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o Section 1: Number Systems

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 1.1 Number Systems 3 Version: Fall 2007 1.1 Number Systems In this section we introduce the number systems that we will work with in the remainder of this text. The Natural Numbers We begin with a definition of the natural numbers , or the counting numbers . Definition 1. The set of natural numbers is the set N = { 1 , 2 , 3 ,... } . (2) The notation in equation (2) 2 is read “ N is the set whose members are 1, 2, 3, and so on.” The ellipsis (the three dots) at the end in equation (2) is a mathematician’s way of saying “et-cetera.” We list just enough numbers to establish a recognizable pat- tern, then write “and so on,” assuming that a pattern has been sufficiently established so that the reader can intuit the rest of the numbers in the set. Thus, the next few numbers in the set N are 4, 5, 6, 7, “and so on.” Note that there are an infinite number of natural numbers. Other examples of nat- ural numbers are 578,736 and 55,617,778. The set N of natural numbers is unbounded; i.e., there is no largest natural number. For any natural number you choose, adding one to your choice produces a larger natural number. For any natural number n , we call m a divisor or factor of n if there is another natural number k so that n = mk . For example, 4 is a divisor of 12 (because 12 = 4 × 3), but 5 is not. In like manner, 6 is a divisor of 12 (because 12 = 6 × 2), but 8 is not. We next define a very special subset of the natural numbers. Definition 3. If the only divisors of a natural number p are 1 and itself, then p is said to be prime . For example, because its only divisors are 1 and itself, 11 is a prime number. On the other hand, 14 is not prime (it has divisors other than 1 and itself, i.e., 2 and 7). In like manner, each of the natural numbers 2, 3, 5, 7, 11, 13, 17, and 19 is prime. Note that 2 is the only even natural number that is prime. 3 If a natural number other than 1 is not prime, then we say that it is composite . Note that any natural number (except 1) falls into one of two classes; it is either prime, or it is composite. Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1 In this textbook, definitions, equations, and other labeled parts of the text are numbered consecutively, 2 regardless of the type of information. Figures are numbered separately, as are Tables. Although the natural number 1 has only 1 and itself as divisors, mathematicians, particularly number 3 theorists, don’t consider 1 to be prime. There are good reasons for this, but that might take us too far afield. For now, just note that 1 is not a prime number. Any number that is prime has exactly two factors, namely itself and 1.
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4 Chapter 1 Preliminaries Version: Fall 2007 We can factor the composite number 36 as a product of prime factors, namely 36 = 2 × 2 × 3 × 3 . Other than rearranging the factors, this is the only way that we can express 36 as a
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o Section 1: Number Systems - Section 1.1 Number Systems 3...

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