数学专业英语(吴炯圻)
-
New Words & Expressions:
algebra
代数学
geometrical
几何的
algebraic
代数的
identity
恒等式
arithmetic
算术
,
算术的
measure
测量，测度
axiom
公理
numerical
数值的
,
数字的
conception
概念，观点
operation
运算
constant
常数
postulate
公设
logical
deduction
逻辑推理
proposition
命题
division
除，除法
subtraction
减，减法
formula
公式
term
项，术语
trigonometry
三角学
variable
变化的，变量
2.1
数学、方程与比例
Mathematics, Equation and Ratio
4
Mathematics
c
omes
from
man’s
social
practice,
for
example,
industrial
and
agricultural
production, commercial activities,
military operations and scientific and
technological researches.
1
－
A
What is mathematics
数学来源于人类的社会实践，比如工农业生产，商业活动，军事行动和科学技术研究。
< br>
And
in
turn,
mathematics
serves
the
practice
and
plays
a
great
role
in
all
fields.
No
modern
scientific
and
technological
branches
could
be
regularly
developed
without
the
application
of
mathematics.
反过来，
数学服务于实践，
并在各个领域中起着非常重要的作用。没有应用数学，
任何一个
现在的科技的分支都不能正常发展。
5
From
the
early
need
of
man
came
the
concepts
of
numbers
and
forms.
Then,
geometry
developed
out
of
problems
of
measuring
land
,
and
trigonometry
came
from
problems
of
surveying. To deal with some more
complex practical problems, man established and
then solved
equation with unknown
numbers , thus algebra occurred.
很早的时候，
人类的需要产生了数和形的概念。
接
着，
测量土地问题形成了几何学，
测量问
题产生了三角学。
为了处理更复杂的实际问题，
人类建立和解
决了带未知数的方程，
从而产
生了代数学。
Before
17th
century,
man
confined
himself
to
the
elementary
mathematics,
i.e.
,
geometry,
trigonometry and algebra, in which only
the constants are considered.
17
世纪前，人类局限于只考虑常数的初等数学，即几何学，三角学和代数学。
6
The
rapid
development
of
industry
in
17th
century
promoted
the
progress
of
economics
and
technology
and
required
dealing
with
variable
quantities.
The
leap
from
constants
to
variable
quantities
brought
about
two
new
branches
of
mathematics----analytic
geometry
and
calculus,
which belong to
the higher mathematics.
17
世纪
工业的快速发展推动了经济技术的进步，从而遇到需要处理变量的问题。从常量到
变量的
跳跃产生了两个新的数学分支
-----
解析几何和微积分，他
们都属于高等数学。
Now there are many
branches in higher mathematics, among which are
mathematical analysis,
higher algebra,
differential equations, function theory and so on.
现在高等数学里面有很多分支，其中有数学分析，高等代数，微分方程，函数论等。
7
Mathematicians
study
conceptions
and
propositions,
Axioms,
postulates,
definitions
and
theorems are all
propositions. Notations are a special and powerful
tool of mathematics and are
used to
express conceptions and propositions very
often.
数学家研究的是概念和命题，
< br>公理，
公设，
定义和定理都是命题。
符号是数学中一个特殊而
有用的工具，常用于表达概念和命题。
Formulas
,figures
and
charts
are
full
of
different
symbols.
Some
of
the
best
known
symbols
of
mathematics
are
the
Arabic
numerals
1,2,3,4,5,6,7,8,9,0
and
the
signs
of
addition
“+”,
subtraction
“
-
” , multiplication “×”,
division “÷” and equality
“=”.
公式，图形和图表都是不同的符号„„
..
8
The conclusions in
mathematics are obtained mainly by logical
deductions and computation. For
a
long
period
of
the
history
of
mathematics,
the
centric
place
of
mathematics
methods
was
occupied by the logical
deductions.
