This article arises because of a lecture I attended where the speaker, who had been heavily involved in the new JMB A level syllabus, was saying that roots of unity had gone, “thank goodness.” Many A level teachers seemed to agree and I began wondering just how “traditional” this topic is. I came to the conclusion that it is the most revealing topic to show the meshing of traditional and modern ideas. I hope to illustrate this by the end of this article.

Since the time of Gutenberg the volume of printed information has been doubling roughly every fifteen years. Only relatively recently, however, has the growth of knowledge reached a point at which it challenges human intellectual capacity. Our society has responded by a progressive specialisation of intellectual roles. Very broadly, the nineteenth century saw the development of specialism in distinct fields; the twentieth, the growth of specialism within those fields. To take just mathematics, it has been estimated that, after allowing for redundancy, our present knowledge fills the equivalent of something of the order of 60 thousand average-sized volumes. While at the start of this century, according to Ostrowski, it was expected that a mathematics graduate would be prepared to deal with any question in any branch of mathematics, by mid century Von Neumann was estimating that a skilled mathematician might be familiar with some 10% of the whole field.

Padé approximations consist of one polynomial divided by another (analogous to the polynomial approximations obtained from Maclaurin series). They can serve, at an elementary level, as a corrective to the idea that a power series is the only form of useful approximation. Indeed, they are more powerful than Maclaurin approximations, since, in general, a suitably chosen Padé approximation of a given order gives greater accuracy and is useful over a wide range than the corresponding Maclaurin approximation.

The relationship in this article was discovered by Alan Tomkins and the proof supplied by David Pitt. The original inspiration was the statistical study of gambling systems—one method of attempting to win being to increase the amount staked each time you lose. The idea of this is that when you eventually win the amount won is sufficient to more than offset the losses on the preceding string of losers and put you back into the black. The snag is that this string of losers can leave you with insufficient funds to keep increasing the stake as necessary. The question which comes to mind is: in a given number of races, on how many occasions are we to expect a run of losers of a certain length?

First it may be necessary to remind some readers of what is meant by the representation of a group by a Cayley graph. Each element of the group is represented by a point, the points being connected by lines (or edges) each of which represents a generating operation of the group. Suppose, for example, that x and y are two elements of a group, while another element;? is taken as a generator, and that these are connected: xp = y. (Throughout this article we shall take the identity element of a group to be 1, inverses, to be indicated x-1, while xp means “first x, thenp”.) This would be graphed:

About four years ago after consulting my copy of Cundy and Rollett [1] and then leaving it on my desk, I noticed that it had fallen open at a diagram of an approximation to Von Koch’s ‘snowflake’ curve. Interest in such curves has increased of late but for most mathematics students they make a fleeting appearance in the analysis course as examples of closed, continuous curves of infinite length and differentiable nowhere.

In order to obtain the area of a curve by the method of the integral calculus the following two conditions should hold:

1. (a) the equation of the curve must be known,

2. (b) the function representing the equation of the curve must be integrable.

MacNab [1] has recently considered the Euler line of a triangle, and derived expressions for the position vectors of certain important points like the orthocentre, the circumcentre and the incentre. In this note I show how these results, and some other ones, can be derived by a more direct use of vector methods than MacNab employed. I believe that such applications provide excellent material for use in a first course on vectors since only elementary properties are needed. Nevertheless, the power of vector methods is well illustrated and, in addition, the results are interesting and the symmetry of the algebra is very appealing. While I suspect that much of the material is not original–for example some of the results are in the well-known book by Weatherburn [2]–I am not aware of the existence of a single concise treatment, and I believe this has potential value.