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1、New Words & Expressions: conversely 反之 geometric interpretation 几何意义 correspond 对应 induction 归纳法 deducible 可推导的 proof by induction 归纳证明 difference 差 inductive set 归纳集 distinguished 著名的 inequality 不等式 entirely complete 完整的 integer 整数 Euclid 欧几里得 interchangeably 可互相交换的 Euclidean 欧式的 intuitive直观的 the f
2、ield axiom 域公理 irrational 无理的,2.4 整数、有理数与实数 Integers, Rational Numbers and Real Numbers,New Words & Expressions: irrational number 无理数 rational 有理的 the order axiom 序公理 rational number 有理数 ordered 有序的 reasoning 推理 product 积 scale 尺度，刻度 quotient 商 sum 和,There exist certain subsets of R which are disti
3、nguished because they have special properties not shared by all real numbers. In this section we shall discuss such subsets, the integers and the rational numbers.,4A Integers and rational numbers,有一些R的子集很著名，因为他们具有实数所不具备的特殊性质。在本节我们将讨论这样的子集，整数集和有理数集。,To introduce the positive integers we begin with t
4、he number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1,2,3, obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers.,我们从数字1开始介绍正整数，公理4保证了1的存在性。1+1用2表示，2+1用3表示，以此类推，由1重复累加的
5、方式得到的数字1,2,3，都是正的，它们被叫做正整数。,Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”.,严格地说，这种关于正整数的描述是不完整的，因为我们没有详细解释“等等”或者“1的重复累加”的含义。,Although the intuitive
6、 meaning of expressions may seem clear, in careful treatment of the real-number system it is necessary to give a more precise definition of the positive integers. There are many ways to do this. One convenient method is to introduce first the notion of an inductive set.,虽然这些说法的直观意思似乎是清楚的，但是在认真处理实数系统
7、时必须给出一个更准确的关于正整数的定义。 有很多种方式来给出这个定义，一个简便的方法是先引进归纳集的概念。,DEFINITION OF AN INDUCTIVE SET. A set of real numbers is called an inductive set if it has the following two properties: The number 1 is in the set. For every x in the set, the number x+1 is also in the set. For example, R is an inductive set. So
8、 is the set . Now we shall define the positive integers to be those real numbers which belong to every inductive set.,现在我们来定义正整数，就是属于每一个归纳集的实数。,Let P denote the set of all positive integers. Then P is itself an inductive set because (a) it contains 1, and (b) it contains x+1 whenever it contains x.
9、Since the members of P belong to every inductive set, we refer to P as the smallest inductive set.,用P表示所有正整数的集合。那么P本身是一个归纳集，因为其中含1，满足(a)；只要包含x就包含x+1, 满足(b)。由于P中的元素属于每一个归纳集，因此P是最小的归纳集。,This property of P forms the logical basis for a type of reasoning that mathematicians call proof by induction, a de
10、tailed discussion of which is given in Part 4 of this introduction.,P的这种性质形成了一种推理的逻辑基础，数学家称之为归纳证明，在介绍的第四部分将给出这种方法的详细论述。,The negatives of the positive integers are called the negative integers. The positive integers, together with the negative integers and 0 (zero), form a set Z which we call simply
11、the set of integers.,正整数的相反数被叫做负整数。正整数，负整数和零构成了一个集合Z，简称为整数集。,In a thorough treatment of the real-number system, it would be necessary at this stage to prove certain theorems about integers. For example, the sum, difference, or product of two integers is an integer, but the quotient of two integers n
12、eed not to ne an integer. However, we shall not enter into the details of such proofs.,在实数系统中，为了周密性，此时有必要证明一些整数的定理。例如，两个整数的和、差和积仍是整数，但是商不一定是整数。然而还不能给出证明的细节。,Quotients of integers a/b (where b0) are called rational numbers. The set of rational numbers, denoted by Q, contains Z as a subset. The reader
13、 should realize that all the field axioms and the order axioms are satisfied by Q. For this reason, we say that the set of rational numbers is an ordered field. Real numbers that are not in Q are called irrational.,整数a与b的商被叫做有理数，有理数集用Q表示，Z是Q的子集。读者应该认识到Q满足所有的域公理和序公理。因此说有理数集是一个有序的域。不是有理数的实数被称为无理数。,The
14、 reader is undoubtedly familiar with the geometric interpretation of real numbers by means of points on a straight line. A point is selected to represent 0 and another, to the right of 0, to represent 1, as illustrated in Figure 2-4-1. This choice determines the scale.,4B Geometric interpretation of
15、 real numbers as points on a line,毫无疑问，读者都熟悉通过在直线上描点的方式表示实数的几何意义。如图2-4-1所示，选择一个点表示0，在0右边的另一个点表示1。这种做法决定了刻度。,If one adopts an appropriate set of axioms for Euclidean geometry, then each real number corresponds to exactly one point on this line and, conversely, each point on the line corresponds to on
16、e and only one real number.,如果采用欧式几何公理中一个恰当的集合，那么每一个实数刚好对应直线上的一个点，反之，直线上的每一个点也对应且只对应一个实数。,For this reason the line is often called the real line or the real axis, and it is customary to use the words real number and point interchangeably. Thus we often speak of the point x rather than the point corresponding to the real number.,为此直线通常被叫做实直线或者实轴，习惯上使用“实数”这个单词，而不是“点”。因此我们经常说点x不是指与实数对应
