Book Description
This book introduces the theory of modular forms with an eye toward the Modularity Theorem. All rational elliptic curves arise from modular forms. The topics covered include: elliptic curves as complex tori and as algebraic curves, modular curves as Riemann surfaces and as algebraic curves, Hecke operators and Atkin-Lehner theory, Hecke eigenforms and their arithmetic properties, the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms, elliptic and modular curves modulo p and the Eichler-Shimura Relation, the Galois representations associated to elliptic curves and to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory. A First Course in Modular Forms is written for beginning graduate students and advanced undergraduates. It does not require background in algebraic number theory or algebraic geometry, and it contains exercises throughout.
Average customer rating:
- Delightful excursion in thinking about how to think
- A great primer and reference to fall back on
- Interesting & valuable, though philosophical > statistical
- not for the technically minded
- Great treatise on critical thinking and organization
|
Turning Numbers into Knowledge: Mastering the Art of Problem Solving
Jonathan G. Koomey
Manufacturer: Analytics Press
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Beautiful Evidence
ASIN: 0970601905 |
Book Description
Mastering the art of problem solving takes more than proficiency with basic calculations; it requires understanding how people use information, recognizing the importance of ideology, learning the art of storytelling, and acknowledging the important distinction between facts and values. Intended for professors, managers, entrepreneurs, and students, this guide addresses these and other essential skills. With clear prose, quotations, and exercises for solving problems in the real world, this book serves as an ideal training manual for those who are new to or intimidated by quantitative analysis and an excellent refresher for those who have more experience but want to improve the quality of their data, the clarity of their graphics, and the cogency of their arguments.
Customer Reviews:
Delightful excursion in thinking about how to think.......2007-04-20
It is different from, and for many purposes, better than a science textbook. More than enough science books have been written, but TNIK is better because it teaches readers how to think about the data on which science is built. Its fresh approach to understanding the natural world as well as human-made systems is a noteworthy improvement over the plug-in, grind-out perspective that academic classes typically offer and that turns off students.
A great primer and reference to fall back on.......2005-11-04
While no doubt I've heard many of nuggets contained in the book over the course of my high school and college days, I found Koomey's book a pleasurable read and useful synthesis of approaches and tips for completing quality research and analyses. Internalizing Koomey's advice is going to help most readers be more discriminating consumers of published research and better authors of their own research. It's a reference source I've already gone back to myself in just a few weeks and a great training resource for new consultants my company hires.
Interesting & valuable, though philosophical > statistical .......2005-02-01
I expected the author to talk much about statistical analysis and related "technical stuff". I had been very wrong. In fact, the book can be regarded as a warning to common people about the "irrelevancy" and "inaccuracy" of data or information we encounter or process so that we can perform better analysis of on our own. As from pg 197, "of primary importance from this book are the following lessons:-"
- Don't be intimidated by anyone (esp those know-it-alls)
- Be a critical thinker
- Don't confuse what's countable with what really counts
- Get organized
- Question authority
- Dig into the numbers
- Focus on the essential
- Document, document, document
- Use the internet
- Remember that others don't care as much about your work as you do
- Synthesis follows analysis
In short, a good read. Dont miss it.
p.s. I like the following quotes from the book very much. (The author did use over 31 quotes with at least one for each chapter)
Just because I use a study to refute another study does not mean my study is right. It just means I believe it. Caveat Emptor. - Cynthia Crossen
Whether or not someone else knows it all isn't really relevant; the only thing that's relevant is what you know and what you do. - Robert Ringer
not for the technically minded.......2003-01-18
This is an entertaining and well written book on some of the do's and don'ts of data analysis. To quote from Dr. Beers review below, "The main emphasis is on the art of data interpretation." Indeed there are useful tools here for performing sanity checks and for asking critical questions about all sorts of data collections. ... The examples are, at best, sketchy and few in number. The anectodes are amusing but not terribly informative. I would have much preferred more concrete examples and further discussion on some technical matters. ....
Great treatise on critical thinking and organization.......2002-12-18
"Turning Numbers Into Knowledge: Mastering the Art of Problem Solving" should be required reading for anyone engaged in producing, reading, or analysing information. Based on the title one might assume that I mean numerical information, but that is not the case at all. The basic principles, such as how to sift through information and the importance of documentation of sources, are important parts of any information product. In fact, except for the sections on graphs, tables, normalizing data and a few others, the rest of the book (fully at least three quarters of it) is dedicated to determining what constitutes good information, good techniques, good analysis, good documentation, etc. This is a book on problem solving techniques and analysis of the information products of others.
Filled with useful tools and tips for problem solving under real-life situations it is one of the most useful books available. "Turning Numbers Into Knowledge: Mastering the Art of Problem Solving" is a masterful work in the area of critical analysis and a highly recommended read for anyone involved in creating or using information of any kind.
Book Description
English translation of standard mathematical work on theory of numbers, first published in Latin in 1801. "Among the greatest mathematical treatises of all fields and periods."--Asger Aaboe.
