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This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline.
Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems." The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study.
Major topics include:
- Sums
- Recurrences
- Integer functions
- Elementary number theory
- Binomial coefficients
- Generating functions
- Discrete probability
- Asymptotic methods
This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them.
- Sales Rank: #41435 in Books
- Brand: Graham, Ronald L.
- Published on: 1994-03-10
- Original language: English
- Number of items: 1
- Dimensions: 9.40" h x 1.60" w x 7.60" l, 3.00 pounds
- Binding: Hardcover
- 672 pages
From the Back Cover
This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline.
Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems." The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study.
Major topics include:
- Sums
- Recurrences
- Integer functions
- Elementary number theory
- Binomial coefficients
- Generating functions
- Discrete probability
- Asymptotic methods
This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them.
About the Author
Donald E. Knuth is known throughout the world for his pioneering work on algorithms and programming techniques, for his invention of the Tex and Metafont systems for computer typesetting, and for his prolific and influential writing. Professor Emeritus of The Art of Computer Programming at Stanford University, he currently devotes full time to the completion of these fascicles and the seven volumes to which they belong.
Excerpt. � Reprinted by permission. All rights reserved.
This book is based on a course of the same name that has been taught annually at Stanford University since 1970. About fifty students have taken it each year juniors and seniors, but mostly graduate students - and alumni of these classes have begun to spawn similar courses elsewhere. Thus the time seems ripe to present the material to a wider audience (including sophomores).
It was dark and stormy decade when Concrete Mathematics was born. Long-held values were constantly being questioned during those turbulent years; college campuses were hotbeds of controversy. The college curriculum itself was challenged, and mathematics did not escape scrutiny. John Hammersley had just written a thought-provoking article "On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities" 176 ; other worried mathematicians 332 even asked, "Can mathematics be saved?" One of the present authors had embarked on a series of books called The Art of Computer Programming, and in writing the first volume he (DEK) had found that there were mathematical tools missing from his repertoire; the mathematics he needed for a thorough, well-grounded understanding of computer programs was quite different from what he'd learned as a mathematics major in college. So he introduced a new course, teaching what he wished somebody had taught him.
The course title "Concrete Mathematics" was originally intended as an antidote to "Abstract Mathematics," since concrete classical results were rapidly being swept out of the modern mathematical curriculum by a new wave of abstract ideas popularly called the "New Math." Abstract mathematics is a wonderful subject, and there's nothing wrong with it: It's beautiful, general, and useful. But its adherents had become deluded that the rest of mathematics was inferior and no longer worthy of attention. The goal of generalization had become so fashionable that a generation of mathematicians had become unable to relish beauty in the particular, to enjoy the challenge of solving quantitative problems, or to appreciate the value of technique. Abstract mathematics was becoming inbred and losing touch with reality; mathematical education needed a concrete counterweight in order to restore a healthy balance.
When DEK taught Concrete Mathematics at Stanford for the first time he explained the somewhat strange title by saying that it was his attempt to teach a math course that was hard instead of soft. He announced that, contrary to the expectations of some of his colleagues, he was not going to teach the Theory of Aggregates, not Stone's Embedding Theorem, nor even the Stone-Cech compactification. (Several students from the civil engineering department got up and quietly left the room.)
Although Concrete Mathematics began as a reaction against other trends, the main reasons for its existence were positive instead of negative. And as the course continued its popular place in the curriculum, its subject matter "solidified" and proved to be valuable in a variety of new applications. Meanwhile, independent confirmation for the appropriateness of the name came from another direction, when Z.A. Melzak published two volumes entitled Companion to Concrete Mathematics 267.
The material of concrete mathematics may seem at first to be a disparate bag of tricks, but practice makes it into a disciplined set of tools. Indeed, the techniques have an underlying unity and a strong appeal for many people. When another one of the authors (RLG) first taught the course in 1979, the students had such fun that they decided to hold a class reunion a year later.
