数学专业英语课后习题2.4

1.

1）integer，rational number，irrational number，real number，negative number，the negative，real line，real axis，scale，to the left/right of 2）sum，difference，product，quotient，power，inequality

3）axiom，the field axiom，the order axiom

4）ordered，entirely complete，Euclidean，appropriate，distinguished，illuminating

5）can be deduced formula，formula interchangeably，using a set of formulas，corresponding to an object，proof by induction，the two set to be distinguished

2.

1）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）The sum, difference, or product of two integers is an integer, but the quotient of two integers need not be an integer.

3）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）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）If a set consisting of real numbers satisfies the following conditions we call it an open interval.

6）The real number a is the negative number of –a and their absolute values are equal. When a ≠ 0, their notations are different.

7）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）In geometry, the ordering relation among the real numbers can be expressed clearly in real axis.

3.

1）一个常见的错误是认为x 是一个负数。但 x 可以为正数、 0 或负数，这取决于x 的值

2）我们在最后一节中介绍的每个属性是唯一的操作，例如，ab =ba和 0 + a= a，我们现在考虑一个连接加法和乘法的属性。它被称为分配律或者乘法对加法的分配，用下面的公式表所示： a（b + c) = ab + ac；（b + c) =ba + ca。

3）考虑小数按自然数顺序写出为：.123456789101112131415…由于自然数数字不会中止或重复，这是一个无限不循环小数，不能转换为两个整数的比的小数叫做无理数，这组数字是指符号H，和H={×：x是无限不循环小数}