部分翻译
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。 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 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 line
The 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 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不是指与实数对应的那个点。
This device for representing real numbers geometrically is a very worthwhile aid that helps us to discover and understand better certain properties of real numbers. However, the reader should realize that all properties of real numbers that are to be accepted as theorems must be deducible from the axioms without any references to geometry.
这种几何化的表示实数的方法是非常值得推崇的，它有助于帮助我们发现和理解实数的某些性质。然而，读者应该认识到，拟被采用作为定理的所有关于实数的性质都必须不借助于几何就能从公理推出。