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Concrete Mathematics: A Foundation for Computer Science (2nd Edition) [Hardcover]

by Ronald L. Graham, Donald E. Knuth, Oren Patashnik
4.5 out of 5 stars  See all reviews (39 customer reviews)

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Book Description

March 10, 1994 0201558025 978-0201558029 2
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. 0201558025B04062001

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Concrete Mathematics: A Foundation for Computer Science (2nd Edition) + The Art of Computer Programming, Volumes 1-4A Boxed Set + Hacker's Delight (2nd Edition)
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Editorial Reviews

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.




Product Details

  • Hardcover: 672 pages
  • Publisher: Addison-Wesley Professional; 2 edition (March 10, 1994)
  • Language: English
  • ISBN-10: 0201558025
  • ISBN-13: 978-0201558029
  • Product Dimensions: 9.4 x 7.8 x 1.4 inches
  • Shipping Weight: 3 pounds (View shipping rates and policies)
  • Average Customer Review: 4.5 out of 5 stars  See all reviews (39 customer reviews)
  • Amazon Best Sellers Rank: #32,295 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
238 of 246 people found the following review helpful
5.0 out of 5 stars Please Be Discrete July 13, 2001
Format:Hardcover
What is "concrete" math, as opposed to other types of math? The authors explain that the title comes from the blending of CONtinuous and disCRETE math, two branches of math that many seem to like to keep asunder, though each occurs in the foundation of the other. The topics in the book, such as sums, generating functions, and number theory, are actually standard discrete math topics; however, the treatment in this text shows the inherent continuous (read: calculus) undergirding of the topics. Without calculus, generating functions would not have come to mind and their tremendous power could not be put to use in figuring out series.
The smart-aleck marginal notes notwithstanding, this is a serious math book for those who are willing to dot every i and cross every t. Unlike most math texts (esp. graduate math texts), nothing is omitted along the way. Notation is explained (=very= important), common pitfalls are pointed out (as opposed to the usual way students come across them -- by getting back bleeding exams), and what is important and what is =not= as important are indicated.
Still, I cannot leave the marginal notes unremarked; some are serious warnings to the reader. For example, in the introduction, one note remarks "I would advise the casual student to stay away from this course." Notes that advise one to skim, and there are a few, should be taken seriously. All the marginal notes come from the TAs who had to help with the text, and thus have a more nitty-gritty understanding of the difficulties students are likely to face. Still, there are plenty of puns and bad jokes to amuse the text-reader for hours: "The empty set is pointless," "But not Imbesselian," and "John .
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72 of 72 people found the following review helpful
5.0 out of 5 stars Great book... some reviewers simply don't get it. June 20, 2007
Format:Hardcover
I have the First Edition and came here to look into the Second Edition. There are several negative reviews and basically those folks have fundamental misunderstandings. So I'll add my review.

First, what kind of book is it? It is not an introductory-level math book with lots and lots of repetition. It is a book on hard math, done in a concise manner by brilliant teachers who assume students are very comfortable with calculus, probability, etc. You really cannot afford to skip around and dabble as if this were an introductory algebra course or something. (I'm not being elitist. I did not attend Stanford and don't consider myself a math genius and am not making this a "we versus the unwashed masses" issue, as I have really struggled with the material myself.)

Second, what is the book about? Several reviewers have theories on where the "Concrete" part of the title comes from, but the bottom line is that it's a book on the discrete math that you need to know for theoretical computer science. (For example, discrete calculus versus the continuous calculus we all learned in school.) Any Analysis of Algorithms course, for example, will confront you with recurrence equations and lots of discrete math.

Third, how is the book organized? At first, it appears rather disjoint. The authors have a sort of, "Hey, look at that flower," and "hey, look under this rock" kind of approach as you walk down a path but the path itself isn't really spelled out. None-the-less, the book does build step-by-step from examples of recurrence equations (Towers of Hanooi, Josephus) in Chapter 1, to Generating Functions in Chapter 7.

Perhaps they could have made the path more explicit, but I can't see how they'd organize it much differently.
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71 of 71 people found the following review helpful
5.0 out of 5 stars I wish every book were written like this! December 13, 2005
Format:Hardcover
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.
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Most Recent Customer Reviews
5.0 out of 5 stars My favorite math text of all time
I've read a lot of math books: I majored in math in college, got a PhD in CS (in cryptography, specifically) and I love doing math recreationally. Read more
Published 3 months ago by JJ
5.0 out of 5 stars Requires me to think for extended periods of time.
Thus far I've read up to chapter two of this book and am currently reading chapter three. The material is extremely interesting and just as difficult (for me anyway. Read more
Published 4 months ago by E. Borota
4.0 out of 5 stars Great reference for aspiring Computer Scientists! Utility is...
This is a wonderful and desk reference, proving holistic coverage of all the math-oriented themes in CS study and practice. Read more
Published 5 months ago by CHAD D WOOD
5.0 out of 5 stars Makes me love discrete math
I was lucky enough to find this book for a dollar. It became very quickly one of my 5 favorite math books (out of 100 or so. Read more
Published 11 months ago by S.Z.
5.0 out of 5 stars amazing book
It simply teaches you how to think in a simple way without complex unproven theorems. I really recommend this book.
Published 14 months ago by M. Majid Khonji
5.0 out of 5 stars For the expert, the eager student, or as a reference
I read this book as my introduction to computer science and discrete mathematics in my re-education as a physicist to a computer scientist. It was invaluable. Read more
Published on April 5, 2012 by Malik Magdon-Ismail
5.0 out of 5 stars great book for developing your math skills
This book isn't about mathematical thinking per se, but it's one of the books I read early on in my studies that really helped me with my mathematical thinking and notation. Read more
Published on January 18, 2012 by Upunder
5.0 out of 5 stars What I was missing
I always felt that there were few topics that I was missing and I needed different books to get the information. This book is perfect for a Computer Scientist!
Published on October 19, 2011 by IBLUES
5.0 out of 5 stars clear and illuminating
I'm currently in a course which covers about half the material in this book. The text the professor selected is a fine work for people who already know the material to skim. Read more
Published on September 20, 2011 by Anne Speck
5.0 out of 5 stars My Favorite Math Book, Hands Down
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). Read more
Published on May 16, 2011 by A Studious Student
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