By John Baylis, Rod Haggarty

'...quite the easiest one i've got had the fortune to read...admirable substitute studying for a beginning path introducing college mathematics.' David Tall, the days greater academic complement

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**Additional resources for Alice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics**

**Sample text**

Check that C z and C 10 are different sets, but that they intersect, so that the distinct Cx's do not partition N. 27J (b) On N, X~ Y means X is a prime factor of y. Here the situation is worse. Not only do distinct classes overlap (6 is in C z and C 3 and C z =1= C 3 ), but the number 1 is not in any class, so we fail on two counts to get a partition of N. 4: ~ is a relation on 7l. defined by x~ y if and only if X - Y is divisible by 4. Co, C 1 , C z and C 3 are clearly the sets written out in that example, and they partition 7l..

From now on we shall use the term nest to denote a sequence of nested closed intervals whose lengths approach zero and whose end-points are rational numbers. In the step from iQ to IR the nests will play the same role as the ordered pairs of naturals in the step from N to 7L.. To be precise, we identify a point (a real number) with the set of all nests having that point as its intersection. To avoid circularity in the definition, we have to be able to say what we mean by two nests representing the same real number in terms of rational numbers only.

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) ... X -, y • x I I I I • I I 0 x+y I (b) Fig. 5 24 y NUMBERS x 's, " x can be defined as the sharp end of x's arrow after giving it a half-turn around its blunt end (zero). In this model such properties as x + 0 = x, x + y = y + x, - (x + y) = ex) + y) and - (- x) = x all become fairly obvious.