20 CHAPTER R Review of Basic Concepts The result of dividing two integers (with a nonzero divisor) is a rational number, or fraction. A rational number is an element of the set defined as follows. ep q ` p and q are integers and q 30f Rational numbers The set of rational numbers includes the natural numbers, the whole numbers, and the integers. For example, the integer -3 is a rational number because it can be written as -3 1 . Numbers that can be written as repeating or terminating decimals are also rational numbers. For example, 0.6 = 0.66666c represents a rational number that can be expressed as the fraction 2 3 . The set of all numbers that correspond to points on a number line is the real numbers, shown in Figure 13. Real numbers can be represented by decimals. Because every fraction has a decimal form—for example, 1 4 = 0.25—real numbers include rational numbers. Some real numbers cannot be represented by quotients of integers. These numbers are irrational numbers. The set of irrational numbers includes 12 and 15. Another irrational number is p, which is approximately equal to 3.14159. Some rational and irrational numbers are graphed in Figure 14. The sets of numbers discussed so far are summarized as follows. –1 0 1 2 3 4 – 2 3 Ë2 Ë5 p Graph of the Set E - 2 3 , 0, 22, 25, p, 4F 22, 25, and p are irrational. Because 22 is approximately equal to 1.41, it is located between 1 and 2, slightly closer to 1. Figure 14 –5 0 1 2 3 4 5 –4 –3 –2 –1 Graph of the Set of Real Numbers Figure 13 Sets of Numbers Set Description Natural numbers 51, 2, 3, 4, c6 Whole numbers 50, 1, 2, 3, 4, c6 Integers 5c , -3, -2, -1, 0, 1, 2, 3, c6 Rational numbers E p q P p and q are integers and q≠0F Irrational numbers 5x x is real but not rational6 Real numbers 5x x corresponds to a point on a number line6 EXAMPLE 7 Identifying Sets of Numbers Let A = E -8, -6, - 12 4 , - 3 4 , 0, 3 8 , 1 2 , 1, 22, 25, 6F. List all the elements of A that belong to each set. (a) Natural numbers (b) Whole numbers (c) Integers (d) Rational numbers (e) Irrational numbers (f) Real numbers SOLUTION (a) Natural numbers: 1 and 6 (b) Whole numbers: 0, 1, and 6 (c) Integers: -8, -6, - 12 4 1or -32, 0, 1, and 6 (d) Rational numbers: -8, -6, - 12 4 1or -32, - 3 4 , 0, 3 8 , 1 2 , 1, and 6 (e) Irrational numbers: 22 and 25 (f) All elements of A are real numbers. S Now Try Exercises 111, 113, and 115.
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