24 Chapter 1 The Set of Real Numbers Skill Practice Plot the numbers on the real number line. 4. 0 51, 34 , 2.5, 10 3 6 TIP: The natural numbers are used for counting. For this reason, they are sometimes called the “counting numbers.” Answer 4. 2.5 1 34 10 3 54321 0 1 2 3 4 5 In mathematics, a well-defined collection of elements is called a set.“Well-defined” means the set is described in such a way that it is clear whether an element is in the set.The symbols 5 6 are used to enclose the elements of the set. For example, the set represents the set of the first five letters of the alphabet. 5A, B, C, D, E6 Several sets of numbers are used extensively in algebra and are subsets (or part) of the set of real numbers. Natural Numbers, Whole Numbers, and Integers The set of natural numbers is 51, 2, 3, . . .6 The set of whole numbers is 50, 1, 2, 3, . . .6 The set of integers is 5. . . 3, 2, 1, 0, 1, 2, 3, . . .6 Notice that the set of whole numbers includes the natural numbers.Therefore, every natural number is also a whole number.The set of integers includes the set of whole numbers.Therefore, every whole number is also an integer. Fractions are also among the numbers we use frequently.A number that can be written as a fraction whose numerator is an integer and whose denominator is a nonzero integer is called a rational number. Rational Numbers The set of rational numbers is the set of numbers that can be expressed in the form where both p and q are integers and q does not equal 0. p q p q, We also say that a rational number is a ratio of two integers, p and q, where q is not equal to zero. Identifying Rational Numbers Example 4 Show that the following numbers are rational numbers by finding an equivalent ratio of two integers. 2 3 a. b. c. 0.5 d. 12 0.6 Solution: a. The fraction is a rational number because it can be expressed as the ratio 2 of and 3. 2 3 b. The number is a rational number because it can be expressed as the 12 12 12 12 1 ratio of and 1, that is, . In this example, we see that an integer is also a rational number.
miller_beginning_intermediate_algebra_4e_ch1_3
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