Customer Reviews:
understandable to all.......2007-03-09
What blows my mind about Gauss's Disquisitiones is that it is understandable to people like me who don't have an advanced background in mathematics. You just need to move through it slowly and carefully. There is a certain joy that comes along with making the discoveries that Gauss walks you through. A book that will change your outlook on life by enhancing what you see as the basic philosophy underlying numbers and mathematics. Mathemtical magicians like Euler and Lagrange will just confuse and dazzle you with their tricks, but Gauss will give you understanding.
Martin Christensen"s Review.......2007-01-08
This book is very good, but not quite excellent. Gauss spent, in my opinion, too much time on the theory of binary forms and too little on the general second-degree equation in two variables that the theory is such a big part of. However, the rest of the book was first-rate, well suited to one of the greatest mathematicians of all time. I considered it money well spent.
Choose your edition carefully.......2006-03-08
This is a great book. It's the place where modern number theory begins. It's also well enough written that it's enjoyable to read today.
There are two editions and you have to choose carefully. I don't know whether my review will appear on both editions or just one. The softcover is only $47. The hardcover is $129, but it is a REVISED translation. A single person, who does not seem to be have been a mathematician, made the first translation from Latin in 1965. That was revised by a team of 4 scholars in 1986.
That turns out to be important, because the original translator got a few things wrong, like the logic of a double negative. So there are some places where the first (cheaper, softcover, Yale) edition is either wrong or unclear. Many of these problems have been fixed in the second (more expensive, hardcover, Springer-Verlag) edition.
I bought the first edition and I have no regrets, but you will get extra value for your extra money if you buy the second edition.
Not bad for 204 years old........2005-07-10
Gauss wrote Disquisitiones between 1795 and 1801. If I can read and understand Clarke's translation in less time than it took Gauss to write the original, I will be doing well. The mathematics has not been superseded and the historical aspect is extremely valuable.
Professor.......2003-09-13
It is the best book for people who takes mathematics seriously
Book Description
I used to think math was no fun
'Cause I couldn't see how it was done
Now Euler's my hero
For I now see why zero
Equals e
[pi] i+1
--Paul Nahin, electrical engineer
In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formula--long regarded as the gold standard for mathematical beauty--and shows why it still lies at the heart of complex number theory.
This book is the sequel to Paul Nahin's An Imaginary Tale: The Story of I [the square root of -1], which chronicled the events leading up to the discovery of one of mathematics' most elusive numbers, the square root of minus one. Unlike the earlier book, which devoted a significant amount of space to the historical development of complex numbers, Dr. Euler begins with discussions of many sophisticated applications of complex numbers in pure and applied mathematics, and to electronic technology. The topics covered span a huge range, from a never-before-told tale of an encounter between the famous mathematician G. H. Hardy and the physicist Arthur Schuster, to a discussion of the theoretical basis for single-sideband AM radio, to the design of chase-and-escape problems.
The book is accessible to any reader with the equivalent of the first two years of college mathematics (calculus and differential equations), and it promises to inspire new applications for years to come. Or as Nahin writes in the book's preface: To mathematicians ten thousand years hence, "Euler's formula will still be beautiful and stunning and untarnished by time."
Customer Reviews:
An excellence introductory book on advanced mathematics such as Euler's Identity, Irrationalioty, Fourier Series.......2007-09-22
The primary topic of Nahin's "Dr. Euler's Fabulous Formula" is the complex number or more appropriately the Euler's identity: e power to (it) = cos(t) + isin(t). Nahin called this book the second half of his complex number series. The first book in the series is named "An Imaginary Tale: The Story of square root of minus one." The second book is called "Dr. Euler's Fabulous Formula." The primary topics of the second book are: Fourier series, which is covered on Chapter 4; Fourier Integrals on Chapter 5; the application of complex numbers on electronics Chapter 6.
The book has six chapters, which contains both pure and applied mathematics materials. Other than the three chapters mentioned above, the other three chapters are (i) Complex Numbers, (ii) Vector Trips, and (iii) The Irrationality of pi square. Chapter one is about the assortment of non elementary complex numbers such as applying complex number on obtaining the sum of a real series. Chapter three provides a detail proof of the irrationality of the number pi square using Euler's Identity. On the applied side: Chapter two demonstrates the application of complex number on mathematical modeling. Since Nahin is an eminent electrical engineering professor, his book also provides plenty of material on (a) partial differential equations (PDE) such as wave equation on chapter four, and (b) electrical engineering material such as baseband, carrying frequencies, antennas, radio receivers and speech scrambler on chapter six.
This is an excellence introductory book not only on pure complex numbers usage in mathematics such as summing a series but also on the usage of PDE, Fourier series, and Fourier Integral in physics and engineering.