But what exactly is Concrete Mathematics? It is a blend of continuous and discrete mathematics. More concretely, it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems. Once you, the reader, have learned the material in this book, all you will need is a cool head, a large sheet of paper, and fairly decent handwriting in order to evaluate horrendous-looking sums, to solve complex recurrence relations, and to discover subtle patterns in data. You will be so fluent in algebraic techniques that you will often find it easier to obtain exact results than to settle for approximate answers that are valid only in a limiting sense.
The major topics treated in this book include sums, recurrences, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods. The emphasis is on manipulative techniques rather than on existence theorems or combinatorial reasoning; the goal is for each reader to become as familiar with discrete operation (like the greatest integer function and finite summation) as a student of calculus is familiar with continuous operations (like the absolute-value function and infinite integration)
Notice that this list of topics is quite different from what is usually taught nowadays in undergraduate course entitled "Discrete Mathematics." Therefore the subject needs a distinctive name, and "Concrete Mathematics" has proved to be as suitable as another
The original textbook for Stanford's course on concrete mathematics was the "Mathematical Preliminaries" section in The Art of Computer Programming 207. But the presentation in those 110 pages is quite terse, so another author (OP) was inspired to draft a lengthy set of supplementary notes. The present book is an outgrowth of those notes; it is an expansion of, and a more leisurely introduction to, the material if Mathematical Preliminaries. Some of the more advanced parts have been omitted; on the other hand, several topics not found there have been included here so that the story will be complete
The authors have enjoyed putting this book together because the subject began to jell and to take on a life of its own before our eyes; this book almost seemed to write itself. Moreover, the somewhat unconventional approaches we have adopted in several places have seemed to fit together so well, after these years of experience, that we can't help feeling that this book is a kind of manifesto about our favorite way to do mathematics. So we think the book has turned out to be a tale of mathematical beauty and surprise, and we hope that our readers will share at least of the pleasure we had while writing it.
Since this book was born in a university setting, we have tried to capture the spirit of a contemporary classroom by adopting an informal style. Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive. The joy and sorrows of mathematical work are reflected explicitly in this book because they are part of our lives.
Students always know better than their teachers, so we have asked the first students of this material to contribute their frank opinions, as "graffiti" in the margins. Some of these marginal markings are merely corny, some are profound; some of them warn about ambiguities or obscurities, others are typical comments made by wise guys in the back row; some are positive, some are negative, some are zero. But they all are real indications of feelings that should make the text material easier to assimilate. (the inspiration for such marginal notes comes from a student handbook entitled Approaching Stanford, where the official university line is counterbalanced by the remarks of outgoing students. For example, Stanford says, "There are a few things you cannot miss in this amorphous .. what the h*** does that mean? Typical of the pseudo-intellectualism around her." Stanford: There is no end to the potential of a group of students living together." Graffito: "Stanford dorms are like zoos without a keeper."
The margins also include direct quotations from famous mathematicians of past generations, giving the actual words in which they announced some of their fundamental discoveries. Somehow it seems appropriate to mix the words of Leibniz, Euler, Gauss, and others with those of the people who will be continuing the work. Mathematics is an ongoing endeavor for people everywhere; many strands are being woven into one rich fabric.
This book contains more than 500 exercises, divided into six categories:
Warmups are exercises that every reader should try to do when first reading the material. Basics are exercises to develop facts that are best learned by trying one's own derivation rather than by reading somebody else's. Homework exercises are problems intended to deepen an understanding of material in the current chapter. Exam problems typically involve ideas from two or more chapters simultaneously; they are generally intended for use in take-home exams (not for in-class exams under time pressure). Bonus problems go beyond what an average student of concrete mathematics is expected to handle while taking a course based on this book; they extend the text in interesting ways. Bonus problems go beyond what an average student of concrete mathematics is expected to handle while taking a course based on this book; they extend the text in interesting ways. Research problems may or may not be humanly solvable, but the ones presented here seen to be worth a try (without time pressure). Answers to all the exercises appear in Appendix A, often with additional information about related results. (Of course the "answers" to research problems are incomplete; but even in these cases, partial results or hints are given that might prove to be helpful.) Readers are encouraged to look at the answers especially the answers to the warmup problems, but only after making a serious attempt to solve the problems without peeking.
