The number 6174 has the well-known property that it equals the difference between the numbers formed by rearranging its digits in descending and ascending orders :

Numbers with this property are called Kaprekar constants after the Indian mathematician D. R. Kaprekar [1]. Similarly, the general process of finding the difference between numbers formed by rearranging digits in descending and ascending orders is called the Kaprekar process.

Like Fibonacci and Lucas numbers, Pell and Pell-Lucas numbers are a fertile ground for creativity and exploration. They also have interesting applications to combinatorics [1], especially to the study of lattice paths [2, 3], as we will see shortly.

Pell numbers Pn and Pell-Lucas numbers Qn are often defined recursively [4, 5]:

where n ≥ 3. They can also be defined by Binet-like formulas:

Transformation geometry is now taught in many schools, and complex numbers are frequently used to describe the geometric transformations. For example, a translation is represented by (where b is a complex number), a rotation by a magnification by and a reflection in the x-axis by

The set of all transformations of the form forms a group and consists of precisely those transformations which preserve distance. Among the subgroups of this group there are the groups which give rise to the well known 17 distinct ‘wallpaper patterns’. These can be described in an elementary and geometric way and are (quite rightly) a popular topic in transformation geometry.

It is proper and desirable, at most of our meetings, for speakers to talk on mathematical topics to which they may themselves have made significant contributions. In general at such meetings we shall all understand the broad outline, but the details are likely to be obscure to most of us. The presidential address, I have always thought, is in rather a different category. The speaker should ensure that his audience, though originally lacking deep knowledge of the chosen topic, should go away with the impression that they not only understand every word but could also reproduce the arguments, say to a class of sixth-form mathematicians. Since our association is mainly concerned with the refreshment of school mathematics and mathematicians, I hope to talk about a part of numerical mathematics which could be taught in school, which uses “standard” mathematics, and which springs from standard mathematics. I start with a rather lengthy introduction on the general principles of error analysis.

We shall try to show that, in some well-defined sense, every right-angled triangle is ‘approximately similar’ to a Pythagorean triangle: and then we’ll ask how large the Pythagorean triangle must be.

“We are all pacifists now”; or if not, it is difficult to see what excuse we have for living in the modern world. I feel therefore that a word of apology is needed for a paper on a military subject. But I hope you will acquit me of any intention to revive the hatred and bloodthirsty panic that poisoned the years 1914 to 1918, because the activity that I have to tell you about is not concerned with the destruction of life but rather with the saving of it.

If a1, …, an and b1, …, bn are two sets of real numbers, then (a1b1 + … + anbb)2 ≤ (a12 + … + an2)(b12 + … + bn2), the equality holding if and only if either 3 a real number λ such that ar — λbr (r = 1, …, n) with some br ≠ 0 or b1 = … = bn = 0. Write a = (a1, …, an) and b = (b1, …, bn). Suppose a ≠ 0 and b ≠ 0, for otherwise the inequality is trivial. Write с = a — λb.

Choose λ so as to make b, c=0, i.e take

the equality holding if and only if e=0, i.e if and only if a=λb,

2. Suppose x1...,xn and y1 are two sets of linearly independent vectors in the space E defined on the previous note.

Following on from Nick MacKinnon’s work in the June 1989 Gazette concerning four different methods of square-rooting 2 × 2 matrices, we may ask if it is possible to compute all the square roots of any given 2 × 2 matrix, and if so, how? From now on by a “matrix” we mean a real or complex 2 ×2 matrix. We know that every non-zero number has precisely two square roots, but the situation is rather more complicated for matrices. In fact we shall see that the zero matrix and any other multiple of the identity matrix has an infinite number of square roots. All other matrices have only a finite number of square roots, with some (rather surprisingly) having none at all! Our method of enquiry is based on Nick MacKinnon's first method, namely matrix diagonalisation.