- Hardcover: 672 pages
- Publisher: Addison-Wesley Professional; 2 edition (March 10, 1994)
- Language: English
- ISBN-10: 0201558025
- ISBN-13: 978-0201558029
- Product Dimensions: 7.6 x 1.6 x 9.4 inches
- Shipping Weight: 3 pounds (View shipping rates and policies)
- Average Customer Review: 62 customer reviews
- Amazon Best Sellers Rank: #60,943 in Books (See Top 100 in Books)
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Concrete Mathematics: A Foundation for Computer Science (2nd Edition) 2nd Edition
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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:
- 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.
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Top customer reviews
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The book is great for self -study. As with TAOCP, problems are graded. Solutions exist to all problems - except research ones- but trying to solve them yourself will be the best way to use this book.
Overall, worth every penny. A classic reference and must have.
Take-Aways (As of Ch 3):
There are many aspects of summations, integer functions, and proofing that: I never saw covered in my CS degree, are unforgettable, and can be immediately applied to most algorithm research. Those alone make this book worth every penny. Further, the problems posed by this book are more than just repeated mechanics, as I have seen in books like those mentioned below. Each problem is carefully chosen, thorough, and exposes multiple aspects of each topic. They really do weed out many faults that I wasn't really exposed to- as a small example: the importance of ensuring validity of n-1 and n-2 hypothesis & base cases during an induction proof.
Students educated through a contemporary CS track at most American uni's, I believe, (e.g. Rosen Discrete Math, Cormen Algorithms) will find this book both terrifyingly terse and frustratingly paced. In many cases, examples are given without derivation. In many cases, important points are made without obvious connection to previous topics. This is not without a solution however, and getting through this book is often an acquired technique of paper noting things as-you-go, as well as a learned hyper-literacy. The terseness is also a double-edged sword, as sometimes I found it useful as an extra opportunity to practice the taught methods to see if I could come to the same result. Further, the reader should be prepared to go back and review propositional logic & university calculus theorems (atleast FTC, definite vs indefinite integrals). For example, the description of sum by parts in the section on finite calculus assumes _much_ from the reader, and being able to use university calc. as a point of reference to get through that is helpful.
A lot of exercises are tersely explained in both problem and solution. Further, many solutions are totally left-field (having little to do with material in the book). This isn't necessarily bad, as even taking the wrong path to a solution is very educational. However, at some point the reader has to make a judgment as to how long to commit to a certain problem. Many terse problems & left-field solutions instill the wrong judgment: quitting too early.
Attention to detail & extra work is necessary to overcome the terseness of this particular beast, but it's worth it. I recommend this book for developers confronted with algorithm optimization problems, as a well as for a different take on parts of discrete math, and definitely for students coming out of a US state school CS program, the last which this book complements very well. Having worked through some of V1 TAOCP, I would also say that the book is effective in expanding upon its math underpinnings (V1 at-least), and incidentally, does give one confidence to tackle Knuth's other works.
The general format for the chapters is some initial, simple motivation, followed by the introduction of some new tool or transformation, then a couple problems to test out the new tool and flesh out some more details. Often the new tools use the old tools introduced earlier in the book (usually "via magic" -- it's often not structurally clear why the old tools are useful in definition of the new tools, they just use them out of nowhere).
Initially things went okay for me, but quickly the book just got incredibly overwhelming because it became difficult to keep track of everything that I had learned thus far, primarily due to the complete lack of abstraction. Maybe others are better at keeping a huge number of concepts in their working memory, but for me, I need some way to organize things, and this book did not provide that at all due to its pedagogically-ordered introduction of the concepts and no appendix or anything on how things fit together from a top-down point of view.
As a concrete example, when generating functions are introduced, they use formal power series, which aren't really introduced. At no point is it really mentioned that these power series that you are working with are not actually functions, but instead just objects which form a ring (there are ways of explaining or hinting out this without actually using the abstract algebra terminology, but none of them were taken). Instead, confusing comments are made regarding divergence, and what looks like illegal operations on functions are applied over and over with no real explanation of what's going on. A tiny bit of abstraction here would have gone a long way, I think.
Overall, I did certainly learn things, but I don't know how many of them I'll be able to remember. Your mileage may vary, but if you are someone who needs a little abstraction to keep things in order, this book can be a frustrating experience.
Most recent customer reviews
This book is for entertainment.
Very good book.
It's extremely dense, which is great for me because I will keep learning from it for months or years to come.Read more