost infinite details to see the general situation. In the long run the methods are the important parts of the course. Again, the specific results that are obtained have been carefully selected to be highly useful, but they are not the heart of the book.

It is not sufficient to know the theory; you should be able to apply it. There are two extremes, both of which are bad: (1) you know what to do but are unable to do it, and (2) you know how to do things but not what to do. From my more than 30 years of experience using mathematics in large scientific-engineering organizations, I believe that if you cannot handle both equally well then it is better to know what to do. You often, but not always, have some time to learn how to do it, or else you can find someone to help you. But if you do not know what to do, then the things that you do will often cause more harm than good. On the other hand, if you are not a master of details, you can seldom create what is needed.

It is currently believed that the human mind progresses from simple to more difficult ideas by “chunking” isolated small results into larger units of thought. Unfortuntely, these elementary results must be thoroughly learned *before* the chunking can occur. If this theory is true, then a lot of this course consists in first mastering many small details and then gradually chunking them into larger units. Still later, when these chunks are mastered, they are combined into even larger chunks. You should be aware of this process and do what you can to facilitate the chunking. In this way you gradually gain a mastery over the sea of details, and thus come to the mastery of the whole as a whole. But you must begin by *mastering* the small chunks thoroughly. It is only then that you can go to the larger chunks.

Because mathematics is abstract, it covers many special cases and, therefore, while studying large, complex systems, mathematics often enables you to cut through the sea of details to get at the essential nature of the problem. It is necessary that you learn both to be comfortable with abstractions and to see the details that are embraced by the sweeping statements of the abstractions. A careful examination of the great contributions to science shows that usually they require both the abstract point of view, so you can see the whole, and mastery of the details, so you can create a relevant theory.

Finally, I refuse to “write down” to you. It would be easy to replace all the long words and sentences with shorter ones, to elide or evade all difficult ideas, and to use only simple examples and simple problems. To do so would not contribute to your general education. At a minimum an educated person should be able to read reasonably sophisticated writing, and this cannot be left solely to the English department. Similarly, I could avoid all references, however slight, that fall outside the strictest bounds of the subject, and again leave you stunted in your intellectual growth. I believe that *every* course has some responsibility to contribute to your general education lest it slip between the individual courses, and from all the specific courses you take you still emerge uneducated. Thus no matter how you look at it, this book is not going to be easy to read and digest. I am ambitious for you, but I can only coach you; you must do the actual work of learning the many things that are being presented.

*1.8 REMARKS ON LEARNING THE COURSE*

Universities are organized into schools, the schools by departments, and the departments often have subdepartments. The subdepartments further divide up knowledge into separate courses. All this (and more) breaks up the unity of knowledge into disconnected pieces, and because everyone, when they were in school, was exposed to this arrangement of knowledge, they tend to believe it and repeat these somewhat artificial divisions in their teaching. However, the applications of knowledge, especially mathematics, reveal the unity of all knowledge. In a new situation almost anything and everything you ever learned might be applicable, and the artificial divisions seem to vanish. One of the main purposes of this book is to show some of the unity of the traditionally separated fields of mathematics and its applications.

In a course the author is forced to present the material in a linear sequence (in time). One topic must follow another in spite of the fact that each may illuminate the other. As a result, if unity is to be emphasized, the author must provide a large number of cross references. It is necessary, therefore, to refer many times to things which are physically remote in the book. To facilitate this matter of referencing things, the book is divided into 26 chapters with each chapter further subdivided into around 10 sections. Within a section the equations, figures, tables, and examples are each individually further labeled for easy reference. The notation

Equation (2.7-5)

means __Chapter 2__, __Section 7__, Equation 5. We use a similar notation for the other kinds of objects. This produces a flavor of excessive numbering, but it is very convenient to refer the reader to exactly what is meant or at least to the immediate region of the book where it can be found. References to books are indicated by [ ] with the letter inside generally the first letter of the author’s name. The references are at the end of this chapter.

Second, elementary mathematical courses are often taught in a form that allows, or even encourages, learning the methods for solving particular classes of problems. The calculus, probability, and statistics can be applied so broadly that it is hopeless for you to try to learn all the “tricks” for solving specific problems. You must learn to create the details of how to solve the problem at hand; you must learn to do mathematics, to create the small tools that you need as you go along. To do this, you must learn to read the language of mathematics, to understand the meaning of the words and symbols, and to listen to the formulas.

Third, at this level of mathematics it is a common remark by students that they learn each course in the following course when they use the mathematics constantly. It is through repetitive use of the ideas that you finally master and make your own the major ideas as well as the skills of doing mathematics. Therefore, this text is organized in the “spiral” for learning (__Figure 1.8-1__). A topic such as probability is returned to again and again, each time higher up on the spiral (and perhaps including some new material). The first time around you may not be completely sure of what is going on, but on the repeated returns to the topic it should gradually become clear. This is necessary when the ideas are not simple but require a depth of understanding, and this includes the three main topics of the course: calculus, probability, and statistics.

