We give an irredundant axiomatization of the complete ordered field of real numbers. In particular, we show that all the field axioms for multiplication with the exception of the distributive property may be deduced as "theorems" in our system. We also provide a complete proof that the axioms we have chosen are independent.

Computers are very good at solving certain types combinatorial problems, such as fitting sets of polyomino pieces into square or rectangular trays of a given size. However, most puzzle-solving programs now in use assume orthogonal arrangements. When one departs from the usual square grid layout, complications arise. The author--using a computer,…

The basis of a good mechanical puzzle is often a puzzling mechanism. This article will introduce some new puzzling mechanisms, like two knots that engage like gears, a chain whose links can be interchanged, and flat gears that do not come apart. It illustrates how puzzling mechanisms can be transformed into real mechanical puzzles, e.g., by…

Hyperbolic geometry occurs on hyperbolic planes--the most commonly cited one being a saddle shape. In this article, the author explores negative hyperbolic curvature, and provides a detailed description of how she constructed two hyperbolic paraboloids. Hyperbolic geometry occurs on surfaces that have negative curvature. (Contains 11 figures and 4…

Responds to R. E. Erard's comments on the current authors' original article which reviewed a number of studies that identified cultural differences in the use and effect of different types of social support among Asians and Asian Americans and European Americans. Essentially, in his comment, Erard denied the validity of research examining cultural…

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (nonradical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment,…

Functional analyses suggested that the disruptive behavior of three preschool children was maintained by escape from demands. While keeping the escape contingency intact, we conducted (a) a density analysis in which the children earned preferred items for task completion according to two schedules that varied in reinforcement density, and (b) a…

The Lorentz velocity addition formula for one-dimensional motion presents a number of problems for beginning students of special relativity. In this paper we suggest a simple rewrite of the formula that is easier for students to memorize and manipulate, and furthermore is more intuitive in understanding the correction necessary when adding…

A problem addressed infrequently in beginning physics courses is that of a moving body with changing mass. Elementary texts often have footnotes referring to jet planes and rockets but rarely do they go further. This omission is understandable because calculations with variable mass generally require the tools of calculus. This paper presents a…

This little problem arose because I was frustrated with the standard electromagnetism texts, which show the magnetic field due to a current-bearing wire outside the wire [proportional to] 1/r and inside [proportional to] r. However, they never point out that the moving electrons must be influenced by the magnetic field created by the other moving…

Mediation has become common in many areas of society, including marital dissolution, community disputes, governmental agencies, and business and industry. Though higher education has been slower than society to adopt mediation services, campus mediation is becoming increasingly more common. This article explains why mediation is a viable…

Academics and activists concerned with race and racism have rightly coalesced around the sociological project to refute biologistic conceptions of race. By and large, our default position as teachers, writers and researchers is that race is a social construct. However, the deconstruction of race and its claims to theoretical intelligibility has…

Visualizing complex roots of a quadratic equation has been a quest since the inception of the Argand plane in the 1800s. Many algebraic and numerical methods exist for calculating complex roots of an equation, but few visual methods exist. Following on from papers by Harding and Engelbrecht (A. Harding and J. Engelbrecht, "Sibling curves and…

Let I be a real interval. We call a continuous function [mu] : I x I [right arrow] [Bold R] a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I [right arrow] [Bold R} be a differentiable and strictly convex or strictly concave function. If a, b [image omitted] I with a [not equal to] b, then there exists a…

Executive functioning (EF) refers to higher order thought processes considered foundational for problem-solving. EF has both "cool" cognitive and "hot" emotional components. This study asks: (a) what are the relative contributions of "hot" and "cool" EF to children's academic achievement? (b) What are the relative contributions of "hot" and "cool"…