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What shall we say of this metamorphosis in passing from finite to infinite? Galileo, Two New Sciences As its title suggests, this book was conceived as a prologue to the study of "Why the calculus works"--otherwise known as analysis. It is in fact a critical reexamination of the infinite processes arising in elementary math­ ematics: Part II reexamines rational and irrational numbers, and their representation as infinite decimals; Part III examines our ideas of length, area, and volume; and Part IV examines the evolution of the modern function-concept. The book may be used in a number of ways: firstly, as a genuine pro­ logue to analysis; secondly, as a supplementary text within an analysis course, providing a source of elementary motivation, background and ex­ amples; thirdly, as a kind of postscript to elementary analysis-as in a senior undergraduate course designed to reinforce students' understanding of elementary analysis and of elementary mathematics by considering the mathematical and historical connections between them. But the contents of the book should be of interest to a much wider audience than this­ including teachers, teachers in training, students in their last year at school, and others interested in mathematics.

Inhaltsverzeichnis

Frontmatter

From Calculus to Analysis

Frontmatter

Chapter I.1. What’s Wrong with the Calculus?

Abstract
The invention (around 1670) of the differential and integral calculus, its development, application and extension during the following two centuries, and the somewhat belated explanation (completed by about 1870) of why this calculus works, together constitute one of the major intellectual achievements of Western European culture.
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Chapter I.2. Growth and Change in Mathematics

Abstract
You may notice that the title (and the contents) of this and the previous chapter contradict the idea that mathematics consists of a fixed and unchallengeable stock of truths. Should this come as a surprise? How are mathematical ideas born? How do they grow? In what sense does mathematics itself “evolve”? There are as one might expect no easy answers. In the long term one simply has to keep such questions permanently in the back of one’s mind, ready to take advantage of any new example which might provide some unexpected insight. But in the short term, the challenge is to begin, somehow or other, to make sense of such questions. We should not perhaps expect at the outset to make much sense of historical examples, but we can at least begin by reflecting how our own view of mathematics has changed. If we do this, then it should soon become apparent that our own private view of mathematics is never rigidly fixed, but changes as we grow.
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Number

Frontmatter

Chapter II.1. Mathematics: Rational or Irrational?

Abstract
Perhaps the most basic idea in all of mathematics is that of counting numbers—the positive whole numbers. If human beings are to get interested in anything mathematical, then we should not be surprised to find them beginning with these counting numbers—their patterns of odd and even; the squares, cubes, and higher powers; the triangular numbers 1, 3, 6, 10, 15,...; the primes; the divisors of a given number and its factorisation as a product of primes; and many other fascinating properties (see, for example, Exercise 2).
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Chapter II.2. Constructive and Non-constructive Methods in Mathematics

Abstract
In the previous chapter we started by tacitly assuming that any given pair of line segments could necessarily be measured exactly by some (suitably small) common measure. But, when this assumption was applied to the pair of segments AB, AC in Figure 6, we derived a contradiction. We were thus forced by the principles of logic to admit that our tacit assumption was untenable: there clearly exist some pairs of segments (such as AB, AC in Figure 6) which have no common measure at all.
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Chapter II.3. Common Measures, Highest Common Factors and the Game of Euclid

Abstract
In Chapter II. 1 we gave an indirect proof of the non-existence of a common measure for certain pairs of line segments (such as the side and diagonal of a square). In this chapter we shall develop a direct, constructive procedure for finding a common measure of two segments when a common measure exists. In the next chapter we shall complement the discussion of Chapter II. 1 by using this constructive procedure to give a second proof that the side and diagonal of a square do not have a common measure.
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Chapter II.4. Sides and Diagonals of Regular Polygons

Abstract
Let us now apply the procedure developed in the previous chapter to try to find a common measure for the diagonal AC and the side BC of a square ABCD.
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Chapter II.5. Numbers and Arithmetic— A Quick Review

Abstract
Some mathematical ideas seem as subtle and difficult today as when they were first introduced; others, which could once upon a time be handled only by those with a very extensive training, now seem entirely elementary to us. For example, addition and multiplication of whole numbers was once a skilled occupation, but is today taught to all children.
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Chapter II.6. Infinite Decimals (Part 1)

