Welcome to Leicester, our first Annual Conference here since it became the home of the Association in 1975 as the result of the centenary appeal in 1971 to buy a permanent headquarters building. The city is, of course, also home of the Association's library. Welcome to the University and to the Annual Conference, one of the many manifestations in which our Association appears. It is a great pleasure and honour to be here tonight to speak on the opening day of our 1990 Conference.

I don’t know how many readers of this journal would have been watching the draw for the opening matches of the 1990 football World Cup, made in Rome on Saturday, 9 December 1989, and televised worldwide. But I do hope that most of you that did—and who were paying attention!—at least thought “Hey, wait a minute, that can’t be right!” Sure enough, the draw was flawed: it was not as fair to all concerned as it might have been. This was a great pity given that the world football authority, FIFA, were at pains to stress how fair these proceedings would be, and the agonies of FIFA trying to be seen to live up to their “fair play” motto were indeed prolonged. The errors lay not in the process of drawing balls from urns itself, but in the algorithm used to define the drawing procedure, as I shall explain below. The problem was caused by FIFA’s decision to keep the four South American countries involved apart and, in particular, by their inability to implement this decision in the most fair manner. There is no allegation of anything being “fixed” here, just one of incompetence on behalf of football’s governing body. That there should be errors of this kind in such an overtly public affair is, however, yet another sad indictment of the general lack of understanding of matters numerical and probabilistic in the public at large.

Some time last year I clipped from the Guardian a report on the bungled draw for the quarter-finals of the Benson and Hedges Cup. The relevant section is

Suppose that a group of friends live along a straight road and that they wish to meet somewhere on that road. They wish to choose the meeting place so that they do not have to travel too far; there are several criteria which they could use:

• (1) Make the total distance travelled as small as possible.

• (2) Make the greatest distance any individual travels as small as possible.

• (3) Minimise the total of the squares of the distances travelled.

The sequence 1, 3, 7, 13, 21, … arises in many ways, for example from the integers placed in a spiral on a square grid, the terms arising where the spiral intersects a diagonal on the grid, and as the central axis of a triangular array of numbers:

In a first year university course on programming and numerical methods (in BBC BASIC), it was decided to give the students the quartic equation:

x4 + 2·88x3 − 19·23x2 − 36·11x + 91·56 = 0

to solve numerically. One root of this equation is an integer—to help the students get started—but a second root is close to the first to make it more interesting. Initially we did not realise quite how interesting this would prove to be.

“Perhaps we do even worse, suppressing the creative instincts of our children at the same time as researchers try to imitate them with machines. A computer is said to have applied a problem solving programme to the proposition that the base angles of an isosceles triangle are equal. Instead of the Euclidean proof which proves two right-angled-triangles [on the left below] to be congruent, the computer produced a more elegant proof from the simpler construction (sic) [on the right below]. The conventional working must have been taught to millions of children: we may reasonably ask why none of them, apparently, have discovered this simple, new proof for themselves.”

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.

Is there a largest number on your calculator? There is on mine; it is 9·9999999 x 1099. It is larger than any number I have ever wanted to calculate, though it puts some known Mersenne primes out of reach. An upper boundary to the numbers with which one can calculate gives one striking difference between our imagined number line and the numbers on a calculator.

A more serious difference between the numbers on a calculator and the real number line is the existence on the calculator of a smallest number, greater than 0. On my calculator, the smallest positive number is 10-99. So, if I imagine the numbers my calculator will display marked on a number line, and zoom in on the number line to magnify it, the calculator numbers will appear spaced out, rather like integers on a conventional number line. We will see shortly that the calculator numbers are not evenly spread out, but the existence of gaps on the line, in any form, is critical. The calculator numbers are not dense on the line.

This article is focussed on a letter that perhaps the greatest mathematician ever, Carl Frederich Gauss, sent to a female mathematician, Sophie Germain. The letter is interesting because it is the first that Gauss wrote to Germain in the knowledge that she was a woman, and it contains a piece of mathematics that reveals better than the text of the letter what Gauss really thought of Germain. To put the letter in context I shall give some of Germain’s biography and also briefly describe female mathematicians that pre-date Germain.

At one time Latin was considered such an essential part of the school curriculum that everyone was required to study it. If justification were needed for the requirement, the argument would likely be given that the study of Latin “trained the mind”. Is secondary school geometry in the same category that Latin formerly was, or are there legitimate reasons for requiring that geometry be studied by most secondary school students? Perhaps one reason for studying geometry is that it improves problem-solving ability in general.

The specific proposition that geometry improves problem-solving ability would seem to be subject to experimental verification, if indeed the study of geometry does improve problem-solving ability. The evidence herein seems to indicate that the kind of reasoning used in studying geometry improves the ability to solve not only geometric problems but other types of problems as well.

Our original source for this investigation was an Open University video on Secondary classroom practice (OU video no. PM644). Since then, however, we have seen the problem mentioned in [1], [2] and [3]. The investigation as presented to pupils is as follows:

Choose two integers between 0 and 9 (not both 0) and write them down. As an example, we shall choose 0 and 2. Write down their sum, doing the addition modulo 10. Then write down the sum of the last two numbers in the sequence, again doing the addition modulo 10. Repeat this process. Our example gives the sequence

Any positive fraction can be represented by a lattice point in the positive quadrant of the coordinate plane, (x, y) being used to represent the fraction y/x. The diagram shows the points representing 2/3,4/6,6/9 and it’s easy to see that equivalent fractions will in general lie on a line through the origin, the gradient of that line equalling the value of the fractions in question.

This device could be a useful teaching aid. It can easily be built by students who like to use their hands; its behaviour can easily be examined quantitatively and compared with theoretical predictions; the mathematics required to analyse its motion is not at all advanced; and it provides a good exercise in mathematical modelling. The problem appears to have been overlooked in books on mechanics, which is surprising in view of the simplicity of the device and of the mathematics associated with it.

In many situations we are faced with two numbers 0 ˂ a ≤ b as values of some variable and we are forced to produce a compromise value which in some way is intermediate between a and b. Such a compromise value can be termed an average or a mean of a and b and we would expect it to lie between a and b, with equality if and only if a = b. There are many different kinds of mean in elementary mathematics (see, for example, “Connected means” in the June 1988 Gazette). Perhaps the following three are the best known and the most important.