Good clear explanation of Fourier series.......2007-04-11
Dr Eulers fabulous formula fits a niche between books for non mathematicians (too simple) and books only understood by mathematicians. It provides the best explanation of Fourier series and integrals that I have read. Its explanation of imaginary numbers is excellent, but not as good as Feynman in his lectures on physics. I reccomend it for those who want to understand how Fourier series work.
excellent for fourrier series and fourrier transform exposition.......2007-03-29
A very readable book. Many concepts developed around Euler's magic formula are clearly explained. Including a lucid exposition on the calculus of the sum of classical series such as the value of zeta function for several positive integer values of its argument. Paul Nahin excels in describing the origin and the development of fourrier series and fourrier integrals from Bernoulli to Fourrier and more. Anyone interested in this field will find something interesting in this book to learn. The reason I didn't rank it five stars is that I found explanations often too lengthy while the addition of a chapter on distribution theory could fill the gaps in mathematical rigor and make the transition from fourrier series to fourrier integrals more logical. I should add that the lack of rigor in transition from fourrier series to fourrier integrals, as described by P. Nahin, is inherent to the more fundamental problem of transition from discrete to continuous. Indeed, in mathematics, this is a very slippery terrain. In functional analysis, mathematicians go round this problem by introducing distribution theory. P. Nahin mentions only the name of distribution theory without any decription. I think a chapter on this theory would make the book a must have.
Excellent expository book.......2007-03-25
Paul Nahin's book, "Dr. Euler's Fabulous Formula," is an excellent expository treatment of Euler's formula (you say, "which one?") e^i*theta = cos(theta) + i*sin(theta) and its profound, and far-reaching, ramifications. Dr. Nahin also gives an extensive informal discussion of Fourier series, Fourier transforms, the Dirac Delta Function, and what electrical engineers would call "signals and systems theory." Some mathematical purists may criticize the lack of pure rigor. However, this book is an "expository" book, not a rigorous "textbook." Ideally, I recommend that you read Dr. Nahin's book in conjunction with your standard college textbook. That way, you will get the best of both worlds. Your textbook will give you the disciplined rigor. Dr. Nahin's book will give you the "Aha... insight!" I read Dr. Nahin's book before taking a graduate level course in electrical engineering (EE) Signals and Systems. I breezed through the EE course with perfect scores on my exams, and I give a lot of credit to Dr. Nahin. When you study mathematics, you really need BOTH disciplined mathematical rigor AND intuitive insight and understanding. Beware, however, that this book has LOTS of mathematics in it. The book is loaded with serious mathematics. Don't read this book if you want something for the intelligent layperson. Read this book if you love mathematics, if you are an engineering or mathematics student, or if you like industrial-strength mathematics. Paul Nahin may single-handedly save Americans from mathematical illiteracy. He does something that the mathematical community does not do well... "market and sell" mathematics.
Errata please.......2007-02-14
Like all of Paul Nahin's books, I really like this one.
However, as with so many books an Errata would help. Mathematical and mathematical finance books are getting so expensive, that unless authors or publishers have a URL for Errata, readers esp. of mathematical books will wait for [sometimes years] for a second corrected edition of books.
I could be wrong about these but it seems these are typos:
p. 30 lines 5 & 6 curly bracket should only be around the 2 * cos(x/2) term
p. 121 second equation should be t=(v+u)/(2*c)
p. 121 '* (1/(2*c)' missing at end of the line
p. 123 line 17, first word should be 'bother' not 'other'
p. 127 line 3 and 4, it seems that the 'icnPI/l' [not the ones in the cos() or sin() terms] term after the 'B' and before the '2*cos' respectively, should not be there. Or am I missing something ?
p. 128 4th line from bottom should be 1753 not 1733
p. 143 2nd line before last equation should be '... (x- i * y)...'
p. 144 equation under 'In summary, then...' cases are reversed
p. 216 seems 1/(2*PI) is missing from right side of first equation, i.e. from "...G(u)G(omega-u)...du"
Book Description
From the Reviews:
"... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. Some of the proofs are classics, but many are new and brilliant proofs of classical results. ...Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... "
Notices of the AMS, August 1999
"... the style is clear and entertaining, the level is close to elementary ... and the proofs are brilliant. ..."
LMS Newsletter, January 1999
This third edition offers two new chapters, on partition identities, and on card shuffling. Three proofs of Euler's most famous infinite series appear in a separate chapter. There is also a number of other improvements, such as an exciting new way to "enumerate the rationals".
Customer Reviews:
this is THE BOOK!.......2007-09-13
One of the most important mathematicians in the last quarter of a century was, arguably, Paul Erdos. At least we could say he was the one who travelled the most, and who had more collaborations with other mathematicians in the areas of Graph Theory, Combinatorics, etc.
One of his dreams was to assemble a collection of all the most beautiful proofs in Mathematics, in which God would maintain the perfect proofs in mathematical theorems. This collection would be called The Book. Unfortunately, he didn't see his dream fulfilled while among us, but after his death Agner and Ziegler wrote this excellent book, "Proofs from THE BOOK", as a nice excerpt of what The Book could have been.
