The pressures currently on teachers of mathematics at every level seem greater than ever, whether that is university mathematics lecturers, school teachers or those who work with very young learners. So, I began my Presidential Address with the hope that no one would leave the Joint Conference of Mathematics Subject Associations 2023 feeling as though they were not good enough and must do more. I did not want anyone to go home with a lengthy to-do guilt list of additional things they must try to incorporate into their practice. Instead, I advocated giving attention to the ancient proverb “less is more”, versions of which are found within many cultures; e.g., “brevity is the soul of wit” (Hamlet), “Sometimes diminishing a thing adds to it; Sometimes adding to a thing diminishes it” (Tao Te Ching). I offered five aspects of mathematics teaching and learning that might benefit from ‘less of’ something, in each case trying to see how this might create space for more of something else. In this article, I present these five suggestions for improving by removing.

We are all familiar with the formulae for finding the sums of the powers of the first n natural numbers. For example,

Throughout this Article, we use the following notations for the triangle ABC

• a, b and c are the lengths of the sides BC, CA and AB, respectively,

• Δ denotes the area of triangle ABC,

• ha, hb and hc are the lengths of the altitudes through the vertices A, B and C, respectively,

• ma, mb and mc are the lengths of the medians through the vertices A, B and C, respectively.

Recall that for a 2π-periodic function f , the Fourier coefficients on the interval (–Π, Π) are

and the Fourier series for f is

On 21 December 2020 the night sky offered a beautiful astronomical treat for stargazers worldwide. An exceptionally rare conjunction of Jupiter and Saturn brought them to 0.1 degrees of angular separation – a fifth of the full moon’s diameter. It marked their closest approach since 1623 and the closest visible conjunction since 1226 (almost 800 years ago!). Astronomy enthusiasts crossed their fingers for clear skies and waited eagerly for this event. The internet and social media were inundated with pictures and news reports, celebrating the great conjunction of our solar system’s two most extensive and majestic planets.

The basic inequalities

The inequalities in question are(1)

and(2)

for x > 1.

Given a polynomial equation x3 + ax2 + bx + c = 0 of degree 3 with real coefficients, we may translate the variable by replacing x with x − to make the quadratic term vanish. We then obtain a simpler equation x3 + px + q = 0 where Therefore, in order to solve a polynomial equation of degree 3, it is sufficient to solve equations of the form x3 + px + q = 0.

Imagine that you go for a walk with your friend. You walk along a closed loop around a university campus, finishing at the point where you started. You and your friend keep the same distance throughout your walk. You walk on the ‘outside’.

The following theorems appear in [1, pp. 62-63]: Theorem 1 If similar triangles A1B0C0, A0B1C0, A0B0C1 are erected externally on the sides of ΔA0B0C0, then the circumcircles of these three triangles have a common point, F.

In teaching mathematics to first-year undergraduates, and thus in the appropriate calculus textbooks, the task of calculating an integral that satisfies a specific first-order or second-order recurrence relation often appears. These relations are obtained mainly by applying the method of integration by parts. Calculating such integrals is usually tedious, especially for an integer n > 2, time-consuming, and presents the possibility of making a large number of errors when computing involves multiple iterative steps. In [1], it is shown that in two cases (Theorems 2.1. and 2.3), the process of calculating integrals satisfying first-order recurrence relations can be performed quickly using easily memorised closed-form formulas for corresponding primitive functions. The question can rightly be asked whether there is a faster way to calculate other integrals of this type. In this paper, our goal is to give an affirmative answer to such a question, though without convering all situations. Since each recurrence relation is equivalent to a difference equation of the same order, the calculation of integrals mentioned above can be reduced to solving the corresponding difference equations. Since every first-order or second-order linear difference equation is solvable, it follows that for every integral which can be reduced to a first- order or second-order recurrence formula, it is possible to find corresponding primitive functions directly. Sometimes such a procedure is much faster than iterative solving of the integral. Closed-form formulas for the integrals discussed in the following sections are not unknown (see [2]). However, here our goal is to present the idea of computing indefinite integrals using difference equations. We will discuss it in more detail in Section 2. In Section 3, we discuss the application of the results obtained to calculate several improper integrals and the application of some of them in different sciences. An exciting example of such an application is the integral , which in the case n = 1 is used in the kinetic theory of gases, particularly in the Maxwell-Boltzmann distribution of gas molecules by energies (see Remark 4). Also, we compare the formulas obtained by the method of difference equations with the formulas obtained using Wolfram Alpha software (see Remark 5).