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Number Theory: A Lively Introduction with Proofs, Applications, and Stories, by James Pommersheim, Tim Marks, Erica Flapan
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Number Theory: A Lively Introduction with Proofs, Applications, and Stories, is a new book that provides a rigorous yet accessible introduction to elementary number theory along with relevant applications.
Readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. Many theorems are preceded by Numerical Proof Previews, which are numerical examples that will help give students a concrete understanding of both the statements of the theorems and the ideas behind their proofs, before the statement and proof are formalized in more abstract terms. In addition, many applications of number theory are explained in detail throughout the text, including some that have rarely (if ever) appeared in textbooks.
A unique feature of the book is that every chapter includes a math myth, a fictional story that introduces an important number theory topic in a friendly, inviting manner. Many of the exercise sets include in-depth Explorations, in which a series of exercises develop a topic that is related to the material in the section.
- Sales Rank: #201650 in Books
- Brand: Pommersheim, James E./ Marks, Tim K./ Flapan, Erica L.
- Published on: 2010-02-15
- Original language: English
- Number of items: 1
- Dimensions: 9.39" h x 1.15" w x 7.81" l, 2.65 pounds
- Binding: Hardcover
- 784 pages
From the Back Cover
Can learning math actually be fun?
Number Theory: A Lively Introduction with Proofs, Applications, and Stories proves it can!
Number Theory: A Lively Introduction with Proofs, Applications, and Stories will introduce you to elementary number theory, helping you develop your proof writing skills while learning the core concepts in number theory as well as advanced topics used in modern applications.
To help you succeed, you'll learn "math myths"—fictional stories starring famous mathematicians that illustrate important number theory topics in a friendly, inviting manner that's easy to understand and remember. "Numerical Proof Previews" give you a concrete way to understand the key ideas of a proof. Illustrations throughout the text add to the fun and your understanding.
As accessible as the book is, it does not skimp on the serious mathematics. You'll explore:
- Methods of proof and the basic properties of the integers, including mathematical induction
- Central concepts and theorems in elementary number theory using a developmental approach, including basic divisibility properties, prime numbers, the Euclidean Algorithm, modular arithmetic (including applications), Fermat's Little Theorem, Euler's Theorem, and RSA (public-key) encryption
- Primitive roots, primality testing, quadratic reciprocity, Gaussian integers, Fermat's Two Squares Theorem, continued fractions, Pell's Equation, and Fermat's Last Theorem, and much more
- Ideal for first-timers in number theory as well as math majors and math education majors, Number Theory: A Lively Introduction with Proofs, Applications, and Stories�makes learning number theory a joy!
About the Author
James Pommersheim is the Katharine Piggott Professor of Mathematics at Reed College. He held post-doctoral positions at the Institute for Advanced Study, M.I.T., and U.C. Berkeley, and he served on the mathematics faculty at New Mexico State University and Pomona College. Pommersheim has published research papers in a wide variety of areas, including algebraic geometry, number theory, topology, and quantum computation. He has enjoyed teaching number theory to students at many levels: college math and math education students, talented high school students, and advanced graduate students. Pommersheim earned his Ph.D. from the University of Chicago, where he studied under William Fulton.
Tim Marks is a Research Scientist at Mitsubishi Electric Research Laboratories in Cambridge, Massachusetts. Previously, he taught high school mathematics and physics for three years, and he worked for three years as a mathematics textbook editor at McDougal Littell/ Houghton Mifflin. Marks and Pommersheim have taught number theory at the Johns Hopkins University's Center for Talented Youth (CTY) summer program for 19 years. Marks earned his A.B. degree from Harvard University. He completed his Ph.D. and postdoctoral research at the University of California, San Diego.
Erica Flapan joined the mathematics department at Pomona College in 1986. She held post-doctoral positions at Rice University and the University of California, Santa Barbara. Flapan has taught a wide range of mathematics courses and has numerous publications in topology and its applications to chemistry, including her book When Topology Meets Chemistry. She developed a course entitled "Problem Solving in the Sciences,” which aims to teach students the mathematics they need in order to succeed in science and economics. Flapan obtained her Ph.D. in knot theory from the University of Wisconsin and is the Lingurn H. Burkhead Professor of Mathematics at Pomona College.
