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.

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:

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.

Among the first examples of groups encountered in group theory are the symmetric and alternating groups and the general linear groups. The purpose of this article is to describe these groups and to describe two unusual functions between groups of this type, in a language which can be understood by those with only an elementary knowledge of group theory and no knowledge of linear algebra or projective geometry.

Prior to the advent of the calculator, divisibility tests were often studied and utilised. Some of these tests were simple and easily remembered; others, more cumbersome and rarely utilised in either practical or theoretical work. The purpose of this article is to present a simple and general test for divisibility which is of applicable and theoretical interest.

Collecting the free picture-cards in packets of cigarettes or tea has always been popular. The equivalent up-to-date example is the gift of a picture—card of a football team with each purchase of petrol. Many a parent has been asked the question ‘how soon will we get the set’. Let us state a fairly general problem based upon this idea of collecting a set. Our problem states:

Given a square matrix A with real or complex entries, there may not exist an invertible matrix S such that S-lAS is diagonal. For instance,

cannot be diagonalised in this way (the contrary assumption leads by easy direct calculation, to a contradition); however, this matrix is already upper-triangular.