# The College Mathematics Journal - March 1999

### Contents for March 1999

Square Roots from 1;24,51,10 to Dan Shanks: a Guide for the Perplexed
Ezra Brown

All about square roots. How to find them (several ways, one even not using a calculator or computer), and how to find them modulo a prime. For example, the square root of 2 (mod 360027784083079948259017962255826129) is 162244492740221711333411667492080568.

Mathematics and the Liberal Arts
Hardy Grant

A survey of the place of mathematics in education from the ancient Greeks to the middle ages. If it weren't for Pythagoras and Plato, mathematics might have had the same status as sanitary engineering.

Interval Arithmetic and Analysis
James Case

Any number that we find by measurement is fuzzy: we know only that it lies in some interval. When we combine such numbers, or solve equations involving them, where do the results lie? In intervals, of course, but where are they and exactly how long? Sometimes it is important to know.

Several Sets of n + 1 Shapes, Each the Similitude Union of the Other n
Allen J. Schwenk

It's possible to find two right triangles and a trapezoid so that each triangle is similar to the other triangle put on top of the trapezoid and the trapezoid is similar to the two triangles put next to each other. It's hard to tell when this is possible in general. In this paper, the problem is solved for rectangles (any number, not just three) and some other cases.

From Euler to Fermat
Hidefumi Katsuura

The definition of e, and no number theory at all, can be used to prove a special case of Fermat's Last Theorem.

An Attempt to Foster Students' Construction of Knowledge During a Semester Course in Abstract Algebra
Thomas G. Edwards and Lawrence Brenton

What happens when students try to learn mathematics? One way of looking at it is that first comes an action--a manipulations of objects. When students can combine actions, they have a process. Later, when a process can be manipulated by some action (for example, reversed or combined with another process), it has been reconstructed to form an object. Finally, a collection of related processes and objects maybe put together to form a schema. This paper shows how this theory was applied in an abstract algebra course.

### Classroom Capsules

Michael J. Bradley, Michael St. Vincent, and David L. Finn

Linear transformations are additive and homogeneous. What transformations have one of these properties but not the other?

On "Rethinking Rigor in Calculus ... ," or Why We Don't Do Calculus on the Rational Numbers
Scott E. Brodie

Many theorems of calculus that students sometime think are so obvious as to not need stating are false if we have only rational numbers, something that it would do students good to think about.

A Far-reaching Formula
Kil S. Lee

The area of a trapezoid is the average of the parallel sides times the height. This formula has other applications.

### Book Review

Mathematics of the 19th Century
by A. N. Kolmogorov and A. P. Yushkevich
reviewed by John Ewing

### Filler Material

Saddam Hussein proved to be a beast, and other items.