The standard tape–recorder consists outwardly of two spools on which the magnetic tape is wound, from the left to the right, between least and greatest radii of a and b. On my machine, a is about 1.05 in. and b is about 2.75 in. Apart from what is on the tape, the only indication at any intermediate stage as to how much playing time is still left is provided by a revolution counter geared to the left hand spool, the one which starts full and finishes empty. The numbers registered are about half the actual number of revolutions made on the left: this arrangement allows some 1400 revolutions to be registered by only three digits. It can be assumed that the tape is fed out at a constant length per unit of time: so the time still to run is proportional to the length still to run. If the total number of revolutions to run is N, how much of the tape ha3 been run off when the reading is n?

Over the years we have come across many different methods of proving the sine function addition formula

and we have long felt it could be both interesting and useful to gather together in one article a list of methods to illustrate how different ideas from geometry, algebra and calculus may be used to prove it.

The “real” in the title refers to real numbers rather than real life. I expect, and hope, that many articles on the future of A-level will expound the cognitive and mathematical necessity of making A-level more practical. While I applaud and try to practise this, classroom teaching and research have convinced me that the notion of the real number line has subtle conflicts that will not go away by simply not mentioning them; they need to be addressed. Moreover teacher-led explorations of the real number line are essential if we are to lead students to an understanding of higher mathematics.

Practitioners of Tantric yoga are said to use certain diagrams, called yantras, as aids to meditation [1]. The most interesting of these, from a geometrical point of view, is the sriyantra (the great yantra). A version of diis, shown below, has been reproduced in various recent books and articles as an example of the mathematical achievements of early non-European cultures. In this article I want to make some observations about the geometric structure of the sriyantra, and to comment on some of the claims being made about it. I also hope, in passing, to be offering an example of the sort of geometric situation that can be explored with the powerful Cabri geometry program [2].

The following theorems are closely connected in various ways; one common thread that runs through them is that in each a critical stage of the proof consists in filling in a missing value in the range of values for which the result is known to hold. Another common feature, of no logical importance but none the less the raison d'être for this article, is that each serves to illustrate important fallacies and omissions in current accounts of elementary algebra and analysis.

The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers. Once introduced, polynomials and matrices open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics. We present an aspect of a rather elementary exploration of polynomials and matrices, which offers a new perspective and an interesting matrix analogue to the concept of a zero of a polynomial. The discussion offers an opportunity for better comprehension of the fundamental concepts of polynomials and matrices. As an application we are led to the meaningful questions of linear algebra and to an easy proof of the otherwise advanced and abstract Cayley-Hamilton theorem.

The ellipse is well known to students of mathematics at an advanced high school level and at an undergraduate level through its associations with, say, conic sections and their deployment as hardware in dual reflector antenna systems. Perhaps less well known is its appearance on the software side, so to speak, for example in RF (Radio Frequency) engineering in the context of polarisation, where it is known as the polarisation ellipse. It is the purpose here to discuss this manifestation and dwell particularly on the concept of a circumscribing rectangle for this ellipse because such aspects are of interest to the RF engineer who may be involved with measurements to determine its properties. This exercise affords an interesting insight into a very practical application of tertiary level mathematics and whilst it is of course helpful to appreciate the physical concepts involved, a lack of such knowledge does not detract from an appreciation of the underlying sums, which should be within the grasp of the above, interested student.

This article grew out of a problem that I saw in an old textbook on sale in a second hand bookshop. It inspired a generalisation that leads to two sets of well known numbers (Bell and Stirling numbers), and then to a new result for one of them.