2.4 整数、有理数与实数4－A Integers and rational numbersThere exist certain subsets of R which are distinguished 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.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 the 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 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 number s is cal led an i n ductiv e set if it has the following two properties:(a) The number 1 is in the set.(b) For every x in the set, the number x+1 is also in the set.For example, R is an inductive set. So 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 d enote t he s et o f a ll p ositive i ntegers. T hen P i s i tself a n i nductive set b ecause (a) i t contains 1, a nd (b) i t c ontains x+1 w henever i t c ontains x. Since the m embers o f P b elong t o e very inductive s et, w e r efer t o P a s t he s mallest i nductive 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 detailed 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，在介绍的第四部分将给出这种方法的详细论述。
归纳证明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 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。
Ina t horough t reatment o f t he r eal-number s ystem, i t w ouldb e n ecessary a t t his stage to prove certain theorems about integers. For example, the sum, difference, or product of two integersis an integer, but the quotient of two integers need 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 b≠0) are called rational numbers. The set of rational numbers, denoted by Q, contains Z as a subset. The reader 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。
4－B Geometric interpretation of real numbers as points on a lineThe 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.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 one 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 customaryto 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”。