Throughout this article, all numerals are assumed to be in base ten notation, unless otherwise stated.

Under many of the new mathematics programmes currently in use in secondary schools, the topic “Numbers to different bases” is introduced. Indeed, provided such an elementary textbook series has chapters on Sets, Counting to different bases, Symmetry and/or Transformation Geometry, and, possibly, Vectors and Matrices, it is then allowed to write “Modern Mathematics” on its title page almost regardless of the methods employed in the book. However, my topic does not concern the vagaries of modern mathematics textbooks, their content and their methodology, but rather the question of why ten is used as a number base. Undoubtedly many of you already feel that you know, and that the reason is concerned with numbers of fingers and toes. However, one might try to decide more rationally whether ten is a sufficient base to represent the wide range of numbers with which we are normally concerned.

In many books on elementary mechanics, it is assumed that velocity is a vector quantity on the grounds that it has magnitude and direction. It is, however, very far from obvious that these are sufficient reasons and a rather more thorough investigation of the nature of velocity may provide not only a clearer idea of what velocity is but may also clear up some of the difficulties associated with “relative” velocity.

Most of us, I suppose, can quote a few phrases read or heard, which have greatly affected our outlook on life in general and our own work in particular. Of about half a dozen such, two came vividly into my mind as I recently attended, at the University of London Institute of Education, a course of lectures in which a closer link between the sixth form and the University courses in mathematics was sought. The first goes back to my youth, and comes from Ruskin—“Know what you have to do—and do it!” Ruskin goes on to say that insufficiency of means is far less frequently a source of failure than an inadequate understanding of the thing to be done. The second is more recent, and came as a gentle rebuke from more than one of the nurses at St. Bartholomew’s Hospital, during my stay there after becoming blind, when I apologetically asked for some small non-routine service—“What are we here for?” Surely Ruskin’s precept and the nurses’ question are very relevant to our work.

In talking about the relations of mathematics and culture or, rather, about the cultural value of mathematics, I think it advisable that I should first say a little about what I suppose the word “culture” to mean. In its general use it contains, I think, two ingredients, as it were. It refers to information and it refers to sensibility An ignorant man, however refined his natural sensibilities may be, is not regarded as cultured. And a man who has amassed an immense knowledge of facts, but is quite unable to distinguish between what is valuable and what is not valuable in his information, is not regarded as cultured. Of these two elements, I am inclined to think that sensibility is the more important. What the old writers on aesthetics used to call taste is, I think, more important than knowledge for the formation of a cultured outlook.

In a recent article in the Gazette (March 1983) Chris Haines described the problems which arose when he was arranging a tenth birthday party for his son. There were seven boys in addition to his son at the party, making eight all together. In order to prevent the party from getting too boisterous, Mr Haines decided to split the party into pairs who would play various boxed games for two players: for example, halma, draughts, snakes and ladders. The plan was to have a sort of tournament in which:

1. (1) each child would play every other child once and once only;

2. (2) in a given session, different games would be played by each pair as only one set of each game was available;

3. (3) each child would play as many of the available games as possible.