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.

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.

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.

In a previous article [1] we examined the concept of blood relations in a family tree and we used mathematics to highlight and simplify the descriptions involved, such as child, parent, great-uncle, second cousin, third cousin once removed.

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.

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.

Five geometrical tasks demonstrating the interesting property of conservation of area that may be used in the mathematics classroom are presented. The initial premise is proved and then changes in attributes are investigated using dynamic geometric software to show that in the cases demonstrated there is conservation of area or specific ratios. These demonstrations illuminate the hidden beauty in mathematics and constitute an intellectual stimulus for discovering additional properties.

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.

It was my nine-year-old daughter who got me interested in the title question. As she appreciates, multiplying an integer by a power of 10 is a cinch. To multiply 34 by 100, simply add two zeros at the end: 34 × 100 = 3400. Dividing 3400 by 100 is the reverse process: remove two zeros to obtain 34. More generally, to multiply an integer by 10N, for non-negative N, add N zeros to the end of its decimal notation, and to divide an integer by 10N remove N zeros from its end — so long as it has them. Easy-peasy; my daughter knows all that.

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.

Mathematical calculation at times may produce erroneous results because of ignoring particular features of the functions involved, such as restricted domains, ranges and discontinuities. In this Article, to highlight the significance of conceptual understanding, we choose six examples and for each we give a flawed solution followed by a correct solution.