We have tried in Appendix C to give proper credit to the sources of each exercise, since a great deal of creativity and/or luck often goes into the design of an instructive problem. Mathematicians have unfortunately developed a tradition of borrowing exercises without an acknowledgment; we believe that the opposite tradition, practiced for example books and magazines about chess (where names, dates, and location of original chess problems are routinely specified) is far superior. However, we have not been able to pin down the sources of many problems that have become part of the folklore. If any reader knows the origin of an exercise for which our citation is missing or inaccurate, we would be glad to learn the details so that we can correct the omission in subsequent editions of this book.
The typeface used for mathematics throughout this book is a new design by Hermann Zapf 227, commissioned by the American Mathematical Society and developed with the help of a committee that included B. Beeton, R.P. Boas. L.K. Durst, D. E. Knuth, P. Murdock, R.S. Palais, P Renz, E. Swanson, S.B. Whidden and W.B. Woolf. The underlying philosophy of Zapf's design is to capture the flavor of mathematics as it might be written by a mathematician with excellent handwriting. A handwritten rather than mechanical style is appropriate because people generally create mathematics with pen, pencil, or chalk. (For example, one of the trademarks of the new design is the symbol for zero, 'O', which is slightly pointed at the top because a handwritten zero rarely closes together smoothly when the curve returns to its starting point.) The letters are upright, not italic, so the subscripts, superscripts, and accents are more easily fitted with ordinary symbols. This new type of family has been named AMS Euler, after the great Swiss mathematician Leonhard Euler (1707-1783) who discovered so much of mathematics as we know it today. The alphabets include Euler Text, Euler Fraktur, and Euler Script Capitals, as well as Euler Greek and special symbols such as
and . We are especially pleased to be able to inaugurate the Euler Family of typefaces in this book, because Leonhard Euler's spirit truly lives on every pare: Concrete mathematics is Eulerian mathematics.
The authors are extremely grateful to Andrei Broder, Ernst Mayr, Andrew Yao, and Frances Yao, who contributed greatly to this book during the years that they taught Concrete Mathematics at Stanford. Furthermore we offer 1024 thanks to the teaching assistants who creatively transcribed what took place in class each year and who helped to design the examination questions; their names are listed in Appendix C. This book, which is essentially a compendium of sixteen years' worth of lecture notes, would have been impossible without their first-rate work.
Many other people have helped to make this book a reality. For examples, we wish to commend the students at Brown, Columbia, CUNY, Princeton, Rice, and Stanford who contributed the choice of graffiti and helped to debug our first drafts. Our contacts at Addison-Wesley were especially efficient and helpful; in particular, we wish to thank our publisher (Peter Gordon), production supervisor (Bette Aaronson), designer (Roy Brown), and copy editor (Lyn Dupr�). The National Science Foundation and the Office of Naval Research have given invaluable support. Cheryl Graham was tremendously helpful as we prepared the index. An above all, we wish to thank our wives (fan, Jill, and Amy) for their patience, support, encouragement, and ideas.
This second edition features a new Section 5.8, which describes some important ideas that Doron Zeilberger discovered shortly after the first edition went to press. Additional improvements to the first printing can also be found on almost every page.
We have tried to produce a perfect book, but we are imperfect authors. Therefore we solicit help in correcting any mistakes that we've made. A reward of $2.56 will gratefully be paid to the first finder if any error, whether it is mathematical, historical, or typographical.
Murray Hill, New Jersey - RLG
and Stanford California DEK May 1988 and October 1993
OP
0201558025P04062001 Most helpful customer reviews
249 of 257 people found the following review helpful.
Beware of great books
By James Street
This book is excellent (5 stars) if you have the mathematical "maturity" that it assumes. If not, it will vary from 4 stars to 0 stars.
