数学专业英语习题答案
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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.
2.2
几何与三角
（
1
）许多专家都认为数学是学习其他科学技术的必备基础和
先决条
件。
Many experts recognize that mathematics
is the necessary foundation and
prerequisite of studying other science
technology.
（
2
）西方国家的专家认为几何起源于巴比伦和埃及人的土地测量技
术，
其实中国古代的数学
家对几何做了许多出色的研究。
The
western
experts
think
that
geometry
had
its
origin
in
the
measurements
by
the
Babylonians
and
Egyptians
of
their
lands.
Infect,
the
ancient
Chinese
mathematicians
made
much
remarkable
study
for
geometry.
（
3
）几何的学习使学生在思考问题时更周密和审慎，他们将不会盲
目接受任何结论。
In studying geometry, the student is
taught to think clearly and critically
and
he
is
led
away
from
the
practice
of
blind
acceptance
of
any
conclusions.
（
4
）数学培养学生的分析问题的能力，使他们能应用毅力、创造性<
/p>
和逻辑推理来解决问题。
Studying
mathematics
can
develop
the
students’
ability
to
analyze
problems and utilizing perseverance,
originality, and logical reasoning in
solving the problem.
（
5
）几何主要不是研究数，而是形，例如三角形，
平行四边形和圆，
虽然它也与数有关。
Geometry
mainly
studies
hot
numbers
but
figures
such
as
triangles,
parallelograms
and circles, though it is related with numbers.
（
6
）一个
立体（图形）有长、宽和高；面（曲面或平面）有长和宽，
但没有厚度；线（直线
或曲线）有长度，但既没有宽度，也没有厚
度；点只有位置，却没有大小。
A
solid (figure) has length, width and height. A
surface (curved surface or
plane
surface) has length and width, but no thickness. A
line (straight line
or
curved
line)
has
length,
but
no
width
and
thickness.
A
point
has
position, but no dimension.
（
7
）射线从某个点出发无限延伸；两
条从同一点出发的射线构成了
角。这两条射线称为这个
角的两边，当这两边位于同一直线上且方
向相反时，所得的角是平角。
A ray starts
from a point and extends infinitely far. Two rays
starting from
one point form an angle,
which are called two edges of the angle. When
two edges lie in the same line and have
opposite direction named plane
angle.
（
8
）平面
上的闭曲线当其中每一点到一个固定点的距离均相等时叫
做圆。这个固定点称为圆
心，经过圆心且其两个端点在圆周上的线
段
称为这个圆的直径，直径的一半叫做半径，这条
曲线的长度叫做
周长。
A
circle
is
a
closed
curve
lying
in
one
plane,
all
points
of
which
are
equidistant
from
a
fixed
point.
The
fixed
point
called
the
center.
A
diameter of a circle is a line segment
through the center of the circle with
endpoints on the circle. Half of the
diameter is called radius. The length
of the circle is called circumference.
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
表示集的包含关系，也就是说，式子
A
B
表示
A
包含于
B
。
The
relation
is
referred
to
as
set
inclusion;
A
B
means
that
A
is
contained in B.
（
4
）命题
A
B
并不排除
B
A
的可能性。
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，
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
“
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
expressions “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).
<
/p>
（
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。