Cyclotomic polynomials, (xp - l)/(x-1), have (p-1) zeros exp(2πiq/p) (q = 1, 2, ... ,p - 1). We consider related polynomials whose zeros are given by cos(qπ/p) (q = 1, 2, ... ,p - 1) and indicate their relation to polynomials Pn (ζ) that satisfy the ‘double angle’ condition Pn(2ζ2 - l) = Pn(ζ)Pn(-ζ). The double angle condition is generalised to a chain of double angle conditions Qi(2ζ2 - 1) = Qi+1(ζ)Qi+1(-ζ), i = 1, 2, ...,n(Q1 ≡ Qn+1) For purposes of illustration, we will focus on P4(ζ), demonstrating how construction of the fourth order polynomial subject to the double angle condition yields an equation for cos . We also discuss more general features of the polynomials Pn (ζ). Furthermore, we illustrate how a simple geometrical construction can be used to derive a polynomial for cos .

The theorems that follow are mostly well known; the object of this note is to show that they all follow simply from a single argument, whereas they are usually reached from different starting-points.

The following note develops ideas in fields outside the normal school syllabus and if its contents are introduced at school, will, it is hoped, serve as a stimulus to encourage the brighter student to carry out further investigations in these fields.

Both terms in the title of this paper are probably suspect. The mathematician, versed in a subject with hundreds of years of noble tradition, tends to be suspicious of a paper on psychology; the teacher of mathematics, immersed daily in psychological relationships with ordinary children, tends to be suspicious of a paper on mathematical ability. The discussion promises to be unsatisfactory, to the one because psychology lacks mathematical precision, to the other because mathematical ability seems to present no urgent psychological problems. The teacher-mathematician may then be pardoned if he reacts sceptically to our title with the petition, “Psychology, if there be a science of psychology, help me to cope with mathematical disability.” But these suspicions are not well founded. Not only is psychology, as we shall see, becoming increasingly mathematical, but also, if we interpret our terms as mathematicians should, disability is seen to be merely the negative aspect of ability. As such, it will not be excluded from our discussion.

There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.

Teaching mathematics has never been easy. There is no reason to imagine that it ever will be. Yet it is still worth asking whether there might not be relatively simple ways of improving our teaching of particular topics and of particular groups of pupils. History of mathematics has much to offer on both counts, and can enrich ordinary teaching in many ways and on many levels. But it also has its problems and its pitfalls. This article focuses on two issues: the temptation to enlist the support of ‘history’ when trying to change social attitudes; and the uncritical way in which intelligent students respond to such pseudohistory.

Augustus De Morgan (1806-1871) was a prolific writer, a notable mathematician and logician, and an excellent teacher. His work in logic and his invention of the logic of relations are well-known. However he also thought and wrote extensively about mathematics teaching pedagogy, and his approach is still relevant to mathematics teachers today. In this short note I offer selected quotations from De Morgan's writings on mathematics education, preceded by a short biography taken from.