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Vol. 40, No. 5, November 2009 i/J^mS ??_ vlHr I COLLEGE_ MATHEMATICS_ JOURNAL_ Bathsheba Grossman: MoonPi IN THIS ISSUE: The Perplex Numbers Draining Cylinders Dynamics of Exponential Functions Differentiating Integers (mod n) An Official Publication of the Mathematical Association of America This content downloaded from 185.205.48.180 on Tue, 15 Nov 2022 21:04:11 UTC All use subject to https://about.jstor.org/terms
EDITORIAL POLICY The College Mathematics Journal serves all who are interested in college mathematics. CMJ seeks lively, well-motivated articles that will enrich undergradu ate instruction and enhance classroom learning, as well as expository papers that stimulate the thinking and broaden the perspectives of those who teach undergraduate-level mathematics. Articles involving all aspects of mathematics are welcome, including his tory, philosophy, problem solving, pedagogy, applica tions, computation, and so on. Emphasis is placed on topics taught in the first two years of college. Contribu tions from teachers in high schools, two-year colleges, four-year colleges, and universities are welcome. In addition to articles, CMJ publishes short pieces containing material suitable for immediate classroom use, problems, solutions to published problems, stu dent research projects, media reviews, and all sorts of mathematical ephemera: proofs without words, ar resting examples of fallacious proofs, mistakes and/or other mathematical anomalies, quotations, poetry, hu mor, cartoons, and doodles. Letters to the Editor on any topic are also welcome, as are all kinds of com ments, criticisms, and suggestions for making CMJ more lively, entertaining, and informative. SUBMITTING AN ARTICLE Manuscripts should be submitted electronically. Please send a pdf file attached to an email to [email protected]. The author's name(s) must not appear in this file, but should be listed in the accompa nying e-mail, along with the postal address of a desig nated corresponding author. In lieu of a pdf, an author may submit a single paper copy of the manuscript, printed on only one side of the paper. Note: The only electronic file format accepted is pdf. Authors who use T^X are asked to use a standard environment with no custom formatting. A T^X file will be requested only if a manuscript is accepted for publication. Contributions to the sections Classroom Capsules, Problems, Solutions, Student Research Projects, and Media Highlights should be submitted to the ap propriate section editor at the address listed inside the journal. All other submissions should be sent to [email protected]. For more information about writing for CMJ, please visit the web site at http://www.maa.org/pubs/ cmj.html and follow the links to Information for Authors and Electronic Production Information. The COLLEGE MATHEMATICS JOURNAL (ISSN 0746-8342) is published by the Mathematical Asso ciation of America at 1529 Eighteenth Street, N.W., Washington, D.C. 20036 and Montpelier, VT, monthly in January, March, May, September, and November. Subscription correspondence and notice of change of address should be sent to the Membership/ Subscriptions Department, Mathematical Association of America, at the address above. Microfilmed issues may be obtained from University Microfilms Interna tional, Serials Bid Coordinator, 300 North Zeeb Road, Ann Arbor, Ml 48106. Advertising correspondence should be addressed to MAA Advertising 1529 Eighteenth St. NW Washington DC 20036 Phone: (866) 821-1221 Fax: (866) 387-1208 E-mail: [email protected] Further advertising information can be found online at www.maa.org. Permission to make copies of individual articles, in pa per or electronic form, including posting on personal and class web pages, for educational and scientific use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear the following copyright notice: Copyright the Mathematical Association of America 2009. All rights reserved. Abstracting with credit is permitted. To copy other wise, or to republish, requires specific permission of the MAA's Director of Publications and possibly a fee. Periodicals postage paid at Washington, DC, and ad ditional mailing offices. Postmaster: Send address changes to Membership/ Subscription Department, Mathematical Association of America, 1529 Eighteenth Street, N.W., Washing ton, D.C. 20036-1385 Printed in the United States of America ABOUT THE COVER Bathsheba Grossman is a self-made, self-invented artist. Her medium, 3D print ing in metal, did not exist when she graduated from Yale College in 1988 or when she completed graduate work at the University of Pennsylvania five years later. She describes her work as being "about life in three dimensions: working with symmetry and balance, getting from a zero point to infinity, and always finding beauty in geometry," or "visions of order in the universe" MoonPi is a puzzle-sculpture in seven parts. Three pieces are 3D prints in steel/bronze composite metal. They interlock with four 3/8" rubber spheres. For more on this piece and picture of it disassembled, see page 344. This content downloaded from 185.205.48.180 on Tue, 15 Nov 2022 21:04:11 UTC All use subject to https://about.jstor.org/terms
