48 Chapter 1 The Set of Real Numbers Example 3 2 5 6 5A, B, C, D, E6 51, 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, Plotting Points on the Real Number Line Plot the points on the real number line that represent the following real numbers. a. b. c. d. Solution: a. Because is negative, it lies three units to the left of 0. b. The fraction can be expressed as the mixed number , which lies halfway between 1 and 2 on the number line. c. The negative number lies unit to the left of on the number line. d. The fraction can be expressed as the mixed number , which lies unit to the right of 3 on the number line. 15 315 16 5 7 4 4.7 10 112 3 2 3 16 5 4.7 3 2 3 6 5 4 3 2 1 0 1 2 3 3 4 5 5 6 16 4.7 3 In mathematics, a well-defined collection of elements is called a set. “Welldefined” means the set is described in such a way that it is clear whether an element is in the set. The symbols are used to enclose the elements of the set. For example, the set represents the set of the first five letters of the alphabet. 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 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 Skill Practice 4. Plot the numbers on a real number line. { 1, , 2.5, 10 3 } 34 0 Classroom Example: p. 56, Exercise 18 TIP: The natural numbers are used for counting. For this reason, they are sometimes called the “counting numbers.” Instructor Note: Memory Device: Rational RATIO-nal can be written as a ratio of two integers. Answer 4. 2.5 1 34 10 3 54321 0 1 2 3 4 5
miller_introductory_algebra_3e_ch1_3
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