数学结论主要由逻辑推理和计算得到
。
在数学发展历史的很长时间内，
逻辑推理一直占据着
数学方法的中心地位。
Now
,
since
electronic
computers
are
developed
promptly
and
used
widely,
the
role
of
computation becomes more
and more important. In our times, computation is
not only used to
deal with a lot of
information and
data, but
also to carry out
some work that merely
could be
done earlier by logical
deductions, for example, the proof of most of
geometrical theorems.
现在，
由于电子计算机的迅速发展和广泛使用，
计
算机的地位越来越重要。
现在计算机不仅
用于处理大量的信息和
数据，
还可以完成一些之前只能由逻辑推理来做的工作，
例如，
证明
大多数的几何定理。
9
回顾：
1.
如果没有运用数学，任何一个科学技术分支都不可能正常
的发展。
2.
符号在数学中起着非常重要的作用，它常用于表示概念和命题。
1
－
A
What is
mathematics
10
An equation
is a statement of the equality between two equal
numbers or
number
symbols.
1
－
B
Equation
等式是关于两个数或者数的符号相等的一种描述。
Equation are of two kinds----
identities and equations of condition.
An
arithmetic
or
an
algebraic
identity
is
an
equation.
In
such
an
equation
either
the
two
members are alike, or become alike on
the performance of the indicated operation.
等式有两种－恒等式和条件等式。
算术或者代数恒等式都是等式。
这种等式的两端要么一样，
要么经过执行指定的运算后变成一样。
11
An identity
involving letters is true for any set of numerical
values of the letters in it.
含有字母的恒等式对其中字母的任一组数值都成立。
An
equation
which
is
true
only
for
certain
values
of
a
letter
in
it,
or
for
certain
sets
of
related
values
of
two
or
more
of
its
letters,
is
an
equation
of
condition,
or
simply
an
equation.
Thus
3x-5=7 is true for x=4 only; and
2x-y=10 is true for x=6 and y=2 and for many other
pairs of values
for x and y.
一个等式若仅仅对其中一个字母的某些值成立，
或对其中两个或者多个字母的若干组相
关的
值成立，
则它是一个条件等式，
简
称方程。
因此
3x-5=7
仅当
x=4
时成立，
而
2x-y=0
，
当
x=6,y=2
时成立，且对
x,
y
的其他许多对值也成立。
12
A root of an equation is any number or
number symbol which satisfies the equation.
To obtain the root or roots
of an equation is called solving an equation.
方程的根是满足方程的任意数或者数的符号。求方程根的过程被称为解方程。
There are various kinds of
equations. They are linear equation, quadratic
equation, etc.
方程有很多种，例如：线性方程，二次方程等。
13
To
solve
an
equation
means
to
find
the
value
of
the
unknown
term.
To
do
this
,
we
must,
of
course, change the terms
about until the unknown term stands alone on one
side of the equation,
thus making it
equal to
something on the
other side. We then obtain the value of
the unknown
and the answer to the
question.
解方程意味着求未知项的值，
为了求未知项的值，
当然必须移项，
直到未知项单
独在方程的
一边，令其等于方程的另一边，从而求得未知项的值，解决了问题。
To
solve
the
equation,
therefore,
means
to
move
and
change
the
terms
about
without
making
the equation untrue,
until only the unknown quantity is left on one
side ,no matter which side.
因此解方程意味着进行一系列的移项和同解变形，
直到未知量被单独留在方程的一边，<
/p>
无论
那一边。
14
Equations are of very
great use. We can use equations in many
mathematical problems. We may
notice
that
almost
every
problem
gives
us
one
or
more
statements
that
something
is
equal
to
something, this gives us equations,
with which we may work if we need to.
<
/p>
方程作用很大，
可以用方程解决很多数学问题。
< br>注意到几乎每一个问题都给出一个或多个关
于一个事情与另一个事情相等的陈述，
这就给出了方程，利用该方程，如果我们需要的话，
可以解方程。
New Words & Expressions:
numerical
数值的，数的
position
位置，状态
cube
n.
