2021年1月23日发(作者：mlm)
2.4
整数、有理数与实数
4
－
A
Integers and rational numbers

There 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)
(b)
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 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。
In
a
t
horough
t
reatment
o
f
t
he
r
eal-number
s
ystem,
i
t
w
ould
b
e
n
ecessary
a
t
t
his 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。