Unformatted text preview: The set Q of all rational numbers is the set of all real numbers of the form pq −1 , where p ∈ Z and q ∈ N. We see that the Field axioms and Order axioms hold good if the set R is replaced by the set Q. On the other hand, the set Q is incomplete. A good illustration is the following well known result.
Chapter 1 : The Number System page 4 of 19 First Year Calculus c W W L Chen, 1982, 2005 PROPOSITION 1A. No rational number x ∈ Q satisﬁes x2 = 2. Proof. Suppose that pq −1 has square 2, where p ∈ Z and q ∈ N. We may assume, without loss of generality, that p and q have no common factors apart from ±1. Then p2 = 2q 2 is even, so that p is even. We can write p = 2r, where r ∈ Z. Then q 2 = 2r2 is even, so that q is even, contradicting that assumption that p and q have no common factors apart from ±1. √ It follows that the real number we know as 2 does not belong to Q. We shall now discuss a property that distinguishes the set R from the set Q. Definition. A nonempty set S of real numbers is said to be bounded above...
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This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.
 Fall '08
 staff
 Math, Calculus

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