立方体
sphere
n.
球
cylinder n.
柱体
cone
圆锥
geometrical
几何的
triangle
三角形
surface
面，曲面
pyramid
菱形
plane
平面
solid
立体，立体的
line
segment
直线段
ray
射线
curve
曲线，弯曲
straight
line
直线
broken line
折线
equidistant
等距离的
2.2
几何与三角
Geometry
and Trigonology
1
New Words
& Expressions:
side
边
angle
角
diameter
直径
circle
圆周，圆
arc
弧
major arc
优弧
right angle
直角
adjacent side
邻边
ra
dius
（
radii
）半径
endpoint
端点
semicircle
半圆
minor arc
劣弧
acute angle
锐角
hypotenuse
斜边
chord
弦
circumference
周长
2
Many leading institutions
of higher learning have recognized that positive
benefits can be gained
by all who study
this branch of mathematics.
2
－
A
Why study geometry?
许多居于领导地位的学术机构承认，所有学习这个数学分支的人都将得到确实的受益。
< br>
This
is
evident
from
the
fact
that
they
require
study
of
geometry
as
a
prerequisite
to
matriculation in those
schools.
许多学校把几何的学习作为入学考试的先决
条件，从这一点上可以证明。
3
Geometry had its origin long ago in the
measurement by the Babylonians and Egyptians
of their
lands inundated by
the floods of the Nile River.
< br>几何学起源于很久以前巴比伦人和埃及人测量他们被尼罗河洪水淹没的土地。
The
greek
word
geometry
is
derived
from
geo,
meaning
“earth”
and
metron,
meaning
“measure”
.
希腊语几何来源于
geo
，意思是”土地“，和
metron
意思是”
测量“。
4
As
early
as
2000
B.C.
we
find
the
land
surveyors
of
these
people
re-
establishing
vanishing
landmarks and boundaries by utilizing
the truths of geometry .
公元前
2000
年之前，我们发现这些民族的土地测量者利用几何知识
重新确定消失了的土地
标志和边界。
One of the most important objectives
derived from a study of geometry is making the
student be
more critical in
his listening, reading and thinking.
In studying geometry he is led away
from the
practice of blind acceptance
of statements and ideas and is taught to think
clearly and critically
before forming
conclusions.
几何的学习使学生在思考问题时更
周密、审慎，他们将不会盲目接受任何结论
.
5
A solid is a three-dimensional figure.
Common examples of solids are cube, sphere,
cylinder, cone
and pyramid.
2
－
B
Some geometrical terms
立体是一个三维图形，立体常见的例子是立方体，球体，柱体，圆锥和棱锥。
A cube
has
six
faces which are smooth and
flat. These
faces are called plane
surfaces or
simply
planes.
<
/p>
立方体有
6
个面，都是光滑的和平的，这
些面被称为平面曲面或者简称为平面。
6
A
plane
surface
has
two
dimensions,
length
and
width.
The
surface
of
a
blackboard
or
of
a
tabletop is an example of
a plane surface.
平面曲面是二维的，有长度和宽度，黑板和桌子上
面的面都是平面曲面的例子。
A
circle
is
a
closed
curve
lying
in
one
plane,
all
points
of
which
are
equidistant
from
a
fixed
point
called the center.
平面上的闭曲线当其中每点到一个固定点的距
离均相当时叫做圆。固定点称为圆心。
7
A line segment drawn from the center of
the circle to a point on the circle is a radius of
the circle.
The circumference is the
length of a circle.
经过圆心且其两个端点在圆周上的线段称为
这个园的直径，这条曲线的长度叫做周长。
One of
the
most important applications of
trigonometry
is the solution
of triangles. Let us
now
take up the solution to right
triangles.
三角形最重要的应用之一是解三角形
，现在我们来解直角三角形。
8
A
triangle is composed of six parts three sides and
three angles. To solve a triangle is to find the
parts not given.
