Number Theory Through Inquiry is an innovative textbook that leads students on a carefully guided discovery of introductory number theory. The book has two equally significant goals. One goal is to help students develop mathematical thinking skills, particularly, theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for an independent study experience, or for a course designed as an introduction to abstract mathematics. Mathematics or related majors, future teachers, and students or adults interested in exploring mathematical ideas on their own will enjoy Number Theory Through Inquiry.
Number theory is the perfect topic for an introduction-to-proofs course. Every college student is familiar with basic properties of numbers, and yet the exploration of those familiar numbers leads us to a rich landscape of ideas. Number Theory Through Inquiry contains a carefully arranged sequence of challenges that lead students to discover ideas about numbers and to discover methods of proof on their own. It is designed to be used with an instructional technique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). Instructors’ materials explain the instructional method. This style of instruction gives students a totally different experience compared to a standard lecture course. Here is the effect of this experience: Students learn to think independently. They learn to depend on their own reasoning to determine right from wrong. They develop the central, important ideas of introductory number theory on their own. From that experience, they learn that they can personally create important ideas. They develop an attitude of personal reliance and a sense that they can think effectively about difficult problems. These goals are fundamental to the educational enterprise within and beyond mathematics.
Table of Contents
1. Divide and Conquer
2. Prime Time
3. A Modular World
4. Fermat’s Little Theorem and Euler’s Theorem
5. Public Key Cryptography
6. Polynomial Congruences and Primitive Roots
7. The Golden Rule: Quadratic Reciprocity
8. Pythagorean Triples, Sums of Squares, and Fermat’s Last Theorem
9. Rationals Close to Irrationals and the Pell Equation
10. The Search for Primes
A Mathematical Induction: The Domino Effect
About the Authors
Excerpt: Ch. 4 Fermat's Little Theorem and Euler's Theorem (p. 53)
Abstracting the Ordinary
One way that mathematics is created is to abstract, change, or generalize some features of familiar mathematical objects and see what happens. For example, we started with the familiar idea of arithmetic with integers and then made some changes to consider modular arithmetic, a sort of cyclical version of arithmetic. Abstract algebra is a mathematical exploration of generalizations of various familiar ideas such as the integers, the rational numbers, the real numbers and their associated arithmetic operations and properties. By selectively focusing on some properties of these examples, abstract algebra constructs categories of algebraic entities including objects called groups, rings, and fields. Modular arithmetic provides us with examples of some of these algebraic structures and illustrates some of the properties that lead to many fundamental ideas in abstract algebra.
Solving the linear congruence
αx ≡ b (mod n)
means finding a number that when added to itself a times results in b modulo n. In studying such congruences we are implicitly studying the results of repeated addition modulo n and patterns that this process might produce. Equally interesting, as well as fruitful, is the study of repeated multiplication modulo n, that is, taking powers of numbers and reducing those powers modulo n. The operations of addition and multiplication are so well understood in the natural numbers that looking at their behavior in modular arithmetic is a natural exploration to undertake.
About the Authors
David Marshall was born in Anaheim, California and spent most of his early life in and around Orange County. After receiving a bachelor’s degree in mathematics from California State University at Fullerton he left the Golden State to attend graduate school at the University of Arizona. David received his Ph. D. in mathematics in 2000, specializing in the field of algebraic number theory. He held postdoctoral positions at McMaster University in Hamilton, Ontario and The University of Texas at Austin before becoming an Assistant Professor at Monmouth University in West Long Branch, New Jersey. David has been an active member of the MAA and AMS for over 10 years and currently serves as the Program Editor for the MAA’s New Jersey Section.
Edward Odell was born in White Plains NY. He attended The State University of New York at Binghamton as an undergraduate and received his Ph.D. from M.I.T. in 1975. After teaching 2 years at Yale University he joined the faculty at The University of Texas at Austin where he has been since 1977, currently as The John T. Stuart III Centennial Professor of Mathematics. He is an internationally recognized researcher in his area, the geometry of Banach spaces, and is a much sought after speaker. He was an invited speaker at the 1994 International Congress of Mathematicians in Zurich. He has given series of lectures at various venues in Spain and recently at the Chern Institute in Tianjin, China. He is the co-author of Analysis and Logic and the co-editor of two books in the Springer Lecture Note series. He is a member of the M.A.A. and the A.M.S.
Michael Starbird is Professor of Mathematics and a University Distinguished Teaching Professor at The University of Texas at Austin. He received his B.A. degree from Pomona College in 1970 and his Ph.D. in mathematics from the University of Wisconsin, Madison, in 1974. That same year, he joined the faculty of the Department of Mathematics of The University of Texas at Austin, where he has stayed except for leaves as a Visiting Member of the Institute for Advanced Study in Princeton, New Jersey; a Visiting Associate Professor at the University of California, San Diego; and a member of the technical staff at the Jet Propulsion Laboratory in Pasadena, California. He served as Associate Dean in the College of Natural Sciences at UT from 1989 to 1997.
Starbird is a member of the Academy of Distinguished Teachers at UT. He has won many teaching awards, including the 2007 Mathematical Association of America Deborah and Franklin Tepper Haimo National Award for Distinguished College or University Teaching of Mathematics; a Minnie Stevens Piper Professorship, which is awarded each year to 10 professors from any subject at any college or university in the state of Texas; the inaugural award of the Dad’s Association Centennial Teaching Fellowship; the Excellence Award from the Eyes of Texas, twice; the President’s Associates Teaching Excellence Award; the Jean Holloway Award for Teaching Excellence, which is the oldest teaching award at UT and is presented to one professor each year; the Chad Oliver Plan II Teaching Award, which is student-selected and awarded each year to one professor in the Plan II liberal arts honors program; and the Friar Society Centennial Teaching Fellowship, which is awarded to one professor at UT annually. Also, in 1989, Professor Starbird was the Recreational Sports Super Racquets Champion.
Starbird’s mathematical research is in the field of topology. He has served as a member-at-large of the Council of the American Mathematical Society and on the national education committees of both the American Mathematical Society and the Mathematical Association of America. He has given more than 150 invited lectures at colleges and universities throughout the country and more than 20 minicourses and workshops to mathematics teachers.
Starbird strives to present higher-level mathematics authentically to students and the general public and to teach thinking strategies that go beyond mathematics as well. With those goals in mind, he wrote, with co-author Edward B. Burger, The Heart of Mathematics: An invitation to effective thinking, which won a 2001 Robert W. Hamilton Book Award. Burger and Starbird have also written a book that brings intriguing mathematical ideas to the public, entitled Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas, published by W.W. Norton, 2005. Starbird has produced four courses for The Teaching Company: Change and Motion: Calculus Made Clear (1st edition, 2001, and 2nd edition, 2007); Meaning from Data: Statistics Made Clear, 2005; What are the Chances? Probability Made Clear, 2007; and, with collaborator Edward Burger, The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas, 2003. These courses bring an authentic understanding of significant ideas in mathematics to tens of thousands of people who are not necessarily mathematically oriented. Starbird loves to see real people find the intrigue and fascination that mathematical thinking can bring.