Topics in this book range from Number Theory to Analysis, and also discussing Combinatorics and Graph Theory. Most of the proofs are accessible to everybody with some mathematical training, they are beautiful and surprisingly simple.
I just love this book, in the last five years or so I have bought several copies of it, as it has been my favorite gift for people who want to discover a little more about the beauty of Mathematics.
Fantastic exploration into mathematics.......2007-01-04
We used it for an undergraduate statistics seminar series; and that was largely a success. PFTB contains many of the most elegant proofs I have ever seen. Reading through it is, by itself, an enjoyable journey through some aspects of many branches of mathematics. A few of its chapters may not address the average undergraduate math population quite well, due to its depth of reasoning and assumption of prior knowledge in Abstract Algebra and Geometry; but those who reads it are likely to be more advanced anyway. In general, an excellent book.
Excellent, but bizarre and atypical book on mathematics.......2006-10-23
I agree with what most of the other authors have said. This is a wonderful book, if only a bit strange. The book is very clearly written and easy to follow, although many of the proofs require a great deal of contemplation in order to fully understand.
This is one of those books that a serious mathematician will probably enjoy picking up and reading from time to time. It is neither a reference nor a textbook, but more a source of mathematical inspiration, a collection of particularly elegant mathematical results. My favourite aspect of this book is the way it focuses on proofs and results that draw connections between different areas of mathematics. The interconnectedness of mathematics is too often ignored by researchers nowadays, who have become specialized to the point that their work is often inaccessible. This book is a healthy step in the opposite direction.
My only complaint about this book (and it's not really about this book at all) is that there are not more books like this. The book is heavily slanted towards number theory, combinatorics, and graph theory. The chapter on analysis is beautiful, but atypical of analysis as a whole. I think that it would be wonderful for people to write similar books in other topics, in particular, analysis, algebra, or probability or topology like the other reviewer mentioned.
As a final note I would like to give credit to the publisher for doing a good job on this one. The typesetting is excellent, the layout is clear with all the illustrations, the paper is of high quality, and the binding is oustanding. It has held up well to moderate use and still looks brand new. I only wish some of my more heavily-used reference texts were bound like this.
Proofs from the Book.......2006-08-24
The famous eccentric mathematician had a theory that God (who he liked to call the SF-Supreme Fascist) had a book with every mathematical theorem and that the book had the most beautiful and elegant proofs for the theorems.
This book has that philosophy and presents great proofs for many beautiful theorems.
It is a masterpiece. A book that shows why mathematicians love what they do.
belong's on any math lover's bookshelf.......2006-08-13
It is a pleasure to explore these polished gems, some prettier than others but all worth looking at. I find it worth having within reach and reading a theorem and proof now and again for the sheer joy of (re)discovery.
Book Description
Has physics gone off in the wrong direction? Peter Woit presents the other side of the growing debate on string theory--arguing that it's not even science
At what point does theory depart the realm of testable hypothesis and come to resemble something like aesthetic speculation, or even theology? The legendary physicist Wolfgang Pauli had a phrase for such ideas: He would describe them as "not even wrong," meaning that they were so incomplete that they could not even be used to make predictions to compare with observations to see whether they were wrong or not.
In Peter Woit's view, superstring theory is just such an idea. In Not Even Wrong, he shows that what many physicists call superstring "theory" is not a theory at all. It makes no predictions, even wrong ones, and this very lack of falsifiability is what has allowed the subject to survive and flourish.
Not Even Wrong explains why the mathematical conditions for progress in physics are entirely absent from superstring theory today and shows that judgments about scientific statements, which should be based on the logical consistency of argument and experimental evidence, are instead based on the eminence of those claiming to know the truth.
In the face of many books from enthusiasts for string theory, this book presents the other side of the story.
Customer Reviews:
The Fall of Strings.......2007-10-06
String theorists have so far been unable to use their results to predict new experimental findings. This book and Smolin's 'The Trouble with Physics' both attempt to document this failure of string theory. Smolin's book is better, but a tougher read. But this book is not bad, and you may want to read them both.
Woit dissects "the only game in town".......2007-10-02
"The fundamental problem with string theory is that, as far as its central goal of unifying physics goes, over the last nearly 25 years it has not only not made any progress toward explaining anything about particle physics, but, quite the opposite. Everything that has been learned about string theory makes it more and more clear that the original hopes for getting unification this way were just misguided and can't work. The derivative here is the wrong sign." Peter Woit, posted on his weblog September 13, 2007.