一个三角形由
6
个部分组成，三条边和三只角。解一个三角形
就是要求出未知的部分。
A triangle may
be solved if three parts (at least one of these is
a side ) are given. A right triangle
has one angle, the right angle, always
given. Thus a right triangle can be solved when
two sides,
or one side and an acute
angle, are given.
如果三角形的三个部分
（其中至少有一个为边）
为已知，
则此三角形就可以解出。
直角三角
形的一只角，即直角，总是已知的。因此，如果它的两边，
或一边和一锐角为已知，则此直
角三角形可解。
New Words & Expressions:
brace
大括号
roster
名册
consequence
结论，推论
roster notation
枚举法
designate
标记，指定
rule out
排除，否决
diagram
图形，图解
subset
子集
distinct
互不相同的
the underlying
set
基础集
distinguish
区别，辨别
universal set
全集
divisible
可被除尽的
validity
有效性
dummy
哑的，哑变量
visual
可视的
even integer
偶数
visualize
可视化
irrelevant
无关紧要的
void set(empty
set)
空集
2.3
集合论的基本概念
Basic
Concepts of the Theory of Sets
1
The concept of a set
has
been utilized so extensively throughout modern
mathematics that an
understanding
of
it
is
necessary
for
all
college
students.
Sets
are
a
means
by
which
mathematicians talk of collections of
things in an abstract way.
3
－
A
Notations for denoting sets
集合论的概念已经被广泛使用，
遍及现代数学，
因此对大学生来说，
理解它的概念是必要的。
集合是数
学家们用抽象的方式来表述一些事物的集体的工具。
Sets
usually are denoted by capital letters; elements
are designated by lower-case letters.
集合通常用大写字母表示，元素用小写字母表示。
2
We use the special
notation
to
mean that “x is an element of S” or “x belongs to
S”.
If x does not belong to S, we write
.
我们用专用记号来表示
x
是
S
的元素或者
x
属于
S
。如果
x
不属于
S
，我们记为。
When convenient, we shall designate
sets by displaying the elements in braces; for
example, the
set of positive even
integers less than 10 is displayed as {2,4,6,8}
whereas the
set of all positive
even integers is displayed as
{2,4,6,…}, the three dots taking the place of “and
so on.”
如果方便，我们可以用在大括号中列出元素的
方式来表示集合。例如，小于
10
的正偶数的
< br>集合表示为
{2,4,6,8}
，而所有正偶数的集合表
示为
{2,4,6,
„
},
三个圆点表示“等等”
。
3
The
dots
are
used
only
when
the
meaning
of
“and
so
on”
is
clear.
The
method
of
listing
the
members of a set within braces is
sometimes referred to as the roster
notation.
只有当省略的内容清楚时才能使用圆点。
在大括号中列出集合元素的方法有时被归结为枚举
法。
The first basic concept that
relates one set to another is equality of
sets:
联系一个集合与另一个集合的第一个基本概念是集合相等。
4
DEFINITION OF SET EQUALITY
Two sets A and B are said to be equal (or
identical) if they consist of
exactly
the same elements, in which case we write A=B. If
one of the
sets contains an element
not in the other, we say the sets
unequal and we write A
≠
B.
集合相等的定义如果两个集合
A
和
B
确切包含同样的元素
,
则称二者相等，此时记为
A=B
。
如
果一个集合包含了另一个集合以外的元素，则称二者不等，记为
A
≠
B
。
5
EXAMPLE 1. According to this
definition, the two sets {2,4,6,8} and {2,8,6,4}
are equal since they
both consist of
the four integers 2,4,6 and 8. Thus, when we use
the roster notation to describe a
set,
the order in which the elements appear is
irrelevant.
