Presents recently developed generalizations to the sieve of Eratosthenes, showing the principles underlying these improvements, which increase its efficiency without changing too much of its simplicity. Offers several possibilities to propose good investigations for students to explore, find patterns, and make generalizations. (JRH)

A method for solving inequalities of degree greater than one is described. (MK)

Facts are given about the ninth-century Arab mathematician Al-Khwarizmi, from whose name the word algorithm is derived. His influence on algebra and arithmetic are discussed. (MNS)

Discusses an algorithm from Vedic mathematics that has similarities to FOIL and the standard algorithm for multiplication. (Author/NB)

An alternate method for factoring quadratic equations is presented as being quicker and easier to use than the trial-and-error method. (MP)

Circuits are described, with discussion on how to help students find the algorithms to solve a variety of problems involving circuits. (MNS)

How set theory, combinatorics, probability, and the study of algorithms can be used in solving two problems is described in detail. Three computer programs are listed. (MNS)

A generalized method of synthetic division where the divisor polynomial may be of any degree equal to or larger than 1, and the dividend polynomial may be of equal or larger degree than the divisor polynomial and a generalization of the familiar remainder theorem, are presented. (Author/MK)

Describes a calculator exercise that can help students develop a better visual and numeric feel for Newton's method. (KHR)

Two mathematical topics are interpreted from the viewpoints of traditional (performing algorithms) and contemporary (creating algorithms and thinking in terms of them for solving problems and developing theory) algorithmic mathematics. The two topics are Horner's method for evaluating polynomials and Gauss's method for solving systems of linear…

A history of perfect numbers is presented, which briefly covers the 27 values known at this time. (MP)

The article presents a general test for divisibility that includes composite numbers and shows that such a test can be used to determine divisibility by several numbers simultaneously. (MK)

An algorithm for multiplying natural numbers is described. The algorithm provides a chance for some unusual drill and might serve as an enrichment topic. (MK)

Described is a simple algorithm that can be used for the input, arithmetic manipulation, and output of large integers in their exact representations. Three BASIC programs are included that apply this method to the problem of multiplication of large integers, computation of factorials, and the generation of palindromic integers. (MNS)

Considered are what discrete mathematics includes, some parallels and differences between new math and discrete mathematics (listed in a table), and lessons to be learned. A list of references is included. (MNS)