Vol. 40, No. 5, November 2009 THE COLLEGE MATHEMATICS JOURNAL Editor Michael Henle, Oberlin College, Oberlin OH 44074 Sally Moffitt, Editorial Assistant Board of Editors Ricardo Alfaro, University of Michigan–Flint, Flint MI 48502 Ed Barbeau, University of Toronto, Toronto, Ontario, Canada M5S 3G3 sarah-marie belcastro, Sarah Lawrence College, Bronxville NY 10708 Curtis Cooper, University of Central Missouri, Warrensburg MO 64093 Susan Goldstine, St. Mary’s College of Maryland, St. Mary’s City MD 20686 Lixing Han, University of Michigan–Flint, Flint MI 48502 Reuben Hersh, University of New Mexico, Albuquerque NM 87131 Donald E. Hooley, Bluffton University, Bluffton OH 45817 Heather A. Hulett, University of Wisconsin–La Crosse, La Crosse WI 54601 Michael A. Jones, Mathematical Reviews, Ann Arbor MI 48103 Gary Kennedy, The Ohio State University-Mansfield, Mansfield OH 44906 Dan King, Sarah Lawrence College, Bronxville NY 10708 Warren Page, 30 Bonnie Way, Larchmont NY 10538 Cecil Rousseau, University of Memphis, Memphis TN 38152 Kenneth E. Schilling, University of Michigan–Flint, Flint MI 48502 Brigitte Servatius, Worcester Polytechnic Institute, Worcester MA 01609 Shing So, University of Central Missouri, Warrensburg MO 64093 Todd G. Will, University of Wisconsin–La Crosse, La Crosse WI 54601 Harry Waldman, Managing Editor
Fundamental Theorems of Algebra for the Perplexes Robert D. Poodiack and Kevin J. LeClair doi:10.4169/074683409X475643 Rob Poodiack ([email protected]) received his Ph.D. in analysis from the University of Vermont in 1999. He is an associate professor of mathematics at Norwich University in Northfield, VT, where he received the Homer L. Dodge Award for Excellence in Teaching in 2005. His research interests are in analysis and the use of technology in the mathematics classroom. He is the Chair-elect of the MAA’s Northeastern Section. He enjoys training to run a half- marathon, and organizing Norwich’s annual Integration Bee. Kevin LeClair ([email protected]) graduated with honors from York (ME) High School in 2003. He graduated from Norwich University, America’s oldest private military college, in 2007 with a degree in mathematics and a minor in business administration. He was a member of Norwich’s Corps of Cadets, completing four years of Reserve Officer Training Corps (ROTC) training. He lives in York Beach, ME and works in the insurance industry. This paper is based on his senior seminar project supervised by Prof. Poodiack. The perplex numbers P (also called the hyperbolic numbers [6, 7], the spacetime num- bers [2, 3], and sometimes the split-complex numbers [8]) are, like the complexes, a two-dimensional number system over the reals. Every perplex number z has the form z = t + xh, where t and x are real numbers. But h, rather than being a square root of minus 1, is a square root of plus 1, an extra such root, supplementing ±1, the preexist- ing, well-known, customary and usual, real square roots of 1. The perplex numbers tend to be rediscovered every few years and put to various uses. They are related, for example, to the hyperbolic geometry Einstein used to define special relativity. As Sobczyk [6] argues, they should get more attention from mathe- maticians and, in particular, deserve to be taught to undergraduates. In this article, we review the basic properties of the perplex numbers, and then state and prove a fundamental theorem of algebra for them. In fact, rather surprisingly, we have a whole series of fundamental theorems for this intriguing, relatively obscure number system. Properties of the perplex numbers If z = t + xh is a perplex number, then t is called the real part of z and x is called the hyperbolic part. Alternatively, t is often referred to as the time component, and x the space component [2]. (Fjelstad [5] dubbed x the hallucinatory part.) Given perplex numbers z1 = t1 + x1h and z2 = t2 + x2h, we have the basic opera- tions: 322 !c THE MATHEMATICAL ASSOCIATION OF AMERICA