根据这个定义，两个集合
{2,4,
6,8}
和
{2,8,6,4}
是相等
的，因为他们都包含了四个整数
2,4,6,8
。
因此，当我们用枚举法来描述集合的时候，元素出现的次序是无关紧要的。
6
EXAMPLE 2. The sets
{2,4,6,8} and {2,2,4,4,6,8} are equal even though,
in the second set, each of
the elements
2 and 4 is listed twice. Both sets contain the
four elements 2,4,6,8 and no others;
therefore, the definition requires that
we call these sets equal.
例
2.
集合
{2,4,6,8}
和
{2,2,4,4,6,8}
也是相等的，虽然在第二个集合中，
2
和
4
都出现两次。
两个集合都包含了四个元素
2,4,6,8
，没有其他元素，因此，依据定义这两个集合相等。
This example shows that we do not
insist that the objects listed in the roster
notation be distinct.
A similar example
is the
set of letters in the word
Mississippi, which
is equal to the set
{M,i,s,p},
consisting of the four
distinct letters M,i,s, and p.
这个例子表明我们
没有强调在枚举法中所列出的元素要互不相同。
一个相似的例子是，
在单
词
Mississippi
中
字母的集合等价于集合
{M,i,s,p},
其中包含了四个
互不相同的字母
M,i,s,
和
p.
7
From a given set S we may
form new sets, called subsets
of S. For
example, the set consisting of
those
positive integers less than 10 which are divisible
by 4 (the set {4,8}) is a subset of the set of
all even integers less than 10. In
general, we have the following definition.
3
－
B
Subsets
一个给定的集合<
/p>
S
可以产生新的集合，
这些集合叫做
S
的子集。
例如，
由可被
4
除尽的并且
小于
10
的正整数所组成的集合是小于
10
的所有偶数所组成集合的子集。
一般来说，
我们有<
/p>
如下定义。
8
In all our applications of set theory,
we have a fixed set S given in advance, and we are
concerned
only with subsets of this
given set. The underlying set S may vary from one
application to another;
it will be
referred to as the universal set of each
particular discourse.
（
35
页第二段）
当我们应用集合论时，
总是事先给定一个固定的集合
S
，
而我们只关心这个给定集合的子集。
基础集可以随意改变，可以在每一段特定
的论述中表示全集。
9
It is
possible for a set to contain no elements
whatever. This set is called the empty set or the
void
set, and will be denoted by the
symbol
. We will consider
to be a subset
of every set.
（
35
页
第三段）
一个集合中不包含任何元素，
这种情况是有可能的。
这个集合被叫做空集，
用符号表示。<
/p>
空
集是任何集合的子集。
Some people find it helpful to think of
a set as analogous to a container (such as a bag
or a box)
containing certain objects,
its elements. The empty set is then analogous to
an empty container.
一些人认为这样的比喻是有益的，
集合类似于容器
（如背包和盒子）
装有某些东西
那样，
包
含它的元素。
10
To avoid logical
difficulties, we must distinguish
between the elements x and the set
{x} whose
only element is x.
In particular, the empty set
is not the same as the set
. <
/p>
（
35
页第
四段
）
为了避免遇到逻辑困难，我们必须区分元素
x
和集合
{x}
，集合
{x}
中的元素是
x
。特别要注
意的是空集和集合是不同的。
In fact, the empty set contains no
elements, whereas the set
has
one element. Sets
consisting of exactly one element are
sometimes called one-element sets.
事实上，
空集不含有任何元素，
而有一个元素。
由一个元素构成的集合有时被称为单元素集。
11
Diagrams often help us visualize
relations between sets. For example, we may think
of a set S as a
region in the plane and
each of its elements as a point. Subsets of
S may then be thought of the
collections of points within S. For
example, in Figure 2-3-1 the shaded portion is a
subset of A and
also a subset of B.
