“ ����������������������������������������������������������������������������������������������������������������������������������������������������������������������������” ���������������������������� �� ���������������� �������� The Set of Real Numbers ��������������������lays the foundation of algebra. In this chapter, we will explore the set of real numbers, operations with real numbers, and numerical and al gebraic expressions. The skills learned in this chapter will be applied in every chapter of this text. These are the skills that will enable us to solve equations, simplify expressions, and solve application problems. How would you classify the following numbers? 12 hr: the number of semester hours to be a full-time student -70°F: the coldest recorded temperature in the continental United States (1971) 7 1 2 ft: the height of one of the tallest NBA players, Yao Ming π: the distance around a circle divided by its diameter In this section, we will learn the different types of real numbers and will classify numbers such as the ones shown. The Set of Real Numbers Numbers are used to represent so many things in our lives—the amount of money in a bank account, hours in a day, temperature, weight, height, number of credit hours, GPA, salaries, measurements, and so on. In addition, computers store all of their data as numbers. The world as we know it would basically cease to exist without numbers. So, it seems logical to begin with a review of the different types of numbers. We first need to define a few terms related to sets. A set is a collection of objects. Each object in a set is called a member or an element. A set is written in braces, { }, and is usually denoted with a capital letter. Sets can be either finite or infinite. A finite set has a specific number of elements. An example of a finite set is A = {1, 2, 3, 4, 5}. An infinite set has infinitely many elements. An example of an infinite set is B = {1, 3, 5, 7, 9, . . . }. The three dots at the end are called an ellipsis and indicate that the set continues indefinitely. In sets A and B, the elements of the sets are listed explicitly. This method of listing each element of the set is called the roster method. Another method, the set-builder notation, states the conditions the elements must satisfy to be included in the set. Setbuilder notation is written in the form: 5x 0 condition x must satisfy6 This is read as “the set of x such that ______.” In the blank, we insert the condition that x must satisfy. The letter x is a variable and represents some unknown number. An example of set-builder notation is C = 5x 0 x is a positive odd number6. The given condition tells us that x can be 1, 3, 5, 7, 9, . . . . Note that set C is the same as set B. ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Classify a number as a natural number, whole number, integer, rational number, or irrational number. 2. Graph real numbers on a real number line. 3. Compare the value of two real numbers. 4. Find the opposite of a real number. 5. Find the absolute value of a real number. 6. Troubleshoot common errors. Objective 1 ▶ Classify a number as a natural number, whole number, integer, rational number, or irrational number. 2
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