Some readers may think that this book gets off to a painfully slow start, given the author's long telling of the history of particle physics, particularly as regards work done with particle accelerators/ colliders. But stay with it [it's worth it!]. Woit holds degrees from Harvard and Princeton (PhD, theoretical physics) and has taught both mathematics and physics at Columbia. He happily describes himself as a mathematician, in large part because that is indeed the career he has chosen but also in large part because he is obviously disgusted with the current state of theoretical physics--in so far as the superstring/ M-theory disciples of Witten have abandoned anything resembling orthodox science. Woit shows no hesitation in acknowledging Witten's great genius, but unlike most theorists of recent decades he is not interested in worshiping at Witten's feet, no matter what the cost. And Woit isn't just some disgruntled nay-saying spoilsport (I can't strictly judge the psychological state of someone I don't personally know, but he doesn't strike me in this way at all). His concern is that there are other prospects for a unified theory that have been summarily brushed aside by the popular mantra that "string theory is the only game in town." [Federal] research funding, positions of influence notably including department chairs, academic and research hirings, increasingly all have played what we are told (by string/brane theorists themselves) is "the only game in town." But after three decades of glowing hype, this "game"--superstrings/'M-theory'/'brane-world'--has failed to move forward. It has essentially demonstrated that it cannot move forward in any scientific sense.
"Superstring theory is to a large degree thought of by mainstream physicists as mathematics and by mainstream mathematicians as physics, with each group convinced that it makes no sense within their frame of reference but presumably does within someone else's." pg 204
Like so many other armchair theorists, I've read and enjoyed books like Greene's `Elegant Universe' and Hawking's glossy `Universe in a Nutshell'. But any astute reader has to notice that no real connection is made between what we are told are compellingly "beautiful mathematics" and the physical world we can examine, and, given a sound theory, even interrogate, to any degree at all. It is particularly instructive to consider strings/ brane-world from the critical perspective of pure mathematics, i.e., Woit's perspective in this volume. It seems that the abstract equations ARE strangely "beautiful" UNTIL the math must be patched to conform to a universe with precisely three large spatial dimensions; as soon as we are forced to manipulate the additional dimensions, the beauty of the mathematics begins to fade. That `beauty' has been fading for 20 years at this writing. Woit finds the equations of strings/branes to be growing uglier at every turn. After decades of contortion, strings/branes are ever becoming less beautiful than advertised. And, as Woit briefly explains with stark, non-glossy frankness, strings/branes are NOT the only game in town, or at least they shouldn't be.
Many mathematicians would cautiously agree that strings/branes make for interesting mathematics--if they can be divorced from physics. There may be sufficient cause to suggest, as some have, that strings/branes make for interesting theology as well. But are strings/branes SCIENCE? At this late date it seems that the answer is probably `no'.
"If someone could come up with a legitimate, distinctive, testable prediction of string theory that gave even the correct order of magnitude for some experimental result, that would be a huge breakthrough."
As I have suggested to anyone willing to listen, read Peter Woit's thought-provoking book, especially if you've read Greene, Hawking, and/or Susskind.
Right on the button.......2007-08-30
What can I say? Like Peter Woit, I am a recovering mathematician, and this book has given me hope. "Not Even Wrong" carries my highest recommendation, especially for those empirically-inclined investigators who have become demotivated by the crisis in science. One proviso, though - don't read it in isolation. Essential companions are Lee Smolin's "The Trouble with Physics", and my own impassioned plea "The Virtue of Heresy - Confessions of a Dissident Astronomer". The Virtue of Heresy: Confessions of a Dissident Astronomer
Hilton Ratcliffe
Astrophysicist
contains interesting information on physics as well as criticism of String Theory.......2007-07-11
As most people know Not Even Wrong is critical of String Theory. What I did not realize going into the book was the detail it delves into in describing events leading up to String Theory, especially events having to do with the Standard Model. I think the book is worth reading just for this information which spans half of the book. I don't know whether String Theory is right or not as I am not a physicist but I do believe that criticism is a healthy thing and that the author is clear in his criticism of String Theory. I would have wished for more information on other Unified Theories but he devotes only one chapter to this. I guess, as the author points out, there is only one game in town and it is String Theory.
A Good Proposal for Using Government Funds More Effectively.......2007-06-23
Woit's book will be very helpful to technical people who do not work daily in the field of physics but want to remain up to date on the progress of this field. Woit's conclusions and recommendations will be widely accepted. Interestingly, in my book review of Leonard Susskind's book on The Cosmic Landscape in December 2005, I said, "I hold hopes for physicists but not much for strings." I made this statement because the length of a string is divisible and cannot be modeled by a zero-point. So, string theory was completely wrong on day one.
Woit gave me a clear view of the histories of particle physics, strings, and the standard model. In Ch. 6, my mind became glued to the Yang-Mills theory and the new behavior named `asymptotic freedom.' This new behavior is consistent with other theories: (1) the infinite gap that separates a creator God from the universe; (2) the Riemann hypothesis on prime numbers; (3) the true atoms (Leibniz's monads); (4) Cantor's transfinite number; (5) and the origin of inertia of Bernard Haisch (see `The God Theory). So, Woit is right. It is time for physicists to return to basics and The Standard Model. But, they might also consider the reality of an active God.
Book Description
The Fifth Edition of one of the standard works on number theory, written by internationally-recognized mathematicians. Chapters are relatively self-contained for greater flexibility. New features include expanded treatment of the binomial theorem, techniques of numerical calculation and a section on public key cryptography. Contains an outstanding set of problems.
