The College Mathematics Journal - September 1999

Contents for September 1999

The Witch of Agnesi
by S. I. B. Gray and Tagui Malakyan

This year marks the 200th anniversary of the death of Maria Gaetana Agnesi (1718-1799), whose 1748 text (in translation, Foundations of Analysis for the Use of Italian Youth) is the oldest surviving mathematical work written by a woman. She was also appointed a lecturer at the University of Bologna, and her name is attached to the curve wrongly called the "witch" of Agnesi. This paper reviews the curve, her life, and her works.

Novel Maclaurin Series-based Approximations to e
by John Knox and Harlan Brothers

Everyone knows that e is the sum of the reciprocals of the factorials from 0! on up and can be approximated by truncating the series. Here we get, using nothing deeper than the series expansion of ln(1 + x), a sequence of ever-better approximations to e.

Casino Gambling, the Ultimate Strategy
by Dennis Connolly

If we go to casinos, and if we gamble there, we are going to lose money in the long run. Of course in the long run we are all dead, but that does not much diminish our strivings in the short run. It is possible, in the short run, to beat the casino.

Suppose we have a stake of $9000 that we wish to turn into $10000. Sitting down at the craps table and making bets of $10 that the shooter will win is not the way to do it: with probability .941 we will lose our stake before reaching our goal, and on the average it will take around 1000 hours (six months of full-time work) before we are cleaned out. Making bets of $100 is better: the probability of ruin is reduced to .262 and the time it takes to 19 hours. Best is the Ultimate Strategy. It gives a probability of ruin of only .1053 (so that almost 90% of the time we will reach our goal) and, better yet, the decision will be reached in an average of only 2.023 minutes.

Relating Geometry and Algebra in the Pascal Triangle, Hexagon, Tetrahedron, and Cuboctahedron II
by Peter Hilton and Jean Pedersen

In Part I, connections were made between the geometry and algebra of the Pascal Triangle. In Part II, we move into three (and higher) dimensions, getting similar results about the Pascal Tetrahedron containing the trinomial coefficients and the Pascal m-simplex, which has in it the m-multinomial coefficients.

Powers as Uniform Sums of Positive Squares
by Robert Wisner

Do you know that every power of 6 is a sum of three (non-zero) squares, every power of 7 is a sum of four squares, and every power of 8 is a sum of five squares? And that, properly viewed, every power of 2 is a sum of negative one squares? Probably you don't.

Teaching the Reasoning of Statistical Inference: A "Top Ten" List
by Allan Rossman and Beth Chance

Basic statistics is being taught, as it should be, to more and more people these days. It is better to teach it well than it is to teach it not well; if it is being taught well, then it is better to teach it better. Here are ten suggestions, all practical, for raising the level of statistics instruction.

Fallacies, Flaws, and Flimflam

Technology goes where people cannot! Chips solve log (x² - 10) = log(10x - 40) = 1 !

Classroom Capsules

Natural Logarithms via Long Division
by Frank Burk

Series for ln 3 (1 + 1/2 - 2/3 + 1/4 + 1/5 - 2/6 + ... ), ln 4, and so on, simply derived.

Simple Geometric Solutions to De l'Hospitals, Pulley Problem
by Raymond Boute

A problem solved in 1696 using calculus solved in 1999 geometrically.

The Derivative of sin q
by Selvaratnam Sridharma.

A quick geometrical method to find it.

Measuring the Curl of Paper
by Haris Tabakovic, Joseph Paullet, and Richard Bertram

Paper manufacturers measure the curl of paper by measuring a distance instead of a curvature. For reasonable sheets of paper, the practical method works fine.

Problems and Solutions

For example, #658: If X and Y are random variables so that (X, Y²), (X², Y), and (X², Y²) are all pairs of independent random variables, must (X², Y²) be independent as well? I'd bet that the answer is "no".

Miscellanea

How to frost an infinite cake with one cup of frosting, how to deal with dyscalculia, and other items.

Media Highlights

Eight pages of abstracts of articles that you would most likely not encounter otherwise.