Professor Forder’s elegantly simple method in Note 3068 (Math. Gazette 47 (1963), 237-8) can be extended to prove a more general result. Let k be any positive integer, ξ any irrational number, ar the (r + l)-th quotient in the simple continued fraction for ξ and pr/qr the (r+ l)-th convergent, so that

“ … And so the impulse of the blow is equal to the change of momentum of the body.”

This phrase, coming at the conclusion of a more or less standard lesson in Elementary Mechanics, must surely have struck many teachers with a feeling of shame for having participated in a piece of not entirely straightforward work, and their pupils with an idea that they have witnessed some downright “wangling”. The usual impulse-momentum treatment cannot but be unsatisfactory to all concerned in the lesson, for it follows on the introduction of a new quantity of measurement of which, as so many of the text-books plaintively remark, “no special name has been given to the unit”.

This problem has been treated in many Geometries (Casey, Forder, Steiner) and in the Mathematical Gazette, and by Frederick Soddy.

Perhaps I can add a little of interest.

In ordinary arithmetic a number ab … jk represents the sum of the place values of its digits a, b, …, j, k, these place values being k, 10j, 102i, etc. Also the representation is unique if the digits are restricted to being positive and less than 10. We are going to consider the representation of whole numbers by means of digits with place values, but with a different definition of place value. The digits a, b, …, j, k + 1 of a given number A are to be positive, and the number of them, n, is to be assigned, the units digit k + 1 (in the standard form) is to be greater than zero, and the place value of the digit a in the nth place counting from k + 1 is to be a. a + 1 … a + n - 1 divided by 1.2. …. n, or the number of homogeneous products of a letters taken n at a time; this number or place value is written (a)n.

Mathematica is a computer software system for symbolic manipulation and numerical computation; it has excellent graphics, a high-level, mathematically-oriented programming language and, very importantly, it takes seriously the need for work to be interactive, properly presented and communicated. This article is in fact prepared as a Mathematica “notebook”. An extensive library of supporting software “packages” has been developed, and several books about using it are now available.