Writing *Numbers and Functions* was difficult. Difficult firstly because the convention in university mathematics is to present the subject in a sequence ‘Axiom’, ‘Definition’, ‘Theorem’, ‘Proof’, ‘Application’ which respects the logic of the argument, but which ignores the ordinary way in which people learn (more like ‘Experience’ followed by testing and ‘Generalisation’) motivated all the way by problems. Difficult secondly because undergraduate analysis does not usually begin at the point where school calculus ends.

These difficulties afflicted me as an undergraduate studying analysis. I loved the power that came from learning about differentiation and integration at school; finding maxima and minima, and through anti-differentiation finding the areas and volumes of strange shapes and their centres of gravity. By contrast, undergraduate analysis began with a construction of the real numbers, which I had not thought were unknown, or with a set-theoretic definition of function. In this new style I could follow the logic, but not grasp what the story was. Ideas taken for granted from what graphs looked like at school, such as the Intermediate Value Theorem or Rolle’s Theorem, were given picky, hard proofs incorporating seemingly gratuitous difficulty.

For years after graduating, I managed to avoid teaching analysis, except for occasional supervisions on a one-to-one basis, but our number one analyst retired from teaching and other colleagues were as reluctant as I was to take up the subject. So the challenge before me was to see whether a problem sequence such as that in my *Pathway into Number Theory* might be constructed for Analysis. In 1982, a reviewer of the *Pathway* said it was all very well to do this for Number Theory, but it *would* be worthwhile for Analysis.

There were really two problems here. The first was how to make undergraduate analysis continue directly on from sixth form calculus. The second was to understand how the subject can grow for those who are ignorant of it. The solution to the first problem was to start with induction of which students have had an experience in school and to include in that first chapter those algebraic properties which will be needed in the rest of the course and to follow this with work on inequalities, again familiar from school but explored with more rigour and again incorporating properties which will be needed later. The second problem was much harder, and I began probing the history of the subject to find out how, in the past, mathematicians had gone from not knowing to knowing. Special functions or series repeatedly provoked new thinking, these often being outside the scope of school calculus.

I came to realise that if the course was to be true to the understanding of Weierstrass (and in fact it had to match the first year analysis course at Cambridge) there were two new building blocks with which students would have to work. These were a formal definition of limit and the notion of completeness. The course consists of applying these two ideas to series, continuity, derivatives and integrals. Whereas the definition of limit emerged over a period of two hundred years from Newton to Weierstrass, completeness was only under discussion after the middle of the 19^{th} century. Yet two thousand years earlier Euclid and Archimedes were finding properties of areas, volumes and centres of gravity that today we would only think of establishing with limits. The base on which Euclid and Archimedes built was Archimedean Order. This is a property which today is often deduced from a completeness axiom. This deduction hides the fact that Archimedean Order can hold (most obviously for the rational numbers) without completeness. Not only is Archimedean Order sufficient to establish the elementary properties of limits, but it is also sufficient to establish the more elementary properties of continuity and derivatives, and a certain amount about integrals. It is however insufficient to establish the characteristic theorems of Analysis which had so troubled me as a student. Knowing, with examples, when these theorems fail (that is, over the rational numbers) gives point to their proof.

The last problem was completeness itself. Until 1960, the leading British text for Analysis was Hardy’s *Pure Mathematics* which started by constructing irrational numbers with Dedekind cuts. Since 1960 British authors have adopted what had long been common practice in America of defining the real numbers by axioms. There are alternative ways in which completeness may be prescribed by axiom, many of which are useful at different points in the development of the subject, so that a single definition, while logically sufficient, does not indicate the multiple significance of the notion. Inspired by a German text of W.F.Osgood (1907) based in turn on the ideas of Du Bois Reymond (1875), I chose what seems the most naïve version, that every infinite decimal is convergent, and devoted a chapter to establishing the various forms of completeness that can be deduced and are needed for the course.

By choosing the format of a sequence of problems, the text may be better exploited by problem classes than by lectures. I have been in correspondence with those working at it alone. The sequence of problems is intended to be accessible enough for success. It is designed so that the student grapples firmly with the big ideas.

Numbers and Functions: Steps into Analysis by R.P. Burn

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