（
35
页第五段）
图解有助于我们将集合之间的关系形象化。
例如，
可以把集合
S
看作平面内的一个区域，
其
中的每一个元素即是一个点。那么
S
的子集就是
S
内某些点的全体。例如，在图
2-3-1
中阴
影部分是
A
的子集，同时也是
B
的子集。
12
Visual aids of
this type, called Venn diagrams, are useful for
testing the validity of theorems in set
theory or for suggesting methods to
prove them. Of course, the proofs themselves must
rely only
on the definitions of the
concepts and not on the diagrams.
这种图解方
法，
叫做文氏图，
在集合论中常用于检验定理的有效性或者为证
明定理提供一些
潜在的方法。当然证明本身必须依赖于概念的定义而不是图解。
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 field axiom
域公理
irrational
无理的
2.4
整数、有理数与实数
Integers, Rational Numbers and Real
Numbers
1
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
2
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.
4
－
A
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。
3
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<
/p>
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。
4
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。
5
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。
6
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 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。
7
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.
Since
the
members
of
P
belong
to
every
inductive set, we
refer to P as the smallest inductive set.
< br>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。
8
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，
数学家称之为归纳证明，
在介绍的第四部分将给
出这种方法的详细论述。
9
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。
10
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。
11
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。
12
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.
4
－
B
Geometric interpretation of
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。
13
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.
< br>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。
14
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<
/p>
x
不是指与实数对应的那个点。
15
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。
16
This
does
not
mean
that
one
should
not
make
use
of
geometry
in
studying
properties
of
real
numbers.
On
the
contrary,
the
geometry
often
suggests
the
method
of
proof
of
a
particular
theorem, and
sometimes a geometric argument is more
illuminating than a purely analytic proof
(one depending entirely on the axioms
for the real numbers).
这并不意
味着研究实数的性质时不会应用到几何。
相反，
几何经常会为证
明一些定理提供思
路，有时几何讨论比纯分析式的证明更清楚。
17
In
this
book,
geometric
arguments
are
used
to
a
large
extent
to
help
motivate
or
clarity
a
particular
discuss.
Nevertheless,
the
proofs
of
all
the
important
theorems
are
presented
in
analytic form.
在本书中，
< br>几何在很大程度上被用于激发或者阐明一些特殊的讨论。
不过，
< br>所有重要定理的
证明必须以分析的形式给出。
3
New Words &
Expressions:
polygonal
多边形的
circular
regions
圆域
parabolic
抛物线的
coordinate axis
坐标轴
the unit
distance
单位长度
the origin
坐标原点
horizontal
水平的
coordinate
system
坐标系
perpendicular
互相垂直的，垂线
vertical
竖直的
an ordered pair
一个有序对
abscissa
横坐标
quadrant
象限
ordinate
纵坐标
intersect
相交
the theorem of Pythagoras
勾股
定理
2.5 basic
concepts of Cartesian geometry
4
As mentioned earlier, one of the
applications of the integral is the calculation of
area. Ordinarily ,
we do not talk about
area by itself ,instead, we talk about the area of
something . This means
that we have
certain objects (polygonal regions, circular
regions, parabolic segments etc.) whose
areas we wish to measure. If we hope to
arrive at a treatment of area that will enable us
to deal
with
many
different
kinds
of
objects,
we
must
first
find
an
effective
way
to
describe
these
objects.
就像前面提到的
，
积分的一个应用就是面积的计算，
通常我们不讨论面积本身，
相反，
是讨
论某事物的面积。
这意味着我们有些想测量的面积的对象
（多边形区域，
< br>圆域，
抛物线弓形
等）
，如果我
们希望获得面积的计算方法以便能够用它来处理各种不同类型的图形，我们就
必须首先找
出表述这些对象的有效方法。
5-A the
coordinate system of Cartesian geometry
5
The most primitive way of
doing this is by drawing figures, as was done by
the ancient Greeks. A
much
better
way
was
suggested
by
Rene
Descartes,
who
introduced
the
subject
of
analytic
geometry
(also
known
as
Cartesian
geometry).
Descartes’
idea
was
to
re
present
geometric
points by numbers.