Customer Reviews:
good book.......2002-01-09
This book (5th edition) cover the topics of undergraduate number theory well. The chapters are -
(1)divisibility
(2)congruences
(3)quadratic reciprocity and quadratic forms
(4)some funtions of number theory
(5)some diophantine equations
(6)farey fractions and irrational numbers
(7)simple continued fractions
(8)prime estimates and multiplicative number theory
(9)algebraic numbers
(10)partition funtion
(11)density of sequences of integers.
It also contains basic cryptography, basic group theory and basic elliptical curves in some of the chapters. The authors give notes on the end of each chapter about some research results, which I enjoy reading.
However, the author give too much hints spoling the fun of solving the problems. Eg 32-36, 40-3, 59-53, 108-36, 136-17, 312-8, and most of the problems in chapter 8. The author should put these hints at the back of the book. I suggest you look up IMO (imo.math.ca) for problems suitable for chapter 1-7 because IMO is well-knowned for its excellent number theory problems (especially 1990-3).
Overall this is an excellent book. I give it a rating of 4.5/5, I don't give it 5 because of the author give too much hints to problems instead of putting hints at back of the book.
Comprehensive.......2000-12-23
This is a fantastic book on number theory. It covers far more ground than most introductory text (comparable to Hardy and Wright in depth with much less concern for the big O). It covers material usually only available in separate texts: Rational points on elliptic curves, the partition function, and Dirchlet series. Quite readable chapters, well motivated theoretically, although the historic motivation for the subject matter comes largely in the end-of-the-chapter notes. It's an excellent refresher and reference for non-specialist who find themselves using an algorithm or formula they've forgotten (number theory now playing a role in physics and CS, like never before). It is well cross-referenced with regards to methods of proofs the can be accomplished in different section by different methods - this again making it an excellent reference.
Alas, it is pre-FLT. So you'll have to look elsewhere for that.
The best intro to the subject!.......2000-09-08
I have started my studies in Number Theory reading this book from the preface to the last word. It is amazing! I think it is a better introduction to the subject than the classical Hardy and Wright...it is "more objective" and almost 100% elementary...a good high school reader could do well with it. The chapter of diophantine equations has some divine proofs, very clever and very beautiful. And there is an easy proof of the irracionality of Pi. The only negative point is the existence of some points where the authors could be less concise and a bit clearer, stating the theorems before giving the demonstrations, instead of saying at the end of the paragraph "we then have proved the theorem of..." Its a good book for self-study. It has many exercises.
I've found a marvellous proof..........1999-11-23
It's a excellent book. Guide you through the simplest proofs until the great ones. If you can follow the book since start until end you'll be prepared for beginning research in this incredible world.
Book Description
Elementary Number Theory, Sixth Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. Written in David Burton’s engaging style, Elementary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history.
Customer Reviews:
A Worthy Number Theory Text.......2006-05-13
We used this book in an number theory course I took recently. Burton is a skilled writer, and his book is extremely easy to read even for those devoid of "mathematical maturity". There is a student solutions manual, but I recommend that you abstain from buying it. Many of the exercises have generous hints provided. In fact, Burton probably overdoes it in the hint department. Some of the exercises are ruined that way. Nonetheless, Burton provides excellent exercise sets. Some of the problems are trivial, some aren't. He is careful to point out certain themes that recur in number theory in the text and the exercises.
As previous reviewers have noted, there are brief biographical sketches of certain mathematicians that were integral to the development of number theory. It is interesting to read about the lives and personalities of the men (and women!) that worked on the subject that Gauss coined as "the queen of mathematics".
Chapters 1-9 are the core of an undergraduate course in number theory. I was not that impressed by Burton's introduction to cryptography in Chapter 10. Chapters 11-13 are a nice read though. I do question the wisdom of wasting an entire chapter (Chapter 14) on Fibonacci numbers. Continued fractions and Pell's equation (or "Fermat's equation", as Pell was a mathematical fraud, according to E.T. Bell) are covered in Chapter 15. Chapter 16 is a delightful (but necessarily brief) introduction to twentieth century innovations in number theory. The reader will definitely be left wanting more after the final pages on the Prime Number Theorem.
All in all, not a bad effort. Burton could raise the level of his work from 4 stars to 5 stars with a couple of modifications. Chapter 14 should probably be condensed to an appendix or inserted in another chapter. Also, Burton goes out of his way not to discuss algebraic concepts (groups, rings, fields). Presumably, this is to make the text more friendly to math education majors. Still, there is a whole other side to the subject that the reader is not exposed to by this regrettable omission. Algebraic number theory is not covered.
For a second number theory read, I recommend one, or several of the following:
(1) "Introduction to Analytic Number Theory" by Tom Apostol. An excellent book. Apostol develops the theory necessary to prove Dirichlet's theorem on primes in arithmetic progressions and of course the Prime Number Theorem (an analytic proof). Apostol's book is noteworthy for its treatment of arithmetical functions, which is extensively developed throughout the text.
