Direct sum decomposition of Abelian groups appears in almost all textbooks on algebra for undergraduate students. This concept plays an important role in group theory. One simple example of this decomposition is obtained by using the kernel and range of a projection map on an Abelian group. The aim in this pedagogical note is to establish a direct…

In this note, an integrated form of some significant means with two variables is provided, and some chains for mean value inequalities are obtained. At the same time, a concise family of algebraic functions appears, which satisfy Mitrinovic's requirements.

This article discusses some interesting physical properties of oscillatory motion of a particle on two joined inclined planes. The geometrical series demonstrates that the particle will oscillate during a finite time. Another detail is the converging path to the origin of the phase space. Due to its simplicity, this motion may be used as a…

A "convex" polygon is one with no re-entrant angles. Alternatively one can use the standard convexity definition, asserting that for any two points of the convex polygon, the line segment joining them is contained completely within the polygon. In this article, the author provides a solution to a problem involving convex lattice polygons.

The purpose of this article is to describe how, within the Dutch approach to mathematics education, called "Realistic Mathematics Education" (RME), models are used to elicit students' growth in understanding of mathematics. First some background information is given about the characteristics of RME related to the role of models in this approach.…

Students learn more when they attempt to make sense of a mathematical situation they face. For example, a question like why square root of 25 is not + or - 5. Providing the intermediate steps and the reasoning of a technique with graphs can help make better sense of mathematics.

One of the things that mathematician like to do is to collect empirical data on some mathematical subject and then to try to discover whether the values indicate that an underlying pattern exists. The conclusion states that almost no pattern is likely to continue unless mathematically shown to do so by rigorous proof.

The walnut sized brain of the African grey parrot may actually be capable of comprehending abstract mathematical concepts. The bird seems to understand a numerical concept akin to zero--an abstract notion that humans don't typically understand until they are three or four years old. Alex, the 28-year-old parrot who lives in a Brandeis University…

It is reasonably well known that the ratios of consecutive terms of a Fibonacci series converge to the golden ratio. This note presents a simple, complete proof of an interesting generalization of this result to a whole family of 'precious metal ratios'.

The national lottery is often portrayed as a game of pure chance with no room for strategy. This misperception seems to stem from the application of probability instead of expectancy considerations, and can be utilized to introduce the statistical concept of expectation.

The familiar property of integral of f (x) = 1/x led to the discovery of other functions with this property.

This article describes some of the work of Jan Hudde who anticipated some results of calculus. Prior to a career as a Burgomaster of Amsterdam, Hudde engaged in mathematics. His method of finding maxima and minima is especially interesting.

The standard reverse-order law for the Moore-Penrose inverse of a matrix product is (AB)[dagger] = B[dagger]A[dagger]. The purpose of this article is to give a set of equivalences of this reverse-order law and other mixed-type reverse-order laws for the Moore-Penrose inverse of matrix products.

In this note, the notion of reverse derivation is studied. It is shown that in the class of semiprime rings, this notion coincides with the usual derivation when it maps a semiprime ring into its centre. However, we provide some examples to show that it is not the case in general.

In this note, we give a method for finding the inverse of a three by three magic square matrix without using the usual methods for finding the inverse of a matrix. Also we give a method for finding the inverse of a three by three magic square matrix whose entries are also matrices. By using these ideas, we can construct large matrices whose…