The procedure for points in a plane is this
：
描述对象最基本的方法是画图，
就像古希腊人做的那样。
R
笛卡儿提出了一种比较好的方法，
并建立了解
析几何（也称为笛卡儿几何）这门学科。笛卡儿的思想就是用数来表示几何点，
在平面上
找点的过程如下：
5-A the coordinate
system of Cartesian geometry
6
Two perpendicular reference lines
(called coordinate axes) are chosen, one
horizontal (called the
“x
-
axis”), the
other vertical (the
“y
-
axis”). Their point
of intersection denoted by
O, is called the
origin.
On
the
x-axis
a
convenient
point
is
chosen
to
the
right
of
O
and
its
distance
from
O
is
called the unit distance.
Vertical distances along the Y-axis are usually
measured with the same
unit
distance
,although
sometimes
it
is
convenient
to
use
a
different
scale
on
the
y
-axis.
Now
each point in the plane (sometimes
called the xy-plane)
is assigned a
pair of
numbers, called its
coordinates. These numbers tell us how
to locate the points.
选两条互相垂直的参考线
（称为坐标轴）
，
一条水平
（称为
x
轴
）
，
另一条竖直
（称为
y
轴）
。
他们的交点记为
O,
称为原点。在
x
轴上，原点的右侧选择一个合适的点，该点与原点之间
的距离称为单位长度，
沿着
y
轴的垂直距离通常用同样的单位长度来
测量，
虽然有时候采用
不同的尺度比较方便。
< br>现在平面上的每一个点都分配了一对数，
称为坐标。
这些
数告诉我们
如何定义一个点。
5-A
the coordinate system of Cartesian geometry
7
A geometric figure, such
as a curve in the
plane , is a
collection of
points satisfying one or
more
special conditions. By translating
these conditions into expressions,, involving the
coordinates x
and y, we obtain one or
more equations which characterize the
figure in question , for example,
consider
a
circle
of
radius
r
with
its
center
at
the
origin,
as
show
in
figure
2-5-2.
let
P
be
an
arbitrary point on this circle, and
suppose P has coordinates (x, y).
一个几何图形是满足一个或多个特殊条件的点集，
比如平面上的曲线。<
/p>
通过把这些条件转化
成含有坐标
x
和
y
的表达式，
我们
就得到了一个或多个能刻画该图形特征的方程。
例如，
如
图
2-5-2
所示的中心在原点，半径为
r
的圆，令
P
是原
上任意一点，假设
P
的坐标为
(x,
y).
5-B
Geometric figure
9
New Words
& Expressions:
prime
素数
displacement
位移
edge
棱，边
domain
定义域，区域
real variable
实变量
schematic representation
图解表示
tabulation
作表，表
mass
质量，许多，群众
absolute-value function
绝对值函数
2.6 function concept and function idea
10
Seldom
has
a
single
concept
played
so
important
a
role
in
mathematics
as
has
the
concept
of
function. It is desirable
to know how the concept has developed.
在数学中，
很少有个概念象函数的概
念那样，
起那么重要的作用的。
因此，
需要知道这个概
念是如何发展起来的。
6-C The concept of function
11
This concept, like many
others ,originates in physics. The physical
quantities were the forerunners
of
mathematical variables. And relation among them
was called a
function relation in the
later
16th century.
这个概念像许多其他概念一样，
起源
于物理学。
物理的量是数学的变量的先驱，
他们之间的
关系在
16
世纪后期称为函数关系。
6-C The concept of function
12
For example , the formula
s=16t2 for the number of feet s a body falls in
any number of seconds t
is
a
function
relation
between
s
and
t.
it
describes
the
way
s
varies
with
t.
the
study
of
such
relations led people in
the 18th century to think of a function relation
as nothing but a formula.