(2) "An Introduction to the Theory of Numbers" by Niven, Zuckerman, and Montgomery. This book gives a nice coverage of the algebraic aspects of number theory. It has an entire chapter on algebraic numbers that is well worth the read. Also, the more recent edition with Montgomery delves into the geometric results in number theory. This is a well rounded book written by mathematicians preeminent in their field.
(3) "An Introduction to the Theory of Numbers" by Dence and Dence. Quite reader friendly, and surprisingly complete. They promote a deep understanding of the relevant algebra, which is covered at a comfortable pace. They provide an easier read than say Niven, Zuckerman and Montgomery with approximately the same coverage of material.
(4) "An Introduction to the Theory of Numbers" by Hardy and Wright. Written by a legendary number theorist, this book is like a history lesson of 20th century number theory (up through Selberg's "elementary" proof of the Prime Number Theorem). Not so fun to read, but worthwhile as a reference.
(5) "An Introduction to Number Theory" by L.K. Hua. Regrettably, this book is out of print. Nevertheless, you should take a look at it. You can read it with no prior knowledge of number theory and go quite far. Has a comprehensive treatment of (elementary) algebraic number theory. Best appreciated after reading Niven, Zuckerman, and Montgomery.
(6) "Number Theory" by George Andrews is recommended for a combinatorial approach to number theory. The Dover publication is very cheap. Also has some nice introductory material to the theory of partitions.
Of course, there are many others. You can probably find all of the above (except maybe #3) in your local university library.
Recommended.
as a start.. perfect.......2006-01-16
I bought this book to study number theory on my own. (but let me say I had great knowledge about the material b4 I got into it). I studied the first three chapters on my own, and it was great experience, but then I had to stop cuz I did not have any free time to continue. From the first three chapters, I rank this book 5 stars!
This book is awesome, written very rigorously!! Its the right way to write any book in mathematics, and I love it.
Adequate introductory text at an outrageous price........2005-02-14
This text has served me through my first course in number theory. It follows the traditional "definition - theorem - proof - example - exercises" format throughout it's sections. For some flavor, it even throws in a little history behind the mathematics it presents. This book, however, IS NOT worth the ridiculous price that McGraw Hill has retailers charging; nothing in it is that spectacular (well, not spectacular at all, really).
Worked for me.......2001-05-04
I must admit, the exposition can get a little hairy at (very few) spots, but the problems are good, and it has served me well as a reference (for certain limited topics). Great introduction.
Fantastic introductory text!.......2000-12-04
I'm a first-year Ph.D. student, taking a graduate-level number theory course, and I still use this book from my undergrad years as a reference. Just about any basic number theory topic you're looking for is in here. I can't recommend it highly enough!
Average customer rating:
- no complaints
- Intermediate Level
- Short and to the point
- Very well written book
- Now I know how beautiful proofs can be
|
An Introduction to Mathematical Reasoning: Numbers, Sets and Functions
Peter J. Eccles
Manufacturer: Cambridge University Press
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ASIN: 0521597188 |
Book Description
This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.
Download Description
This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.
Customer Reviews:
no complaints.......2007-09-30
In less than a week I had the book and I'm happy with my purchase
Intermediate Level.......2007-08-21
I'm biased. This is the sort of thing U.S. schools should require. Can't go wrong with this. This has 2X as much as "How to Read @ Do Proofs" (Daniel Solow). Wich is better? No Answr. I Luv em both. Former is more expository the other more mechanical. Go For It! Recomended.
Short and to the point.......2006-04-11
This book is excellent! It chapters are broken down into short sections and the content in each section is to the point! I also bought the book Proofs and Fundamentals by Ethan D. Bloch but found it to be long and drawn out. If you liked The Nuts and Bolts of Proofs by Antonella Cupillari then this book is for you!
Very well written book.......2004-09-30
I have a mathematics degree. Like most math majors, I struggled with proofs all through college. This book really has help me understand the art of writing proofs. The book is very well written and easy to read. This is just an awesome book!!!
Now I know how beautiful proofs can be.......2002-12-13
This book provides a nice introduction to mathematical reasoning and proofs. My intention on purchasing this book was to learn how to perform mathematical proofs. I believe it has achieved that purpose. The text is easy to follow and the author presents the work clearly.
Book Description
Mathematicians solve equations, or try to. But sometimes the solutions are not as interesting as the beautiful symmetric patterns that lead to them. Written in a friendly style for a general audience, Fearless Symmetry is the first popular math book to discuss these elegant and mysterious patterns and the ingenious techniques mathematicians use to uncover them.
Hidden symmetries were first discovered nearly two hundred years ago by French mathematician Évariste Galois. They have been used extensively in the oldest and largest branch of mathematics--number theory--for such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack well-known problems such as Fermat's Last Theorem, Pythagorean Triples, and the ever-elusive Riemann Hypothesis. Mathematicians are still devising techniques for teasing out these mysterious patterns, and their uses are limited only by the imagination.