Most helpful customer reviews
16 of 16 people found the following review helpful.
Genuinely funny! (Not to mention clear and well-written.)
By Mike
This is an extraordinary book. A model of clarity, painstaking attention to detail, and all of it in a light-hearted, engaging manner. With cartoons! That are actually funny!
I compared this book's TOC to the catalog description of the Number Theory course at my school. It contains every topic in our syllabus. So it passes my first test.
Every dozen pages or so, one comes across a real gem. For example, the book discusses the AKS primality test and notes the significant contributions of undergraduates to that breakthrough. The illustrations of pineapples exhibiting Fibonacci numbers are both beautiful and illuminating.
Do not skip the "Math Myths"! They are not mere add-ons. They introduce the topics well, via engaging and well-thought-through analogies, and they are genuinely entertaining. The puns are of Daily-Show-headline calibre. It is one thing to ask a class for their ideas on how to solve the linear Diophantine equation aX+bY=1. It is another thing altogether to introduce the topic by asking the class to help Diophantus the "farmer" weigh potatoes using only a balance scale and the bricks he has on hand. The context provided by the latter provides the traction needed for students to *think* about the question for themselves.
I think this book would be especially well-suited to schools that offer Number Theory as a sophomore-level course, where students have not yet had experience with writing abstract proofs. No other book on this subject will soften that shock nearly as well, helping students make the transition to advanced mathematics by spelling out in detail the habits of mathematical thinking that we practioners of math tend to simply perform without explanation as autonomously as tying our shoes.
I will certainly be using this book for my class the next time I teach Number Theory.
26 of 29 people found the following review helpful.
monumental work of patience and love
By electron0511
When I first started reading this book, I was almost brought to tears at the appallingly elementary level that the book starts at: is the average American College Student so ignorant, unprepared, and childish to the point that one must start by teaching them material that would be covered in primary school in Asia and Europe, in a tone appropriate for kindergarten? However, as I kept on reading, I was amazed at the care and thought that the authors put into this book to keep the reader entertained and engaged, while at the same time making sure that no intermediate step, however small, was ever overlooked so that even the most unprepared students could be brought up to speed (as long as they work through the book diligently).
Starting from the Pythagorean theorem and the irrationality of the square root of 2, by the end of the book the reader will be able to understand the proof of Fermat's Last Theorem for the case n=4. Proofs of theorems are always preceded by numerical examples to give the reader a feel of the logic being used. Detailed instructions on how to perform proofs (including some basic English grammar) are also provided so that the reader can write proofs themselves. Various real world applications of the material are provided showing the reader the relevance of it all.
The problems are also very entertaining. The lack of solutions is not a drawback since for most problems one can easily tell whether the solution is correct or not by simply checking to see if one's solution satisfies the condition provided in the problem. Also, the joy of math is in trying to figure out the solution by oneself. If you cannot figure out an answer, just keep on thinking. You will learn nothing by trying to peek at the answer and trying to memorize it.
The authors' patience and their love of the field emanates from every page. The book is so well written that I would say that a gifted 6th or 7th grader will be able to understand it. This is a monumental piece of work that all future textbooks in math should be judged against. If you are interested at all in number theory, this book is a must have.
Postscript: as is often the case with first-editions, there are a few unfortunate typos which may be immensely confusing to the reader:
page 25, second equation: the left-hand-side should be x
page 29, first equation: the left-hand-side should be 2
page 275, table: the numbers in the second row under 5 and 8 should be 2
Hope these will be corrected in the second printing.
16 of 17 people found the following review helpful.
One of the most helpful guides I have read.
By Alex
Though it starts out at a nearly absurdly fundamental level, the authors do a fantastic job at guiding the reader through the material, guaranteeing that almost no prior knowledge is necessary to get the full experience. The stories and historical anecdotes, true or not, are almost a reason to read the book by themselves. Like one of the other reviewers said, an advanced middle-schooler would be able to go through this book and understand it, as would any undergrad who has yet to take number theory. The examples and exercises are fun and engaging even to someone who has learned the subject.
Be warned, however, that the book is oriented at those who want to learn the subject, and who are willing to take the time for a deep understanding. The lack of answers is not an accidental omission, it is meant to force readers to think through the problems and come up with their own solution. The book teaches not only number theory, but mathematical reasoning and logic as well.
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