By John Baylis, Rod Haggarty

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

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

**Example text**

Finally, by inserting an extra 5 in the factorisation of b2 , we get a factorisation of 5b2 which contains 2n + 1 5s. So we have prime factorisations of the same number (a 2 = 5b2 ) one of which contains an even number (2m) of 5s and the other an odd number (2n + 1). This is impossible by UPF (Chapter 2). To recap, we have made a supposition, one of the consequences of which was the denial of something we know to be true. Hence, that supposition must be wrong. That is, 5 cannot be rational. 1. /5 as a fully respectable 'number'.

So when we translate them back to their original positions, this property of having at least one point in common should be preserved. 11 I~ n 11 I~ I~ I~ ~ ] E E I 3 t 3 t t E-----J I Fig. 6 Having made it reasonable to believe that nested closed intervals have at least one point in common, let us now restrict consideration to nested sequences whose intervals become arbitrarily small. In this case it is virtually obvious that the sequence has only one point common to all its intervals. To see this, argue by contradiction and suppose that Xl and X 2 are both common points (and suppose that Xl is the smaller) .

Shapes are 30 NUMBERS classified into regular, irregular, convex, concave, . . ; numbers into prime, composite, odd , even, squares, cubes, .... It is often convenient for our classification to involve only non-overlapping classes (like primes and composites but unlike squares and cubes), and in this case the classification is called a partition. The following are examples of partitions. (a) { { - 8, - 4,0,4,8,12, ... }, {... - 7, - 3, 1,5,9, - 6, - 2,2,6,10, ... }, {. . - 9, - 5, - 1,3,7, } } is a partition of Z into four classes.