The first popular book to address representation theory and reciprocity laws, Fearless Symmetry focuses on how mathematicians solve equations and prove theorems. It discusses rules of math and why they are just as important as those in any games one might play. The book starts with basic properties of integers and permutations and reaches current research in number theory. Along the way, it takes delightful historical and philosophical digressions. Required reading for all math buffs, the book will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life.
Customer Reviews:
Let me Inject Some Reality into Discussion.......2007-08-05
In spite of some of the comments posted already and in spite of what is on the book's back cover - this is a math book - this is a serious math book. I personally don't see that average person getting anything out of this if they hadn't had say Linear Algebra in particular. Calculus is not required but higher alegra is.
The reason I bought this book is that I read Ian Stewert's book on Symmetry and Beauty and found it lacking as it was not very mathematical.
I was not dissapointed in the level of math in this book. If anything, I got overwhelmed by the end.
I call this type of book "drill deep" but not wide. I like that idea.
The author's have a real ambitious goal. It's laid out on pages 11 and 12:
"in this book we explore ..representations...we consider sets, groups, matrices and functions between them. We show you in detail in one particular case that we develop throughout the book that sets us to our goal: mod p linear representations of Galois groups."
THIS IS THE GOAL OF THIS BOOK. They are not kidding this is what the book sets out to do and I belive accomplishes.
The authors are true to this goal in the "drill deep" mode. Example: Chapter 2 is Groups - not everything about Group Theory is presented but enough that is needed for the rest of the book. In a similar manner one chapter is on so called reciprocity laws. Chapter 4 is on Modular Arithmetic a crucial aspect to this book.
One prior reviewer indicated that each chapter is far more difficult than the last; this is sortof the general tenure of the book - but with exceptions if you know that material. Example, Chapter 5, Complex Numbers, for me was a relief sandwiched in between Modular Artimetic and Equations and Varieties. I can attest that for the subject "Complex numbers" - that they treated it at a relativley elementary level and focused on just those aspects needed later on. I am sure that for all subjects like "Quadratric reciprocity" that was the case. However, if you hadn't been exposed to quadratic reciprocity and Legendre symbols it is a tough slog.
For me the high point of the book was Chapter 8, I felt that I understood the difficult concept of the the Absolute group of the field of algebraic numbers by the end of the chapter. It is an infinite group that only elements can really be enumerated - Identity and complex conjugation. It fills in some (but not all) of the points in the number line between the group of rational numbers and the line with no gaps the field of real numbers.
Chapters 13 to 22 my ability to follow went way downhill and I just skimmed to get some highpoints.
I might return to this book in the future. I like the idea of not having to learn every aspect of something like alebraic ring theory , then every aspect of permutation theory etc. but just learning enough to accomplish some higher level of understanding like ultimatley how Fermat's Last Therom was solved.
I would recomend Stwert's book on Symmetry and Beauty first if you feel you want a more general understanding of this subject as opposed to a real math book which this is.
From the Earth to the stars.......2007-07-29
The book has a goal which is very difficult to reach: introducing people without every mathematical background to the contemporary research in Galois Theory, Number Theory and Diophantine Geometry. In such a situation it is always very hard to choose the best proofs to be written in the book, the best examples and the best way... Maybe, if was the author, I would have made more proofs in the second part of the book, and have chosen other examples for this part. However --- it is so light to criticize --- and the author achieved his goal in proportion of at least 80%, which is not less for an impossible goal!
outstanding.......2007-06-27
This is a very good introduction to arithmetic that everyone wanting to be initiated in this important branch of mathematics should read.
The authors achieved something remarkable: they were able to communicate with accuracy the deepest concepts of arithmetic without the boring style of many mathematics textbooks. The book is very engaging, with nice reflections about the nature of mathematical thought, as well as the motivations behind the concepts.
The authors managed to have a gradual build up of difficulty of topics all the way
to the proof of Fermat's last Theorem. Unlike other introductory texts that let you down
because in their effort to be more engaging, end up too elementary, this one is perfectly balanced. I will also recommend the "Calculus Gallery" as a second outstanding book introducing Analysis.
It would be great if other branches of mathematics like mathematical physics, algebra, mathematical informatics etc had the privilege of such well balanced and insightful introductions.
Well done!
Rare - a well written book about math .......2007-05-30
Unlike most math books, Fearless symmetry is well written. Key concepts from prior chapters are reemphasized in subsequent chapters so readers are less likely to get lost. This is the first book on groups and representation theory that made clear sense to me. I can see where galois therory is going and now have an understanding of the basic form of the proof of Fermat's last theorem.
Definition Dump from Freshman Abstract Algebra Course.......2007-05-16
The book is simply a compilation of definitions from a second rate textbook for Abstract Algebra 101 pasted together with high-five chit-chat from the mathematics-is-really-neat school of math edutainment. There are only a handful of illustrations in the book, none of which have anything to do with symmetry or patterns of numbers. The book Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity by M.R.Schroeder is a much better choice for the subject.
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