o Section 1: Number Systems

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

This preview shows pages 1–3. Sign up to view the full content.

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: 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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 product of prime factors.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern