November 2005 Contents
When the Pope was a Mathematician
At the end of the first millennium, at a time when neither mathematics nor the papacy required much specialized knowledge, a mathematician became pope. This article describes the life and times of Gerbert of Aurillac, also known as Pope Sylvester II.
Ramanujan's Continued Fraction for a Puzzle
This article describes a method of solution that Ramanujan may have used in solving the following puzzle: The number of a house is both the sum of the house numbers below it on the street and the sum of those above it. (The houses on a street are numbered consecutively, starting with 1.)
Centers of the United States
This article is a discussion of some geographical problems related to geographic and population centers of the U.S. The focus is on modeling, using concepts from calculus.
A Paper-and-Pencil gcd Algorithm for Gaussian Integers
As with natural numbers, a greatest common divisor of two Gaussian (complex) integers a and b is a Gaussian integer d that is a common divisor of both a and b. This article explores an algorithm for such gcds that is easy to do by hand.
How to Avoid the Inverse Secant (and Even the Secant Itself)
The primary use of the inverse secant in calculus is as an antiderivative. In this article, the author advocates taking advantage of properties of the hyperbolic functions instead.
Differentiability of Exponential Functions
Philip M. Anselone and John W. Lee
The authors give a rigorous treatment of the differentiability of the exponential function that uses only differentiable calculus. It can thus make "early transcendental" courses complete.
Fallacies, Flaws, and Flimflam
Ed Barbeau, editor
Michael Kinyon, editor
A Non-Visual Counterexample in Elementary Geometry
The author presents a method for showing in an elementary geometry class that triangles with equal perimeters may have different areas.
Can You Paint a Can of Paint?
Robert M. Gethner
The author considers ways of resolving the Gabriel's horn paradox, focusing on modeling the paint.
A Paradoxical Paint Pail
An example is given of a bounded container with the same paradoxical property as Gabriel's horn.
Differentiate Early, Differentiate Often!
This capsule argues for differentiating implicitly early in max-min problems in calculus as well as in related-rates problems.
A Two-Parameter Trignonometry Series
The problem examined in this capsule is that of determining whether a trig series that looks like a Fourier series is actually the Fourier series of some function, and if it is, finding such a function.
Problems and Solutions