By John Baylis, Rod Haggarty

'...quite the simplest one i've got had the fortune to read...admirable replacement analyzing for a origin direction introducing collage mathematics.' David Tall, the days larger academic complement

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**Sequences of Numbers Involved in Unsolved Problems**

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9J The required hcf is the last non-zero remainder. 10J-one with two memories is particularly convenientor on a home computer. 44 THE REAL NUMBERS Euclid's algorithm is not just a pleasant curiosity. It is a starting point for many important topics in number theory, such as continued fractions and Diophantine equations, and it provides a method of simplifying significantly our proof of UPF given in Chapter 2. Tempting as it would be to digress to these topics now, it would lead us away from the main business of this chapter, which is to explore some of the issues involved in getting from Q to IR, so we resist the temptation.

Choose any b in S for which a~b. Then by symmetry b~a. Now that we have a~b and b~a, transitivity ensures that a~a. Where is the fallacy? J If ~ is a relation on S and XES, we use C, to denote the subset of elements y of S for which X i:Jt y. Formally, C x = {Y :YES A x~y}. The following three examples lead us back to partitions. (a) On N, x~y means that X and y differ by at most 5. Check that C z and C 10 are different sets, but that they intersect, so that the distinct Cx's do not partition N.

Why not? 4] Now the set of common measures of AB and CD has at least one memberremember that this was the given information. 5], and then convince yourself that although this set, like M, is infinite and bounded above (there is no member longer than CD), it must have, unlike M, a longest member. 6] Digression 2 One of the pleasures of mathematics is finding connections between ideas where no such connection was previously suspected. As an example of this consider the case of two sets which may have cropped up in quite different contexts and which are therefore defined or described in quite different ways, but which turn out to be identical.