5.5 Real Numbers and Their Properties 259 Now that we have discussed both the rational and irrational numbers, we can discuss the real numbers and the properties of the real number system. The union of the rational numbers and the irrational numbers is the set of real numbers , symbolized by R. Fig. 5.8 illustrates the relationship among various sets of numbers. It shows that the natural numbers are a subset of the whole numbers, the integers, the rational numbers, and the real numbers. For example, since the number 3 is a natural or counting number, it is also a whole number, an integer, a rational number, and a real number. Since the rational number 1 4 is outside the set of integers, it is not an integer, a whole number, or a natural number. The number 1 4 is a real number, however, as is the irrational number 2. Note that the real numbers are the union of the rational numbers and the irrational numbers. Real numbers Rational numbers Integers Fractions Terminating or repeating decimal numbers Natural numbers Zero Negative integers Irrational numbers Figure 5.9 The relationship between the various sets of numbers in the real number system can also be illustrated with a tree diagram, as in Fig. 5.9. Fig. 5.8 and Fig. 5.9 show that, for example, the natural numbers are a subset of the integers, the rational numbers, and the real numbers. We can also see, for example, the natural numbers, zero, and the negative integers together form the integers. Properties of the Real Number System We are now prepared to consider the properties of the real number system. The first property we will discuss is closure. Did You Know? An Important Number Some years ago, a group of French mathematicians who worked under the collective pseudonym “Monsieur Nicholas Bourbaki” embarked on the development of an encyclopedic description of all mathematics. They devoted 200 pages simply to introduce the innocent-looking concept, the number 1. Figure 5.8 Real numbers Rational numbers Integers Whole numbers Natural numbers ... 23, 22, 21 0 1, 2, 3, . . . Irrational numbers ⎯2, ⎯3, p, ⎯7 2 – 7.82, 20.096, , , 3 5 2 , 7 8 123 25 1 4 1 , 0.3 √ √ √ Elena Schweitzer/Shutterstock
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