[MUSIC] The essential elements of every mathematical model are variables and equations. It is obvious that each economical variable takes numerical value. This is why it is important to know the main characteristics of the number system. Let's start with the real numbers. Whole numbers such as 1, 2, 3 are called positive integers. These are the numbers most frequently used in counting y, the negative counterparts- 1,- 2,- 3, are called negative integers. The number 0 is neither positive nor negative. This is why we can consider it unique. All the positive and negative integers and the number zero can be grouped into a single category, referring to them collectively as the set of all integers. Along from the integers, the number system refers to fractions such as 1/3, 1/4, 4/3. In order to understand the position of a fraction into the number system, let's think about finding a fraction in a ruler. The fraction will fill in the gaps between the integers. Also, there exist negative fractions such as -3/5 and -5/2, and so forth. Together, this make up the set of all fractions. What characterize all fractional numbers is that each of them can be expressed as the ratio of two integers. Also, it is important to know that if a number can be expressed as a ratio of two integers, it's called a rational number. Notice that since any integer n can be considered as a ratio of n/1, therefore, also integers are rational. The set of all integers, in the set of all fractions together from the set of all irrational numbers. Moreover, an additional characteristic of a rational number is that it can be expressed as either a terminating decimal, like 1/4 equal to 0.25, or repeating decimal like 1/3 equal to 0.3333, where some number or series of numbers to the right of the decimal points is repeated indefinitely. The opposite of the concept of a rational number is the one of irrational numbers which are numbers that can affirm a ratio of two integers numbers. In particular, an irrational number cannot be neither terminating decimal, nor a repeating decimal. For example, the square root of 2 is equal to 1.4142. We can notice that it is a no repeating and non-terminating decimal. The same must be said for the special constant pi which is equal to 3.1415, which represent the ratio of the circumference of any circle to its diameter. And again, is a no repeating, non-terminating decimal like all irrational numbers. If we think of an irrational number placed on a ruler, the number will fill a gap between two rational numbers. Does the whole ruler represent a continuum which is the set of all real numbers? That set is indicated with the symbol R. In the picture that you find in your slides, I listed all the number sets. If we read from the bottom to the top, this figure represents a summary of the structure of the real number system. In our lectures, we use and we are interested only in the real numbers. However, we should know that they are not the only numbers that we use in mathematics. In fact, along with the real number system, we have the imaginary numbers, which refer to the square roots of negative numbers. [MUSIC]
