Showing papers in "The Mathematical Gazette in 2021"
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TL;DR: In the May 1954 issue of the Gazette Daniel F. Ferguson challenged readers to devise their own proof for what he described as a curious and somewhat pleasing sum (see [1]) as discussed by the authors.
Abstract: In the May 1954 issue of the Gazette Daniel F. Ferguson challenged readers to devise their own proof for what he described as a curious and somewhat pleasing sum (see [1])
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TL;DR: In this article, the golden ratio, Fibonacci numbers, groups, and Pythagorean triples are discussed, and there are some analogies with the fourth triples.
Abstract: Some favourite topics of both mathematics teachers and mathematics students are the golden ratio, Fibonacci numbers, groups, and Pythagorean triples. The material of this paper involves the first three, and there are some analogies with the fourth.
TL;DR: In this paper, the authors give an account of two appealingly simple techniques to generate monotonic sequences that were developed by the author and Grahame Bennett, who died in 2016 after a lifetime of distinguished contributions in the field of inequalities.
Abstract: Here we give an account of two appealingly simple techniques to generate monotonic sequences that were developed by the author and Grahame Bennett. Sadly, Bennett died in 2016 after a lifetime of distinguished contributions in the field of inequalities. This article is dedicated to him.
TL;DR: A notable occurrence, and a sign of things to come, was the publication in 1962 of a joint paper with his parents which gave rise to the concept of an NNN group as discussed by the authors.
Abstract: Peter Neumann was a major figure in UK mathematics during a career spanning over 50 years. His parents, Bernhard and Hanna Neumann, were group theorists, so it was perhaps inevitable that Peter would become a specialist in the same area. A notable occurrence, and a sign of things to come, was the publication in 1962 of a joint paper with his parents which gave rise to the concept of an NNN group. There can be few instances in mathematics of three members of the same family co-authoring a paper. His parents spent many years in Australia and Peter maintained strong links with that country throughout his own professional life.
TL;DR: In this paper, the authors address the issue by providing examples from the realms of antenna reflector theory and the use therein of conic sections, which are a force to be reckoned with.
Abstract: According to reports in the media, there is a dearth of practical examples that students of mathematics en route to their qualification can feast upon, at either sixth form level or an undergraduate level. Despite these alleged shortages, it is this author’s opinion that there are numerous examples that can be drawn from the literature and it is the purpose of this article to address the issue by providing examples from the realms of antenna reflector theory and the use therein of conic sections. Some students will be familiar with conic sections and others might not, but the numerous instances of their manifestation in the real world would suggest that they are a force to be reckoned with, and this is certainly true from a mathematical perspective.
TL;DR: The world is amply supplied with probability teasers and these can mystify and intoxicate. as mentioned in this paper provides a good selection of the classic probability teases and presents some classics.
Abstract: The world is amply supplied with probability teasers. These can mystify and intoxicate. Here are some classics.
TL;DR: In this paper, the sine and cosine rules for a quadrilateral were derived and applied to the problem of solving a set of problems involving a triangle with respect to the sines and cosines.
Abstract: We aim to produce formulae for a quadrilateral which correspond to the sine and cosine rule for a triangle. Subsequently, we apply these results to solve problems involving a quadrilateral.
TL;DR: In this article, it was shown that a convex quadrilateral is a Pitot quadra if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair.
Abstract: It is well known that a convex quadrilateral is a cyclic quadrilateral if, and only if, the sum of each pair of opposite angles is π. This result (which gives a necessary and sufficient condition for the existence of a circle which circumscribes a given quadrilateral) is beautifully complemented by Pitot’s theorem which says that a given convex quadrilateral has an inscribed circle if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair. Henri Pitot, a French engineer, noticed the easy part of this result in 1725 (see Figure 1), and the converse was first proved by J-B Durrande in 1815. Accordingly, we shall say that a convex quadrilateral is a Pitot quadrilateral if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair.
TL;DR: The trisector theorem of as discussed by the authors states that the three intersections of the trisectors of the angles of a triangle, lying near the three sides respectively, form an equilateral triangle.
Abstract: Morley’s trisector theorem says that the three intersections of the trisectors of the angles of a triangle, lying near the three sides respectively, form an equilateral triangle. See Figure 1. Morley discovered his theorem in 1899, and news of it quickly spread. Over the years many proofs have been published, by trigonometry or by geometry, but a simple angle-chasing argument is elusive. See [1] for a list up to 1978. Perhaps the easiest proof is that of John Conway [2], who assembles a triangle similar to the given triangle by starting with an equilateral triangle and surrounding it by triangles with very carefully chosen angles.