**Figure 1.8-1** The spiral of

learning

Fourth, unfortunately, besides the theory there are a lot of small technical details that must be learned so well that you can recall them almost instantaneously, such as the trigonometric identities that you once knew and forgot as soon as possible. One way of learning this kind of small detail for immediate recall when needed is to put one part of the identity on one side of a 3 by 5 inch card and the other part on the other side. Using these flash cards you can, in odd moments of your daily life, learn the mechanical parts of the course (small flash cards are often available locally in book stores). If you must stop and look up such details in the middle of a large problem, or even pause to try to recall them, then you will not be able to create the mathematics you will need to solve the problem at hand.

Flash cards may be used to learn many of the formulas and definitions that arise in this course and in later years. It is well known that *for this kind of low-level material* many short learning sessions are much more efficient than a few long, intense ones; but this is not necessarily true for larger ideas. No matter how obvious it is, most students will not use such trivial devices as flash cards; it seems to be beneath their dignity. They suffer accordingly.

You are advised to read this book with paper and pen handy so you can frequently check your understanding of what you are reading. This advice is repeatedly given to mathematics students, and it is repeatedly ignored by them. Remember, the book is only the coach; *you* must learn to do the mathematics. As in so many other situations, you only learn by doing it yourself; you must *involve* yourself in mathematics if you are to master it as a way of thinking about the universe.

Whenever possible, the exercises are arranged as follows:

1.Check that you have the general ideas.

2.Practice on the more mechanical parts.

3.Doing some mathematics using the ideas.

4.Extensions of these ideas (which, if hard, will be starred).

You cannot learn to do mathematics without effort. “There is no royal road to geometry” was said in ancient Greek times, (Menaechmus to Alexander, around 300 B.C.). Because we expect you to make progress, the later parts of the book are deliberately arranged to require greater effort on your part. Often it is not the particular result that is important; it is the development of your abilities that matters.

Measurements on outstanding mathematical and scientific students show that the successful students regularly go over their successful problems and examine why they succeeded. Thus you are *strongly* advised to ask yourself at the end of a problem, “Why did it work?”, “What other problems can be solved in this way?”, and, “How else could it be solved?” The author believes that the old saying, “You should learn from your mistakes,” is less important than, “You should learn from successes.” After all, there are so many ways of being wrong and so few of being right that it is much more economical to study successes. The constant recommendation of successful scientists is to

Go to the masters, not the commentators.

It is the master who, by definition, has the right style, and often the commentators give the results without the essence—style! A common error of students is to do too many problems and too little study of why things went right or wrong. Chunking must be accomplished at each level if progress to higher levels of thought is to be made, and reflection on the details is a good way of accomplishing this goal.

Fifth, although textbooks (and professors) like to make definite statements indicating that they know what they are talking about, there is in fact a great deal of uncertainty and ambiguity in the world. Contrary to normal practice, we will not evade this question but rather explore (overexplore?) it. It is hard not to be told definitely that something is or is not so, but great progress is often made when what was long believed to be true is now seen to be perhaps not the whole truth. Thus the text often uses words like “It is currently believed that …” and “Mathematicians believe that …” rather than the conventional dogmatic statements that you usually see. They are there to cause you to think about the uncertainess and even the arbitrariness of much of our current conventions and definitions, to ponder about your acceptance of them. Again, this book is concerned with your total education, not just a small part of your mathematical training, and this makes it hard on you as you go along. It is not easy to become an educated person.

**REFERENCES**

The following references are for general reading. [__N__] gives, in four volumes, selected writings of great mathematicians and scientists, translated when necessary into English. Both [__B–2__] and [__K–1__] are histories of mathematics. [__B–1__] refers especially to the history of the calculus, and [__T__] to the history of probability. [__P__] is the classic on how to solve problems and has not been improved on in spite of recent computer-based books.

[__D__] gives a survey of what it is to be a mathematician. [__L__] is a powerful dialogue on the actual nature of mathematics as opposed to the official versions that most people have in their minds, and [__K–2__] is devoted to the subtitle, “the loss of certainty” in mathematics since the early Greek times, when mathematics was regarded as certain knowledge.

Any reasonable library can supply many other books on mathematics, generally, and calculus, probability, and statistics specifically. [__C__] is a classic on the calculus and is well worth consulting, and [__A__] is an elegant calculus book at a slightly more advanced level; both are in two volumes. [__F__] is the classic in probability, especially the first volume. It is hoped that this course arouses your interest enough for you to look beyond it for further information of interest to you.

[ |
Apostol, T. M. |

[ |
Boyer, C. B. |

[ |
Boyer, C. B. |

[ |
Courant, R. |

[ |
Davis, P., and R. Hersh. |

[ |
Feller, W. |

[G-R] |
Gradshteyn, I., and I. Ryzhik. |

[ |
Kline, Morris. |

[ |
Kline, Morris. |

[ |
Lakatos, Imre. |

[ |
Newman, James. |

[ |
Polya, George. |

[ |
Todhunter, I. |