When Johann Bernoulli published his lectures on integrals in 1742, integral calculus had become very advanced since the time of their composition in 1692. Nevertheless, these lectures are of excellent clarity and simplicity even when the book deals with major problems of Mathematical Physics. Just to pique some interest, we offer a commented…

This paper is concerned with developing a new class of generalized numbers. The main advantage of this class is that it generalizes the two classes of generalized Fibonacci numbers and generalized Pell numbers. Some new identities involving these generalized numbers are obtained. In addition, the two well-known identities of Sury and Marques which…

In this paper, the use of guided hyperlearning, unguided hyperlearning, and conventional learning methods in mathematics are compared. The design of the research involved a quasi-experiment with a modified single-factor multiple treatment design comparing the three learning methods, guided hyperlearning, unguided hyperlearning, and conventional…

We present a research report on addition and subtraction conducted with Down syndrome students between the ages of 12 and 31. We interviewed a group of students with Down syndrome who executed algorithms and solved problems using specific materials and paper and pencil. The results show that students with Down syndrome progress through the same…

This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos--satisfying an axiom sin[superscript 2] + cos[superscript 2] = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two…

This article deals with a brief history of Fibonacci's life and career. It includes Fibonacci's major mathematical discoveries to establish that he was undoubtedly one of the most brilliant mathematicians of the Medieval Period. Special attention is given to the Fibonacci numbers, the golden number and the Lucas numbers and their fundamental…

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two…

We present a general formula for a triple product involving four real numbers. As a particular case, we get the sum of a triple product of four odd integers. Some interesting results are recovered. We derive a general formula for more than four odd numbers.

This classroom note includes the following sections: Introduction; The General Case; and Related Formulas.

In a thesis written for the Doctor of Arts in Mathematics, the connection between Euclid's algorithm and continued fractions is developed and extended to n dimensions. Applications to computer sciences are noted. (SD)

Hypothesizing that subtraction of (negative) integers requires formal operations, the author describes a study of children's understanding of related concepts. The study involves the use of apparatus embodying properties of the number line. (SD)