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Vol. 40. No. 5, pp. 322-399

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ARTICLES

**Fundamental Theorems of Algebra for the Perplexes**

Robert Poodiak, Kevin LeClair

The fundamental theorem of algebra for the complex numbers states that a polynomial of degree *n* has *n* roots, counting multiplicity. This paper explores the *perplex number system* (also called the *hyperbolic number system* and the *spacetime number system*) In this system (which has extra roots of +1 besides the usual ±1 of the reals) every degree-*n* polynomial has exactly *n*^{2} roots. In the case of multiple roots, we find a satisfying method of assigning multiplicity.

**The Draining Cylinder**

James Graham-Eagle

This article explores the time it takes for a liquid to drain from a cylindrical container through a hole in the bottom. Using dimensional analysis and some thought experiments this time is determined and Torricelli's law derived as a consequence. Finally, the effect of pouring liquid into the container as it drains is considered.

**How to Differentiate an Integer Modulo ***n*

Caleb Emmons, Mike Krebs, and Anthony Shaheen

A number derivative is a numerical mapping that satisfies the product rule. In this paper, we determine all number derivatives on the set of integers modulo *n*. We also give a list of undergraduate research projects to pursue using these maps as a starting point.

**Minimal Solutions to the Box Problem**

Jer-Chin Chuang

The *box problem* from introductory calculus seeks to maximize the volume of a tray formed by folding a strictly rectangular sheet from which identical squares have been cut from each corner. In posing such questions, one would like to choose integral side-lengths for the sheet so that the excised squares have rational or integral side-length. Building on the work of Philip Hotchkiss that earlier appeared in this JOURNAL, we find the dimensions of the smallest sheet with distinct integral side-lengths such that the side-lengths of the excised squares are rational or integral.

**Dynamics of Exponential Functions**

Jiu Ding, Zizhong Wang,

Using calculus, we explore the long term dynamical behavior of exponential functions under iteration for all initial points.

**CLASSROOM CAPSULES**

**Correlation of the Union of Two Bivariate Data Sets**

Robert A. Fontenot

Given two bivariate data sets with the same nonnegative correlation, such that one data set has larger means for both variables, the problem solved here is to ?nd (additional) conditions guaranteeing that the union of the two data sets has correlation greater than the common correlation of the two data sets.

**On the Remainder in Taylor's Theorem**

Eli Leher, Lior Bary-Soroker

The Taylor polynomial is used to approximate a real function in the neighborhood of a point. Here we give a short straightforward proof for the bound of the remainder term of this approximation. The proof uses only induction and the fact that non-negativity of the derivative implies monotonicity of the function, thus it is suitable for presentation to undergraduates.

**PROBLEMS AND SOLUTIONS**

**BOOK REVIEWS**

*Strange Attractors, Poems of Love and Mathematics*

Edited by Sarah Glaz and JoAnne Growney

Reviewed by: Deborah Bacharach

**MEDIA HIGHLIGHTS**

**REFEREES IN 2008 **

**ADDITIONS, CORRECTIONS, EMENDATIONS, AND REVISIONS**