Abstract
Those who introduce us to counting numbers usually go to great lengths both to invest these mathematical entities with ordinary meaning and to clarify the mathematical idea behind the usual base 10 representation of numbers (that is, the idea of place value corresponding to increasing powers of 10). Similar efforts are made when the time comes to introduce negative whole numbers, fractions, and the decimal representation of decimal fractions. But though the long division process, which we use to transform an ordinary fraction into a decimal fraction, frequently gives rise to infinite decimals, little if any time or effort is devoted either to investing these curious entities with ordinary meaning, or to clarifying the mathematical idea which justifies their representation as never ending decimals. Experience suggests that many undergraduates complete their studies of sequences, series, and limits in the calculus without ever realising the light they shed on infinite decimals. But since the very essence of the calculus lies in the careful use it makes of infinite processes to supplement the familiar processes of ordinary arithmetic, and since infinite decimals constitute the most familiar example of such infinite processes, it seems rather obvious that these should be the very first candidates for analysis: this is precisely the aim of the next few chapters.
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Chapter II.7. Infinite Decimals (Part 2)

Abstract
In Chapter II.6 we stressed the tension between the familiar division process used to generate the decimal representation of a fraction like 3/40, and the (as yet) meaningless string of digits which results when we apply this same familiar division process to a fraction like 3/39 = 1/13, namely
$$ \frac{1}{{13}} = .07692307692307692307 \ldots $$
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Chapter II.8. Recurring Nines

Abstract
We learn to compare ordinary finite decimals more or less by eye: thus .09 and .1 not only look different, they really are different—.09 being less than .1, since 9/100 ( = .09) is less than 10/100 ( = .1). But we have gone out of our way to stress the fact that, unlike ordinary finite decimals, infinite decimals do not correspond to decimal fractions; instead they have to be interpreted in a completely new way as endless sums. We therefore have to resist any temptation to assume that procedures which work with finite decimals will automatically carry over to infinite decimals.
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Chapter II.9. Fractions and Recurring Decimals

Abstract
Each time we have worked out the infinite decimal corresponding to a fraction, the string of decimal digits has always ended with a repeating block: for example,
$$ \begin{array}{*{20}{c}} {1/3 = .\dot{3}} \\ {8/70 = .1\dot{1}4285\dot{7}} \\ {1/13 = .\dot{0}7692\dot{3}} \\ {1/11 = .\dot{0}\dot{9}.} \\ \end{array} $$
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Chapter II.10. The Fundamental Property of Real Numbers

Abstract
Suppose I write down the first two infinite decimals that come into my head:
$$ \begin{array}{*{20}{c}} {123 \cdot 45678967896789678967 \ldots ,} \\ {123 \cdot 45678910111213141516 \ldots .} \\ \end{array} $$
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Chapter II.11. The Arithmetic of Infinite Decimals

Abstract
To end our protracted encounter with infinite decimals we should at least answer the question which started it all off:
Given that the familiar arithmetical procedures for addition, subtraction, multiplication and division simply do not work for infinite decimals, how can we possibly calculate
$$ \alpha + \beta ,\alpha - \beta ,\alpha \cdot \beta ,\frac{\alpha }{\beta } $$
where ∝ and β are real numbers given in the form of infinite decimals?
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Chapter II.12. Reflections on Recurring Themes

Abstract
We began Part II by examining (in Chapters II.1, II.3 and II.5) the relationship between three ideas:
(1)
common measures for pairs ofline segments AB, CD in geometry;
 
(2)
ordinary rational numbers a/b; and
 
(3)
(highest) common factors of a and b.
 
