An Alternate Approach to Alternating Sums: A Method to DIE For
Arthur T. Benjamin and Jennifer J. Quinn
Positive sums count. Alternating sums match. Alternating sums of binomial coefficients, Fibonacci numbers, and other combinatorial quantities are analyzed using sign-reversing involutions. In particular, we Describe the quantity being considered, match positive and negative terms through an Involution, and count the Exceptions to the matching rule (the method of D.I.E.). Careful use of this technique often results in nice generalizations. Any sum arising from the Principle of Inclusion-Exclusion (P.I.E.), such as the number of derangements, can be understood using D.I.E. too.
Dinner Tables and Concentric Circles: A Harmony of Mathematics, Music, and Physics
Jack Douthett and Richard J. Krantz
How should men and women be seated around a dinner table to maximize conversation between members of the opposite sex? What can be said about the distribution of points around two concentric circles? How are the white and black keys on the piano keyboard organized? What spin configuration in the Ising model minimizes energy? These four problems have remarkably similar solutions.
Squaring a Circular Segment
Consider a circular segment (the smaller portion of a circle cut off by one of its chords) with chord length c and height h (the greatest distance from a point on the arc of the circle to the chord). Is there a simple formula involving c and h that can be used to closely approximate the area of this circular segment? Ancient Chinese and Egyptian records indicate the use of a formula based on a trapezoid to approximate this area, namely h(c+h)/2. Several centuries later, Archimedes discovered a formula (based on a triangle) that gives the exact area of a parabolic segment. Since a parabola can be used to approximate a circular arc, Archimedes' result yields 2ch/3 as another formula to approximate the area of the circular segment. A search for a better estimate, one that continues to rely on a quadratic function of c and h, reveals a much better approximation for this area than either of the ones mentioned thus farand generates some interesting elementary mathematics.
Dependent Probability Spaces
William F. Edwards, Ray C. Shiflett, and Harris Shultz
The mathematical model used to describe independence between two events in probability has a non-intuitive consequence called dependent spaces. The paper begins with a very brief history of the development of probability, then defines dependent spaces, and reviews what is known about finite spaces with uniform probability. The study of finite dependent spaces with non-uniform probabilities is then introduced. The paper concludes with an investigation of infinite spaces.
From Mixed Angles to Infinitesimals
Jacques Bair and Valérie Henry
In different ways, both Euclid (with his “horn angles”) and Newton (with his “angles of contact”) had a more general concept of angle than we do today. Giving a measure to these angles, as well as being of interest in its own right, leads to infinitesimal numbers.
The Perimeter of a Polyomino and the Surface Area of a Polycube
Wiley Williams and Charles Thompson
In this note, we develop a formula for the perimeter of a polyomino in terms of the number of tiles and interior vertices, and also a formula for the surface area of a polycube in terms of the number of cubes, interior edges, and interior vertices..
The Cross Product as a Polar Decomposition
The cross product of two vectors a and b in 3-space can be given as a product Tab, where Ta is a matrix that depends only on a. In this note, we show that this result is a natural consequence of a “polar decomposition” of the matrix Ta.
The Right Theta
William Freed and Athanasios Tavouktsoglou
The formula θ = arctan (y/x) gives the angle associated with a point (x,y) in the plane, valid for |θ| < π/2. This note presents a formula that is valid for |θ| < π.
College Mathematics Journal Homepage
Read the the College Mathematics Journal online. (This requires MAA Membership.)