This edition of the Gazette is the first that bears my name as editor (although it will be several editions before the change-over is complete). So this is an appropriate time to pause and take stock. First, I would like to thank Douglas Quadling for his brilliant years as editor. No-one who has met him can fail to marvel at the enthusiasm and devotion which he has applied to his task; and no-one who reads the Gazette can fail to appreciate how the journal, under his guidance, has kept its sense of balance through almost a decade which has seen some considerable upheaval in the teaching of mathematics. Second, as a newcomer I really do need to know who reads the Gazette. You do, presumably; so you are the obvious person to tell me what you like or dislike about it, and to tell me about any particular aspects which you would like to be introduced/abolished/expanded/reduced. I will welcome your comments and do my best to steer the Gazette on a course which’

It has been common in recent years, if not entirely obligatory, for your President to address you on this occasion about the difficulties and opportunities he has found in the practice of teaching, and this year will be no exception. But following the excellent example of all past Presidents, as far back as I have investigated, I intend to confine my address to that part of the activity about which I am practically conversant. So while I hope that the general ideas and conclusions of my talk will be enjoyable and possibly useful to everyone here, I shall deal in detail only with the narrow problem of first-year undergraduates. I say simply ‘problem’ because I believe that the various different questions (the ‘interface’, the content of first-year pure/applied courses, how should the sixth form teacher best prepare the future university entrant,…) are all aspects of a single problem. For the sake of precision I shall deal with only one aspect of this problem, as it arises in analysis. Knowing me, you might have expected rather that I would have discussed the difficulties of teaching mechanics, or of persuading young people of 18 plus (although many of them have been brought up since five years of age on a practical approach to mathematics!) that there is any possibility at all of making an application of mathematics in the external world. My depression about that aspect is too strong, and my involvement in it is probably too deep, for me to see it clearly. But my experiences supervising small groups of students who are making their first genuine acquaintance with the real number field suggest to me the possibility of improving matters.

Interest rates are playing a large role in many people’s lives these days. The creation of a property owning democracy is part of the reason; another lies in their rapid increase. Yet in spite of this phenomenon the actual method of calculation of interest is almost wholly ignored by the mass of the population, and the fluctuations in the rate are looked upon in qualitative terms. In fact the formula for simple interest is easy to write down and, from this, compound interest can be deduced in a straight-forward fashion. That is, provided you remember what compound interest is. I am short enough of breath to have been at Grammar School when O-level Mathematics was divided into three parts, Arithmetic Algebra and Geometry; and the most boring of these was Arithmetic. So boring indeed thatcompound interest, stocks and shares and other useful tricks were retained until the examination and then promptly forgotten.

For many years extra-mural classes have been held on Wednesday afternoons at the University of Birmingham for the benefit primarily of teachers of mathematics. Part of one of these courses for sixth form teachers involved taking A level questions and using them as starting points for further enquiries. The intention was to illustrate how almost any A level question can yield something fruitful, beyond its straightforward solution, if one perseveres with asking ancillary questions and exploring alternatives. The approach was presented as a possible means of encouraging students to adopt more divergent thinking than the solution of A level questions generally produces.

If formulae are known for

1 + 2 +… + n and l2 + 22 +… + n2,

then, by a standard school method, a formula for

l3 + 23 +… + n3

can be deduced in the following fashion:

Calculus made easy by “F.R.S.” was published seventy years ago. Its anonymous author wrote explicitly for engineering students rather than for future mathematics teachers. He ended by asking those who had enjoyed the book “not to give the author away nor to tell the mathematicians what a fool he really is”.

Decimal expansions

Every real number x has a decimal expansion of the form

x=±N.n1n2n3…,

where N is either zero or a positive integer and each ni is one of the digits from 0 to 9. The notation N.nln2n3… is an abbreviation for the infinite series

When a function in arithmetic modulo n is represented through a cartesian graph, the resulting set of points reveals little information about the form of the function (for example, see Fig. 1). It is not easy to discern connections between the points; a restriction to the values 0, 1, 2,…, n — 1 would have to be placed on x and y; and the fact that n — 1 is as ‘near’ to 0 as 1 is, is not apparent in the graph.

Formulae for Σl=1n ik when k is a positive integer are well known, and many methods for finding them have been described. This article gives an account of an approximate, but remarkably accurate, formula which I have found for the case k =1/2.

Most students in their undergraduate careers ensure that they have many examples of groups at their finger tips. These include not only whole systems such as cyclic groups and symmetric groups but also specific examples, in particular groups of small order. These latter groups are useful for trying out new ideas as soon as they are encountered, such as the bogeys of cosets, normal subgroups and factor groups. If we consider only groups of order 7 or less, then we already are furnished with nine examples of cyclic, non-cyclic, abelian and non-abelian groups.

It seems that, ever since first meeting the ham sandwich theorem fourteen years ago, I have had a lurking interest in halving shapes. Somehow a few months ago I turned my attention more specifically to the problem of halving triangles and tetrahedra—a problem with which I am still wrestling. This problem was going to be the subject of an article for the Gazette, but it did not really work out well. In a letter about this abortive article Douglas Quadling drew my attention to the significance of hyperbolae for halving triangles—something which I had previously not noticed.

Let D and R be two non-empty sets. A function f: D → R is a rule for assigning to each element x belonging to D a single element y = f(x) belonging to R. The set D is called the domain of the function f and the set R is called its range.

The differential triangle of Leibniz for a real function f is found by taking an increment dx in the variable x, finding the corresponding increment dy in y = f(x) and drawing the ‘triangle’ in Fig. 1. Here ds is the increment in the length of the graph, where

and the derivative of f is

Nowadays no mathematician is surprised when he hears someone speaking about an n-dimensional space: in fact, work involving vector spaces and functions of several variables makes every student familiar with this concept.