数学专业英语课后答案.doc 2

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.

2.3 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 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.

（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

（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 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.