1.1 INTRODUCTION

INTRODUCTION

    A satisfactory discussion of the main concepts of analysis (such as convergence, continuity, differentiation, and integration) must be based on an accurately defined number concept. We shall not, however, enter into any discussion of the axioms that govern the arithmetic of the integers, but assume familiarity with the rational numbers (i.e., the numbers of the form 1.1 Introduction-Publish-web_1, where 1.1 Introduction-Publish-web_2 and 1.1 Introduction-Publish-web_3 are integers and 1.1 Introduction-Publish-web_4).

    The rational number system is inadequate for many purposes, both as a field and as an ordered set. (These terms will be defined in Secs. 1.6 and 1.12.)
For instance, there is no rational 1.1 Introduction-Publish-web_5 such that 1.1 Introduction-Publish-web_6. (We shall prove this presently.) This leads to the introduction of so-called "irrational numbers" which are often written as infinite decimal expansions and are considered to be "approximated" by the corresponding finite decimals. Thus the sequence
1.1 Introduction-Publish-web_7
"tends to 1.1 Introduction-Publish-web_8." But unless the irrational number 1.1 Introduction-Publish-web_9 has been clearly defined, the question must arise: Just what is it that this sequence "tends to"?

    This sort of question can be answered as soon as the so-called "real number system" is constructed.

1.1 Example

We now show that the equation

(1)

1.1 Introduction-Publish-web_10

is not satisfied by any rational 1.1 Introduction-Publish-web_11. If there were such a 1.1 Introduction-Publish-web_12, we could write 1.1 Introduction-Publish-web_13 where 1.1 Introduction-Publish-web_14 and 1.1 Introduction-Publish-web_15 are integers that are not both even. Let us assume this is done. Then (1) implies

(2)

1.1 Introduction-Publish-web_16

This shows that 1.1 Introduction-Publish-web_17 is even. Hence 1.1 Introduction-Publish-web_18 is even (if 1.1 Introduction-Publish-web_19 were odd, 1.1 Introduction-Publish-web_20 would be odd), and so 1.1 Introduction-Publish-web_21 is divisible by 1.1 Introduction-Publish-web_22. It follows that the right side of (2) is divisible by 1.1 Introduction-Publish-web_23, so that 1.1 Introduction-Publish-web_24 is even, which implies that 1.1 Introduction-Publish-web_25 is even.

    The assumption that (1) holds thus leads to the conclusion that both 1.1 Introduction-Publish-web_26 and 1.1 Introduction-Publish-web_27 are even, contrary to our choice of 1.1 Introduction-Publish-web_28 and 1.1 Introduction-Publish-web_29. Hence (1) is impossible for rational 1.1 Introduction-Publish-web_30.

    We now examine this situation a little more closely. Let 1.1 Introduction-Publish-web_31 be the set of all positive rationals 1.1 Introduction-Publish-web_32 such that 1.1 Introduction-Publish-web_33 and let 1.1 Introduction-Publish-web_34 consist of all positive rationals 1.1 Introduction-Publish-web_35 such that 1.1 Introduction-Publish-web_36. We shall show that 1.1 Introduction-Publish-web_37 contains no largest number and 1.1 Introduction-Publish-web_38 contains no smallest.

    More explicitly, for every 1.1 Introduction-Publish-web_39 in 1.1 Introduction-Publish-web_40 we can find a rational 1.1 Introduction-Publish-web_41 in 1.1 Introduction-Publish-web_42 such that 1.1 Introduction-Publish-web_43, and for every 1.1 Introduction-Publish-web_44 in 1.1 Introduction-Publish-web_45 we can find a rational 1.1 Introduction-Publish-web_46 in 1.1 Introduction-Publish-web_47 such that 1.1 Introduction-Publish-web_48. To do this, we associate with each rational 1.1 Introduction-Publish-web_49 the number

(3)

1.1 Introduction-Publish-web_50

Then

(4)

1.1 Introduction-Publish-web_51

    If 1.1 Introduction-Publish-web_52 is in 1.1 Introduction-Publish-web_53 then 1.1 Introduction-Publish-web_54, (3) shows that 1.1 Introduction-Publish-web_55, and (4) shows that 1.1 Introduction-Publish-web_56. Thus 1.1 Introduction-Publish-web_57 is in 1.1 Introduction-Publish-web_58.     

    If 1.1 Introduction-Publish-web_59 is in 1.1 Introduction-Publish-web_60 then 1.1 Introduction-Publish-web_61, (3) shows that 1.1 Introduction-Publish-web_62, and (4) shows that 1.1 Introduction-Publish-web_63. Thus 1.1 Introduction-Publish-web_64 is in 1.1 Introduction-Publish-web_65.

1.2 Remark

    The purpose of the above discussion has been to show that the rational number system has certain gaps, in spite of the fact that between any two rationals there is another: If 1.1 Introduction-Publish-web_66 then 1.1 Introduction-Publish-web_67. The real number system fills these gaps. This is the principal reason for the fundamental role which it plays in analysis.

    In order to elucidate its structure, as well as that of the complex numbers, we start with a brief discussion of the general concepts of ordered set and field.

    Here is some of the standard set-theoretic terminology that will be used throughout this book.

1.3 Definitions

    If 1.1 Introduction-Publish-web_68 is any set (whose elements may be numbers or any other objects), we write  1.1 Introduction-Publish-web_69 to indicate that 1.1 Introduction-Publish-web_70 is a member (or an element) of 1.1 Introduction-Publish-web_71. If 1.1 Introduction-Publish-web_72 is not a member of 1.1 Introduction-Publish-web_73, we write: 1.1 Introduction-Publish-web_74.

    The set which contains no element will be called the empty set. If a set has at least one element, it is called nonempty.
    If 1.1 Introduction-Publish-web_75 and 1.1 Introduction-Publish-web_76 are sets, and if every element of 1.1 Introduction-Publish-web_77 is an element of 1.1 Introduction-Publish-web_78, we say that 1.1 Introduction-Publish-web_79 is a subset of 1.1 Introduction-Publish-web_80, and write 1.1 Introduction-Publish-web_81, or 1.1 Introduction-Publish-web_82. If, in addition, there is an element of 1.1 Introduction-Publish-web_83 which is not in 1.1 Introduction-Publish-web_84, then 1.1 Introduction-Publish-web_85 is said to be a proper subset of 1.1 Introduction-Publish-web_86. Note that 1.1 Introduction-Publish-web_87 for every set 1.1 Introduction-Publish-web_88.
    If 1.1 Introduction-Publish-web_89 and 1.1 Introduction-Publish-web_90 , we write 1.1 Introduction-Publish-web_91. Otherwise 1.1 Introduction-Publish-web_92.

1.4 Definition

     Throughout Chap. I, the set of all rational numbers will be denoted by 1.1 Introduction-Publish-web_93.

1.1 Introduction-Publish-web_94

1.1 Introduction-Publish-web_95

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posted on 2013-07-07 19:31  HyperGropus  阅读(164)  评论(0)  编辑  收藏  举报

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