1. In the real number system, different kinds of numbers are determined and named. Numbers used for counting such as 1, 2, 3, 4, … are called natural numbers. Natural numbers and-1, -2, -3, -4, 0 are all called integers. Since 1, 2, 3, 4, … are all greater than 0, these numbers are also called positive integers; -1,-2,-3,-4, ... are all less than 0, so these numbers are called negative integers. If a real number can be expressed by the ratio of two integers (the denominator cannot be 0), then it is a rational number. Integers are included in rational numbers because any integer can be expressed as the ratio of itself to 1. If a real number cannot be expressed by the ratio of two integers, it is irrational.

2. Graph theory is a rapidly developing branch of mathematics. The graph discussed in this chapter is different from the function graph we have learned before, but a brand-new graph. Like many important discoveries and new learning fields, graph theory also comes from an interesting physical problem, the so-called Konigsberg Bridge problem (discussed in Section 2). The outstanding Swiss mathematician leonhard euler (1707- 1783) solved this problem in 1736, thus laying the foundation for this branch of mathematics. Therefore, Euler is called the father of graph theory.

3. Operations research appeared in the background of the British war in World War II, and was quickly studied by the United States in the name of operations research. After World War II, operational research and industrial organizations developed together, and many technologies of operational research broadened the application fields of the United States. However, it is not easy to give an accurate definition of operational research. There are three representative definitions.