数学专业英语课后答案.doc 2
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2.1
数学、方程与比例
（
1
）数学
来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术
研究等活动。
Mathematics
comes
from
man’s
social
practice,
for
example,
industrial
and
agricultural
production,
commercial activities, military operations and
scientific and
technological
researches.
（
2
）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。
No
modern
scientific
and
technological
branches
could
be
regularly
developed
without the application of mathematics.
（
3
）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。
Notations
are
a
special
and
powerful
tool
of
mathematics
and
are
used
to
express
conceptions and
propositions very often.
（
4
）
17
世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常
< br>数。
Before
17th
century,
man
confined
himself
to
the
elementary
mathematics,
i.
e.
,
geometry,
trigonometry and algebra, in which only the
constants were considered.
（
5
）方程与算数的等式不同在于它含有可以参加运算的未知量
。
Equation
is
different
from
arithmetic
identity
in
that
it
contains
unknown
quantity
which can join
operations.
（
6
）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。
Equipment is called an
equation of condition in that it is true only for
certain values of
unknown quantities in
it.
（
7
）方程很有用，可以用它来解决许多实际应用问题。
Equations
are
of
very
great
use.
We
can
use
equations
in
many
mathematical
problems.
（
8
）
解方程时要进行一系列移项和同解变形，
最后求出它的根，
即未知量的值。
To solve the equation means to move and
change the terms about without making the
equation
untrue,
until
the
root
of
the
equation
is
obtained,
which
is
the
value
of
unknown
term.
2.3
集合论的基本概念
（
1
）由小于
10
且能被
3
整除的正整数组成的集是整数集的子集。
The set consisting of those
positive integers less than 10 which are divisible
by 3 is a
subset of the set of all
integers.
（
2
）如果方便，我们通过在括号中列举元素的办法来表示集。
When convenient, we shall
designate sets by displaying the elements in
braces.
（
3
）用符号
B
。
The relation
表示集的包含关系，也就是说，式子
A
B
表示
A
包含于
is referred to as set inclusion;
A
B means that A is
contained in B.
B
并不排除
B
A
的可能性。
（
4
）命题
A
The statement
A
B does not rule out the
possibility that B
A.
（
5
）基础
集可根据使用场合不同而改变。
The
underlying
set
may
vary
from
one
application
to
another
according
to
using
occasions.
（
6
）
为了避免逻辑上的困难，
< br>我们必须把元素
x
与仅含有元素
x
< br>的集
{x}
区别
开来。
To avoid logical
difficulties, we must distinguish between the
element x and the set {x}
whose only
element is x.
（
7
）图解法有助于将集合之间的关系形象化。
Diagrams often help using
visualize relationship between sets.
<
/p>
（
8
）定理的证明仅仅依赖于概念和已知
的结论，而不依赖于图形。
The
proofs of theorems rely only on the definitions of
the concepts and known result,
not on
the diagrams.
2.4
整数、有理数与实数
整数
（<
/p>
1
）严格说，这样描述整数是不完整的，因为我们并没有说明
“
依此类推
”
或<
/p>
“
反
复加
1”
的含义是什么。
Strictly
speaking,
this
description
of
the
positive
integers
is
not
entirely
complete
because we have not explained in detail
what we mean by the express
ions “and so
on”,
or “repeated addition of 1”.
（
2
）两个
整数的和、差或积是一个整数，但是两个整数的商未必是一个整数。
The sum, difference, or
product of two integers is an integer, but the
quotient of two
integers need not be an
integer.
（
3
）
这种用几何来表示实数的办法对于帮助我们更好地发现与理解实数的性质
是非常有价值
的。
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.
（
4
）几何经常为一些特定的定理提供证明思路（建议）
，而且，有时几何的论
证比纯分析的
（完全依赖于实数公理的）证明更清晰。
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).
（
5
）一个
由实数组成的集若满足如下条件则称为开区间（
open
interval
）
。
If
a set consisting of real numbers satisfies the
following conditions we call it an open
interval.
（
6
）实数
a
是
-a
的相反数，它们的绝对值相等，且当
a ≠ 0
时，其符号不同。
The real number a is the
negative number of
–
a and
their absolute values are equal.
When a
≠ 0, their notations are different.
（
7
）每个实数刚好对应着实轴上的一点，
反之，对实轴上的每一点，有且只有
一个实数与之
对应。
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.
（
8
）在几何上，实数之间的次序关系可以在数轴上清楚地表示出
来。
In geometry,
the ordering relation among the real numbers can
be expressed clearly in
real axis.
2.5
笛卡儿几何学的基本概念
（
1
）计算图形的面积是积分的一种重要应用。
The calculation
of figure area is the important application of the
integral.
（
2
）在
x-
轴上
O
点右边选定一个适当的点，并把它到
O
点的距离称为单位
长度。
On the x-axis a convenient
point is chosen to the right of O and its distance
from O is
called the unit distance.
（
3
）对
<
/p>
xy-
平面上的每一个点都指定了一个数对，称为它的坐标。
Each point in the
xy-plane is assigned a pair of numbers, called its
coordinates.
（
4<
/p>
）选取两条互相垂直的直线，其中一条是水平的，另一条是竖立的，把它们
的交点记作
O
，
称为原点。
Two perpendicular reference lines are
chosen, one horizontal, the other vertical. Their
point of intersection, denoted by O, is
called the origin.
（
5
）当我们用一对数（
a, b
）来表示平面的点时，商定要把横坐标写在第一个位
置上。
When we write a pair of
numbers such as (a, b) to represent a point, we
agree that the
abscissa or
x-coordinate, a, is written first.
（
6
）微积分与解析几何在它们的发展史上已
经互相融合在一起了。
Throughout
their
historical
development,
calculus
and
analytic
geometry
have
been
intimately intertwined.
（
7
）如果想拓展微积分的范围与应用，需要
进一步研究解析几何，而这种研究
需用到向量的
方法。
A
deeper study of analytic geometry is needed to
extend the scope and applications of
calculus, and this study will be
carried out using vector methods.
（
8
）
今后我们要对三
维解析几何做详细研究，
但目前只限于考虑平面解析几何。
We shall discuss three-
dimensional Cartesian geometry in more detail
later on; for the
present we confine
our attention to plane analytic geometry.
2.6
函数的概念与函数思想
（
1
）常用英语字母和希腊字母来表示函数。
Letters of the
English and Greek alphabets are often used to
denote functions.
（
2
）若
f
是一个给定的函数，
x
是定义域里的一个元素，那么记号
f
(x)
用来表
示由
f
确定的
对应于
x
的值。
If f is a given function and if x is an
object of its domain, the notation f(x) is used to
designate that object in the range
which is associated to x by the function f.
（
3
）该射
线将两个坐标轴的夹角分成两个相等的角。
The ray makes equal angles with the
coordinates axes.
（
< br>4
）可以用许多方式给出函数思想的图解说明。
The function idea may be
illustrated schematically in many ways.
（
5
）容易
证明，绝对值函数满足三角不等式。
It is easy to proof that the absolute-
value function satisfies the triangle inequality.
（
6
）对于
实数
x>0
，函数
g(x)
表示不超过
x
的素数的个数。
For a given real number
x>0, the function g(x) is defined by the number of
primes