5.2 The Integers 225 Another important set of numbers, the whole numbers , helps to answer the question, How many? Whole numbers {0, 1, 2, 3, 4, } = … Note that the set of whole numbers contains the number 0 but that the set of counting numbers does not. Although we use the number 0 daily and take it for granted, the number 0 as we know it was not used and accepted until the sixteenth century. If the temperature is ° 12 F and drops ° 20 F, the resulting temperature is − °8 F. This type of problem shows the need for negative numbers. The set of integers consists of the negative integers, 0, and the positive integers. Integers { , 4, 3, 2, 1, 0, 1, 2, 3, } = … − − − − … Negative integers Positive integers The term positive integers is yet another name for the natural numbers or counting numbers. An understanding of addition, subtraction, multiplication, and division of the integers is essential in understanding algebra (Chapter 6). To aid in our explanation of addition and subtraction of integers, we introduce the real number line (Fig. 5.5). The real number line contains the integers and all the other real numbers that are not integers. Some examples of real numbers that are not integers are indicated in Fig. 5.5, namely, − , , 2, 5 2 1 2 and π. We discuss real numbers that are not integers in the next two sections. 25 24 23 22 21 0 1 2 3 4 5 5 2 2 1 2 p 2 Figure 5.5 The arrows at the ends of the real number line indicate that the line continues indefinitely in both directions. Note that for any natural number, n, on the number line, the opposite of that number, −n, is also on the number line. This real number line was drawn horizontally, but it could just as well have been drawn vertically. In fact, in the next chapter, we show that the axes of a graph are the union of two number lines, one horizontal and the other vertical. The number line can be used to determine the greater (or lesser) of two integers. Two inequality symbols that we will use in this chapter are > and .< The symbol > is read “is greater than,” and the symbol < is read “is less than.” Expressions that contain an inequality symbol are called inequalities. On the number line, the numbers increase from left to right. The number 3 is greater than 2, written > 3 2. Observe that 3 is to the right of 2 on the number line. Similarly, we can see that > − 0 1 by observing that 0 is to the right of −1 on the number line. Instead of stating that 3 is greater than 2, we could state that 2 is less than 3, written < 2 3. Note that 2 is to the left of 3 on the number line. We can also see that − < 1 0 by observing that −1 is to the left of 0. The inequality symbol always points to the smaller of the two numbers when the inequality is true. Did You Know? Triangular Numbers O ne very simple reason that the ancient Greek mathematicians thought of mathematics in terms of whole numbers and their ratios (the rational numbers) was that they were still working with numbers that were represented by objects, such as squares and triangles, not number symbols. Still, that did not prevent them from drawing some conclusions about number theory. Consider two consecutive triangular numbers (introduced in Section 1.1), for instance, 3 and 6. You can see from the diagram that the sum of the two triangular numbers is equal to the square number 9, which is the square of a side of the larger triangle. Example 2 Writing an Inequality Insert either > or < in the shaded area between the paired numbers to make the statement correct. a) −7 8 b) −7 −8 c) −7 0 d) −7 4− Solution a) − < 7 8, since −7 is to the left of 8 on the number line. b) − > − 7 8, since −7 is to the right of −8 on the number line. c) − < 7 0, since −7 is to the left of 0 on the number line. d) − < − − 7 4, since 7 is to the left of 4− on the number line. 7 Now try Exercise 9
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