There is some interesting mathematics associated with the integration of the function

for positive integers n. For n = 1 we have

to within an arbitrary constant. Expanding the integrand with the binomial theorem and integrating term by term gives the well-known result

The question of the convergence of the sequence defined recursively by

for a given complex value of c, and a fixed determination of the square root was posed by Ogilvy [3], and partial solutions have been given by Rohde [4] and Gerber [5].

Although the approximation of elementary functions is a well-researched area of mathematics, very little has been published on built-in routines in any particular calculator or computer. Computer manufacturers are, understandably perhaps, reluctant to publicise details about their algorithms. Hewlett-Packard have been less secretive than most and allowed an occasional glimpse behind the scene in their own Journal and various other sources. The following square root routine for one of their machines, the now extinct 2000F series, has appeared in an Open University text on numerical computation [1]. The process of evaluating √x is carried out in four steps:

1. Introduction

Since Poincaré introduced algebraic or geometric topology in a series of papers during the years 1895-1901, the subject has grown rapidly to become one of the major branches of mathematics with ramifications in many other branches. However, apart from some recreational level ‘rubber-sheet’ geometry, very few students, even at first degree level, meet algebraic topology.

This article is about a powerful method of carrying out tests of statistical significance. Its rationale depends on simple ideas of probability and is suitable for pupils at school. Practical applications require a considerable amount of computation but now that schools are equipped with computers this should not be a problem.

School mathematics is customarily taught in three stages which can be described as practical, pure and applied respectively. They differ from each other in the way that they make use of scientific data; [1]. The first stage involves teaching mathematics through its applications; pure mathematics makes no use of applications and at the third stage known pure mathematics is applied to other scientific problems.

The interest of a problem, to me at least, depends on the simplicity of its statement and the subtlety of its solution. The four-colour problem is a good example. When the solution of the problem also brings surprise, the interest becomes fascination. Good paradoxes particularly fascinate me for they even start with a surprise, and in this case the resolution brings another.

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.

1. Introduction

A question of the following type recently appeared in a mathematics contest for high school students in the U.S.A.

The apparent paradox posed by Stephen Ainley (Gazette, December 1982, note 66.43) arises from two interconnected causes.

Example 1

Take a cardboard equilateral triangle. It is well known that this has six symmetries. To display these symmetries we colour the vertices of the triangle on both sides of the card with three different colours.

“What is the probability that a randomly drawn triangle is acute-angled?” This question has been discussed in recent Gazettes (for example by Trevor Easingwood in [2], followed up by Stephen Ainley in [1]). As Easingwood pointed out, the answer to this question depends on what properties of triangles are assumed to be random and on what one means by random. And as Ainley pointed out some of these assumptions can lead to a contradiction. A statistical model is necessary to define the sense in which the chosen properties (lengths of sides, angles, co-ordinates of vertices,…) are assumed random. This model may (and preferably will) relate to a possible physical ‘random’ mechanism for generating the random triangle.

In elaborating John Ward’s method for the calculation of pi, Cohen and Shannon drew attention to Ward’s assumption that the smallest positive root of a polynomial equation can be approximated by a root of the truncated equation in which higher powers are ignored. (The interested reader can find full details in [1] and [2].) The mathematical basis of this is justified in a plausible style in this note, and it can be readily used by the interested teacher as a basis for further computational work with students. It is likely that Ward employed a Newtonian fluxional kind of analysis when he obtained the result, rather than the purely algebraic considerations employed here. The historical significance of Ward is that his calculation of pi was possibly the last attempt to use geometric methods. Its mathematical significance was his use of successive trisection rather than the traditional bisection of angles.