By Raymond Ayoub

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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.

29] Verify that the relation ~ defined on ~ x ~ by (a, b) ~ (e, d) if and only if a + d = b + e is an equivalence relation. 30] Do the same for (a, b)fYi(e,d) if and only if ad = be, where fYi is a relation on 7L. x (7L. \ {0 }). 3 should recognise the equivalence classes involved in the two examples above! 31] How many different equivalence relations are there on a threeelement set? 32] A relation fYi on {a, b, c, d} consists of the ordered pairs (a, b), (a, c), (a, a), (b, d), (e, c). 0// (i) reflexive, (ii) symmetric, (iii) transitive.

For division 15/3 and - 15/3 can be pictured as sharing a profit or loss of £ 15 between three shareholders, but that sort of model can't make sense of 15;-3, or of 15/(14/11). Our final model of addition and multiplication of numbers returns to the number line . 5(a). 5(b). This works for any combination of signs of x and y. The definition of - x can also be incorporated as part of the specification of the model: if we have only defined the position of zero and the positive ... y • x • I x 0 y (a) ...