例如，代表一物体在若干秒
t
中下落若
干英尺
s
的公式
s=16t2
就是
s
和
t
之间的函数关系。
它描述了
s
< br>随
t
变化的公式，对这种关系的研究导致了
18
世纪的人们认为函数关系只不过
是一个公式罢
了。
6-C The concept of
function
13
Only after the
rise of modern analysis in the early 19th century
could the concept of function be
extended. In the extended sense , a
function may be defined as follows:
if a variable y depends
on
another variable x in such a way that to each
value of x corresponds a definite value of y, then
y is a function of x. this definition
serves many a practical purpose even
today.
只有在
19
世纪初期现代分析出现以后，函数的概念才得以扩大。在扩大的意义上讲
，函数
可定义如下：
如果一变量
y
随着另一个变量
x
而变换，
即
x
的每一个值都和
y
的一定值相对
应，那么，
y
< br>就是
x
的函数。这个定义甚至在今天还适用于许多实际的
用途。
6-C The concept of
function
14
Not specified by
this definition is the manner of setting up the
correspondence. It may be done
by
a
formula
as
the
18th
century
mathematics
presumed
but
it
can
equally
well
be
done
by
a
tabulation such as a statistical chart,
or by some other form of description.
至于如何建立这种对应关系，
这个定义并未详细规定。
可以如
18
世纪的数学所假定的那样，
用公式来建立，但同样也可以用统计表那样的表格或用某种其他的描述方式来建立。
6-C The concept of function
15
A
typical
example
is
the
room
temperature,
which
obviously
is
a
function
of
time.
But
this
function admits
of
no formula
representation, although it can be recorded
in a tabular form or
traced
but graphically by an automatic device.
典型的例子是室温，
这显然是一个时
间的函数。
但是这个函数不能用公式来代表，
但可以用
表格的形式来记录或者用一种自动装置以图标形式来追踪。
6-C The concept of function
16
The modern definition of
a function y of x is simply a mapping from a space
X to another space Y
.
a
mapping is defined when every point x of X has a
definition image y, a point of Y
. the
mapping
concept is close to intuition,
and therefore desirable to serve as a basis of the
function concept,
Moreover,
as
the
space
concept
is
incorporated
in
this
modern
definition,
its
generality
contributes
much
to the generality of the function
concept.
现代给
x
的一个函数
y
所下的定义只是从一个空间
X
到另一个空间
Y
的映射。
当
X
空间的每
一个点
x
有一个确定的像点
y
，即
Y
空间的一点，那么，映射就确定
了。这个映射概念接近
于直观，
因此，
很可能作为函数概念的一个基础。
此外，
由于这个现代的定义中
体现了空间
的概念，所以，它的普遍性对函数概念的普遍性有很大的贡献。
New Words &
Expressions:
alphabet
字母表
prime
素数，质数
displacement
位移
proportional
成比例的
domain
定义域
the
real-valued function
实值函数
edge
棱，
边
spring constant
弹性系数
graph
图，
图形
limit
极限
stretch
拉伸
volume
体积，
容积，卷
2.6
函数的概念与函数思想
Function concept and function
idea
1
New Words
& Expressions(
二
)P59:
admit
准许
mapping
映射
forerunner
先行者
presume
假定
incorporate
并入，结合
trace
追踪
2
Various fields of human
have to do with relationships that exist between
one collection of objects
and another.
6-A Informal description of
functions
各行各业的人们都必须处理一类事物与另一类事物之间存在的关系
。
Graphs, charts, curves,
tables, formulas, and Gallup polls are familiar to
everyone who reads the
newspapers.
几乎每个人都熟悉图形，图表，曲线，公式和盖洛普民意测验。
3
These
are
merely
devices
for
describing
special
relations
in
a
quantitative
fashion.
Mathematicians
refer to certain types of these relations as
functions.
这些只是以定量的方式描述特定关系
的方法。数学家将这些关系中的某些类型视作函数。
In
this
section,
we
give
an
informal
description
of
the
function
concept.
A
formal
definition
is
given in Section 3.