4 Chapter 1 Real Numbers and Algebraic Expressions We can use the calculator to approximate square roots to determine if the number is rational or irrational. If the value on the calculator is a decimal that repeats or terminates, then the number is rational. If the value on the calculator is a decimal that is nonterminating and nonrepeating, then the number is irrational. Some irrational numbers are 13 = 1.7320508075 . . . 18 = 2.8284271247. . . A real number is either rational or irrational. It cannot be both. The next diagram illustrates how the sets of numbers relate to one another. Real Numbers Irrational Numbers p = 3.14159 . . . 5 e = 2.71828 . . . 5.616116111 . . . 14 1 100 –24, – , 0, , p , 7 Rational Numbers Integers –3 –2 –1 Natural Numbers 1 2 3 25 7 0.25 0.9292 . . . 11 – Whole Numbers 0 This diagram shows us the following facts. • All natural numbers are whole numbers, integers, and rational numbers. • All whole numbers are integers and rational numbers. • All integers are rational numbers. • Natural numbers, whole numbers, integers, and rational numbers are not irrational numbers. • Natural numbers, whole numbers, integers, rational numbers, and irrational numbers are all real numbers. In Chapter 9, another set of numbers, complex numbers, is introduced. This set contains all of the real numbers. ���������������������� Classifying a Real Number Step 1: If the number is 1, 2, 3, . . . , then the number is a natural number. Step 2: If the number is 0, 1, 2, 3, . . . , then the number is a whole number. Step 3: If the number is . . . , -3, -2, -1, 0, 1, 2, 3, . . . , then the number is an integer. Step 4: Determine if the number is rational or irrational. a. If the number is equivalent to a decimal that terminates or repeats, then the number is rational. b. If the number is equivalent to a decimal that does not terminate or repeat, then the number is irrational. Step 5: If the number is rational or irrational, it is a real number.
hendricks_beginning_algebra_1e_ch1_3
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