The problem is, the book looks as if it might be an entry level text and it is tempting to think that with a little extra hard work any intelligent, reasonably well-grounded mathematics undergraduate student could prove that he is a genius by mastering the content. A fair number, of course, will do just that. But many more will unnecessarily bloody their noses and egos.
Most people skip prefaces but this one shouldn't be skipped. The preface says that most of the people who have taken the course that the book is based on have been graduate students and alumni and (some) have been juniors and seniors.
To give an example of the difficulty an unwary student might find: The chapter on probability looks straightforward and well-written and it is! But it is truly useful only to students who have already studied probability theory and mastered the basic theory. The trap is that the book does, in fact, provide introductions to most of the topics covered. But in reality, they are reviews, introductions to the symbols and notation to be used and repositories for results that will be referenced throughout the book.
The prerequisites for having a profitable encounter with this book are : a good understanding of elementary number theory, probability theory and linear algebra and two years of calculus with a very good understanding of infinite series. A good knowledge of generating functions and recursive functions is also necessary. A few juniors and seniors will always be dedicated and smart enough to achieve this level of maturity but it usually takes more than four years.
In addition, while any reasonably intelligent mathematics student can learn the topics covered in this book, it is written by three master programmers and discrete mathematicians and inevitably also contains enough to challenge just about anyone (even them.) After all, the book is dedicated to Leonard Euler, possibly implying that the authors think he is among the very few persons who could have solved most (all?) of the problems.
79 of 80 people found the following review helpful.
I wish every book were written like this!
By Anthony Widjaja To
This book is perhaps one of the most beautifully written books I have ever read. All the proofs presented here are elegant. When reading the proofs in this book, you can feel that one sentence logically and smoothly follows from the previous sentence. This is partly because of the elegant and effective notations adopted by the authors. [Note: Donald Knuth, one of the authors, has been one of the biggest proponents of good mathematical notations. See his book titled "Mathematical Writing".]
Other reviewers have provided a summary of this book. So, I will only say that every computer scientist and combinatorialist should read at least chapters 1, 2, 5, 7, and 9. Chapter 5 is very highly recommended. Trust me: once you have mastered these chapters, you will be able to do things your colleagues just can't. Even just familiarizing yourself with the notations in this book will help you produce proofs that you probably won't be able to otherwise. [Great ideas are of course always important in every proof - but without good notations, you probably won't be able to come up with the ideas in the first place.]
There is pretty much nothing bad about this book that I am aware of. I will just say though that it takes a lot of time and effort to acquire mastery of the material. As for my own story, I started reading chapter 1 and 2 when I just got interested in discrete mathematics. It took me about 1/2 year (part time) to get through this. I came back to this book again when I took a course on "generatingfunctionology". I found that chapter 5 and 7 were indispensable. I was also forced to reread chapter 2 again because the lecturer, as most people do, just waived his hands when it comes to manipulating sums and binomial coefficients. However, all the effort that I put in paid off in the end as I could solve problems in the final exam which all my other friends could not.
In summary, I strongly recommend this book to every computer scientist and combinatorialist. I will finally remark that, if you are serious about learning concrete mathematics, you will probably find that generating functions pop up pretty much everywhere. To understand these beasts, I highly recommend Sedgewick and Flajolet's "Introduction to Analysis of Algorithms" and "Analytic Combinatorics" (not yet published, but next-to-final draft is available at Flajolet's web site), and Wilf's "Generatingfunctionology".
47 of 49 people found the following review helpful.
My Favorite Math Book, Hands Down
By A Studious Student
This is by far my favorite math book. I was introduced to it in a Putnam preparation course and didn't buy it at first (seemed too over the top). I found it in the library and used the school's book for some assignments. After a few nights studying out of the book I went off and bought it for myself, (half price, what a steal!) Granted, the book isn't for everyone, but if you even a glint of interest in discrete mathematics, you should definitely pick this book up.
