英语翻译1.Within the realnumber system,numbers of various kinds

英语翻译
1.Within the real
number system,numbers of various kinds are identified and named.The numbers
1,2,3,4,… which are used in the counting process,are called natural
numbers.The natural numbers,together with–1,-2,-3,-4,…and zero,are called
integers.Since 1,2,3,4,…are greater than 0,they are also called positive integers;
-1,-2,-3,-4,…are less than 0,and for this reason are called negative integers.
A real number is said to be a rational number if it can be expressed as the
ratio of two integers,where the denominator is not zero.The integers are
included among the rational numbers since any integer can be expressed as the
ratio of the integer itself and one.A real number that cannot be expressed as
the ratio of two integers is said to be an irrational number.
2.Graph theory is a
rapidly growing branch of mathematics.The graphs discussed in this chapter are
not the same as the graphs of functions that we studied previously,but a
totally different kind.Like many of the important discoveries and new areas of
learning,graph theory also grew out of an interesting physical problem,the
so-called Konigsberg bridge problem.(this problem is discussed in Section 2)
The outstanding Swiss mathematician,Leonhard Euler (1707-1783) solved the
problem in 1736,thus laying the foundation for this branch of mathematics.
Accordingly,Euler is called the father of graph theory.
3.The start of
operations research took place in a military context in the United Kingdom
during World War Ⅱ,and it was quickly taken up under the name operations
research (OR) in the United States.After the war it evolved in connection with
industrial organization,and its many techniques allowed for expanding areas of
application in the United States,the United Kingdom,and in other industrial
countries.It is,however,not easy to give a precise definition of operationsresearch,There are three different representative definitions.

1. 在实数系中,不同种类的数字得以确定和命名.在数数时所用到的1,2,3,4, … 等数字被称为自然数.自然数以及-1, -2, -3, -4 和0都被称为整数.由于1,2,3,4,…都大于0,这些数字也被称为正整数；-1,-2,-3,-4,…都小于0,因此这些数字被称为负整数.如果一个实数能用两个整数之比来表示（分母不能为0）,那么这个实数就是有理数.整数包括在有理数之中,因为任何一个整数都可以表示为该整数自己与1之比.如果一个实数不能用两个整数之比来表示,则该实数就是无理数.
2. 图论是发展迅速的一个数学分支.本章所讨论的图形不同于我们之前所学过的函数图形,而是一种全新的图形.就如同很多学问的重要发现和新领域一样,图论也出自一个有趣的物理问题,即所谓的哥尼斯堡桥问题（该问题在2小节予以讨论）.杰出的瑞士数学家莱昂哈德*欧拉(1707-1783)于1736年解决了这个问题,并因此而为数学的这个分支奠定了基础.因此,欧拉被称为图论之父.
3. 运筹学出现在二战时期英国的战争背景下,并很快被美国以运筹学的名义开始进行研究.二战结束以后,运筹学与行业组织共同发展,并且运筹学的很多技术使得美国的应用领域得以拓宽.但是,要对运筹学给出一个准确的定义却并非易事.具有代表性的有三个不同定义.