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Chapter II.13. Continued Fractions

Without Abstract
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Geometry

Frontmatter

Chapter III.1. Numbers and Geometry

Abstract
In Parts I and II we have gone out of our way to stress the enormous difference between the finite procedures of ordinary arithmetic, and those mathematical concepts whose very meaning depends on the introduction and interpretation of infinite processes. In contrast, you have in the past been encouraged to use real numbers (whether rational or irrational) in a naive, unquestioning way—especially in geometry: for example, you have been quietly encouraged to assume that, if we measure the length of a line segment AB in terms of some given unit segment CD, then its length AB/CD can obviously be expressed as a real number. While this is obvious when CD fits into AB a whole number of times leaving no remainder, or when CD and AB have some common measure MN which fits into CD precisely b times with no remainder and into AB precisely a times with no remainder (in which case AB/CD = a/b), it is not at all obvious in general. In Chapter 11.13 we saw one way of justifying the belief that AB can always be measured in terms of CD, but it was not exactly obvious!
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Chapter III.2. The Role of Geometrical Intuition

Abstract
The fact that our mental picture of real numbers is all tied up with the geometrical number line certainly helps our intuitive understanding of their properties: inequalities (a < b) correspond naturally to the relative position of points on the number line (to the left of/to the right of); addition and subtraction correspond to shifts (to the right and to the left); and multiplication (× a) can be thought of either in terms of enlarging (a times), or in terms of areas of rectangles. It makes good psychological sense to choose a framework for real numbers which exploits our intuitive geometrical understanding of points on a line—much of which derives from our experience of drawing and measuring with a ruler. But when the time comes to examine the precise nature of real numbers, this kind of dependence on geometrical intuition is logically indefensible, unless geometry itself can be shown to be in some sense a simpler, or a more natural starting point—in which case we should instead begin by analysing the details of our geometrical intuition as carefully as we can. The examples we discuss in Part III are intended to convince you that in a first course in analysis it is simpler to abstain completely from the use of geometrical arguments, and to develop both the real numbers and the calculus itself purely arithmetically.
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Chapter III.3. Comparing Areas

Abstract
In Part II we considered the problem of measuring the length of line segments—or, to be more precise, we considered the problem of comparing two line segments AB, CD by looking for a common measure. We discovered that a given pair of line segments AB, CD may have no common measure at all. However in this case, though our procedure for finding common measures could not possibly produce a genuine common measure as it was designed to do, it could nevertheless be interpreted to obtain a value for, or a sequence of better and better approximations to, the irrational ratio AB/CD (see Chapter II.13).
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Chapter III.4. Comparing Volumes

Abstract
In Chapter III.3 we discovered that comparing shapes in 2-dimensions was noticeably more complicated than comparing plain line segments. The proverbial optimist might of course declare that we should have expected 1-dimension to be rather special, and that, now we know (more or less) how to make the jump from 1- to 2-dimensions, we shall probably find that 3-, 4- and higher dimensions are really no more difficult than 2-dimensions. The pessimist, on the other hand, might point out that, since 2-dimensions gave rise to so many unexpected difficulties, we must surely expect 3-, 4- and higher dimensions to become steadily more complicated.
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Chapter III.5. Curves and Surfaces

Abstract
Like rulers themselves, length and distance are essentially “straight,” 1-dimensional concepts; similarly area is an essentially “flat,” 2-dimensional concept. Yet both concepts are used in other contexts: we do not restrict our attention solely to lengths of straight line segments, but are also interested in “lengths of curves”, such as the circle, or the cycloidal arch (Exercise 1); and we refer to “the area” not just of flat 2-dimensional shapes, but also of curved surfaces such as the cone, cylinder,1 and sphere. We shall end Part III by inquiring a little more closely how these extended notions of length and area can be justified, though we shall not come to any final conclusion.
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Functions

Frontmatter

Chapter IV.1. What Is a Number?

Abstract
In Chapter II.5 we stressed the distinction between the intuitive idea that the counting process is endless, and the actual fact that we have English names for only finitely many of these counting numbers.
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Chapter IV.2. What Is a Function?

Abstract
When we asked the question What is a number? we certainly did not expect an answer such as 1 and π are numbers.
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Chapter IV.3. What Is an Exponential Function?

Abstract
We shall end Part IV by examining very briefly one particular class of functions: namely powers x α , otherwise known as exponential functions. Our aim in so doing is simply to indicate the richness and the complexity of our own mathematical experience of such functions, and to consider how this complex experience might lead us eventually to appreciate the way exponential functions are usually treated in an analysis course.
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Backmatter

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