Neater representations for variance are given for small sample sizes, especially for 3 and 4. With these representations, variance can be calculated without a calculator if sample sizes are small and observations are integers, and an upper bound for the standard deviation is immediate. Accessible proofs of lower and upper bounds are presented for…

It is proved that if the differential equations "y[(n)] = f(x,y,y[prime],...,y[(n-1)])" and "y[(m)] = g(x,y,y[prime],...,y[(m-1)])" have the same particular solutions in a suitable region where "f" and "g" are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical…

A factorisation of a general second order ordinary differential equation is introduced from which the full solution to the equation can be obtained by performing two integrations. The method is compared with traditional methods for solving these type of equations. It is shown how the Green's function can be derived directly from the factorisation…

Eigenvalue is an important concept in Linear Algebra. It is well known that the eigenvectors corresponding to different eigenvalues of a square matrix are linear independent. In most of the existing textbooks, this result is proven using mathematical induction. In this note, a new proof using Vandermonde determinant is given. It is shown that this…

This paper investigates a general identity which expresses an apparently complicated and intriguing sum of fractions as an elegant and straightforward sum of simple terms.

One of the most astonishing properties when studying numerical series is that the sum is not commutative, that is the sum may change when the order of its elements is altered. In this note an example is given of such a series. A well-known mathematical proof is given and a MATLAB[C] program used for different rearrangements of the series…

It is shown that the symbols for the fundamental units of mechanics, namely length, time and mass, are capable of a meaningful interpretation as positive real parameters. Then a suitable parameter domain allows one to take the derived units into account. The formal manipulations usually carried out with symbols of physical quantities, involving…

One of the most important principles in reform mathematics is allowing students to be challenged by mathematics problems. The National Council of Teachers of Mathematics (NCTM 2000) has called for changes in the problems that are typically posed to students, asserting that such problems should instead require students to reason, communicate,…

In this article, the author explores the concept of "moving one place to the left." A recent investigation by Thompson and Bramald explored the relationship between young children's understanding of place value and their ability to add two-digit numbers. The study took the form of a series of one-to-one interviews with a sample of 144 children…

This article employs the Archimedean method of estimating the value of pi within an inscribed pentagon. We show how to write these approximations in terms of the golden ration.

An interesting new partitioning and bounded variable algorithm (PBVA) is proposed for solving linear programming problems. The PBVA is a variant of the simplex algorithm which uses a modified form of the simplex method followed by the dual simplex method for bounded variables. In contrast to the two-phase method and the big M method, the PBVA does…

This article describes a development of conic surfaces that connects different definitions of the conics. A classroom project using Dandelin Spheres is also described.

This article describes five representations and the connections between and among them. It also describes how to use multiple representations to support students' mathematical understanding.

The phenomenon under the name "resonance," which, is based on the mathematical analogy between mechanical resonance and the behavior of wave functions in quantum mechanical exchange phenomena was described. The resonating system does not have a structure intermediate between those involved in the resonance, but instead a structure which is further…

In this article we present a segment of Sulinet Digital Knowledgebase curriculum system in which you can find the sections of subject-matter which aid educating the database management. You can follow the order of the course from the beginning when some topics appearance and raise in elementary school, through the topics accomplish in secondary…