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Chap 1 The Number System

Chap 1 The Number System - FIRST YEAR CALCULUS W W L CHEN c...

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FIRST YEAR CALCULUS W W L CHEN c W W L Chen, 1982, 2005. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 1 THE NUMBER SYSTEM 1.1. The Real Numbers The purpose of the first four sections of this chapter is to discuss a number of the properties of the real numbers. Most readers will be familiar with these properties, or have at least used most of them, perhaps sometimes unaware of their generality. We do not propose to discuss here these properties in great detail, and shall only give a brief introduction. Throughout, we denote the set of all real numbers by R , and write a R to indicate that a is a real number. The first collection of properties of R is generally known as the Field axioms. We offer no proof of these properties, and simply treat and accept them as given. FIELD AXIOMS. (A1) For every a, b R , we have a + b R . (A2) For every a, b, c R , we have a + ( b + c ) = ( a + b ) + c . (A3) For every a R , we have a + 0 = a . (A4) For every a R , there exists a R such that a + ( a ) = 0 . (A5) For every a, b R , we have a + b = b + a . (M1) For every a, b R , we have ab R . (M2) For every a, b, c R , we have a ( bc ) = ( ab ) c . (M3) For every a R , we have a 1 = a . (M4) For every a R such that a = 0 , there exists a 1 R such that aa 1 = 1 . (M5) For every a, b R , we have ab = ba . (D) For every a, b, c R , we have a ( b + c ) = ab + ac . Remark. The properties (A1)–(A5) concern the operation addition, while the properties (M1)–(M5) concern the operation multiplication. In the terminology of group theory, not usually covered in first Chapter 1 : The Number System page 1 of 19
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First Year Calculus c W W L Chen, 1982, 2005 year mathematics, we say that the set R forms an abelian group under addition, and that the set of all non-zero real numbers forms an abelian group under multiplication. We also say that the set R forms a field under addition and multiplication. The property (D) is called the Distributive law. The set of all real numbers also possesses an ordering relation, so we have the Order axioms. ORDER AXIOMS. (O1) For every a, b R , exactly one of a < b , a = b , a > b holds. (O2) For every a, b, c R satisfying a > b and b > c , we have a > c . (O3) For every a, b, c R satisfying a > b , we have a + c > b + c . (O4) For every a, b, c R satisfying a > b and c > 0 , we have ac > bc . Remark. Clearly the Order axioms as given do not appear to include many other properties of the real numbers. However, these can be deduced from the Field axioms and Order axioms. For example, suppose that x > 0. Then by (A4), we have x R and x + ( x ) = 0. It follows from (O3) and (A3) that 0 = x + ( x ) > 0 + ( x ) = x , giving x < 0.
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