I have not yet worked by way through the whole book (I've only solidly covered a sixth of it, but I've referenced about three quarters of it over the course of owning it). Here's what I like the most about the book:
- Good density / readability trade-off. Some math books are dense to the point that each page should take you an hour or more to fully understand. This isn't THAT bad, but it is definitely denser than your first year calculus texts. If you're used to breezing through a chapter that covers 2-6 concepts followed by some practice problems, you're going to have to get used to slowing down and doing some serious re-reading. I would estimate that getting a good grasp of each chapter takes about 20 to 60 hours of work, depending on how strong you are with the material. This includes time for solving the problems at the end of the chapter. This may sound like a lot, but trust me when I say you will know all the tricks of the trade when you're finished.
- Intuitive approach. I find many texts will spend a lot of time walking through the material in a very linear fashion. In the sense that the organization of the material makes sense once you've absorbed it, but seems random and un-ordered as you learn it. This book is ordered very intuitively the first time you read it, in the sense that the ideas seem to flow where your mind does when you first pick up the material (does this make sense?) Most chapters open with a (refreshingly tough) problem and you are walked through the solution as you learn the techniques you need to solve it. You'll pick up some very powerful problem solving techniques as you read this book.
- Good depth. Instead of telling you how to manipulate certain classes of sums, the authors go deep into the powers at work behind the problems, as well as "tricks of the trade". Some of the sum manipulations I've learned in this book I have yet to see elsewhere, and they're very handy! For example, a type of "discrete calculus" is defined, which will let you find the closed form for the sum of integers (1 through n) to the power of k with remarkable ease (it uses stirling numbers, I'm sure you can find it on the net.)
What I don't like about it:
- Doesn't take a very deep plunge into number theory. Covers the basics very well, but wont get you into RSA.
- Does not blend in continuous mathematics as well as I'd hoped.
In short, this book covers basic topics in discrete math very well. After reading this book you will be extremely comfortable manipulating sums, discrete probabilities, and number theory. It also gives you a surprisingly in-depth look at discrete math you don't often encounter in your undergraduate courses, including:
- Floor / ceiling functions and manipulating them.
- Binomial coefficients (n choose k).
- Generating functions.
- Special numbers (stirling, fibonacci, harmonic, etc.)
- O-notation (this is where the computer science finally kicks in).
Do read this book if you are:
- A math undergraduate.
- Someone who enjoyed math but lost touch with it long ago.
- Someone raised on continuous mathematics (calculus, high school math, analysis) looking to diversify their thinking.
- Into prime numbers, fibonacci numbers, or curious mathematical patterns.
Don't bother with this book if you are:
- Dumb.
- A math genius.
- A computer whiz but don't enjoy math (despite the "foundation for computer science" line, this is purely a math book).
- An engineering student who only cares about the answer, not the process (see the one star guy).
- Note: This book doesn't use the lemma - proof - example format you may be used to. You've been warned.
This book is extremely well written, and I guarantee that if you're in the "read this book if" list you will find this book impossible to get rid of, it will haunt your bookshelf for life.
Also, concerning the title. There seems to be some confusion as to why it is called "Concrete Mathematics" (see one star guy). It isn't called that because it presents, everyday, formula-solution math. The reason it's called "Concrete Mathematics" (and I quote from the preface) is because "It is a blend of CONtinuous and disCRETE mathematics." That's by far the silliest book-title formula I've ever encountered, but to each their own I guess.
And finally, I didn't see a table of contents on the order page, so I will summarize the table of contents here, in order the topics are presented:
- Recurrence
- Sums
- Integer Functions (floor, ceiling, modulo)
- Number Theory
- Binomial Coefficients (n choose k)
- Special Numbers (Stirling, Eularian, Bernoulli, Fibonacci, and others)
- Generating Functions
- Discrete Probability
- Asymptotics (O notation)
There's just under 500 pages of good stuff before you hit the answer to exercises. Oh yeah, there's also no flimsy "answer to odd number exercises" back page. Each problem is solved ENTIRELY, some solutions take over a page. If that's not the best way to learn, I don't know what is. Enjoy.
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