I had the pleasure of taking the course "Concrete Mathematics" with Ron Graham as an undergrad engineering student at Princeton University in the late 1980s, and in retrospect it was quite possibly the highlight of my academic career. Likewise, this text has a cherished spot in my bookcase even to this day. The material is challenging, sometimes downright frustrating, but good stuff nonetheless... and as others have noted, very useful! The skills gained are specifically tailored for application in fields such as computer science, really a number of disciplines; and complement what one learns in the more "standard" math courses. How many academic texts have become must-haves even beyond your college years? For me, only this one. I'd give it fifteen stars if I could.

This book's title can be misleading. I would say it is more of an advanced textbook on the mathematics that is a foundation for computer science than a foundational book on the mathematics of computer science. I think this misreading of the title and thus the book's content is what is behind much of the heartache that readers have when trying to tackle it. This book expands on the "Mathematical Preliminaries" portion of "The Art of Computer Programming" series of books by Knuth, and thus this book has a style much like that series of books. The book is complete and clear, but it is also densely packed with lots of theory and proofs and will require much effort and time to understand well. It is really not meant to be an applied mathematics textbook at all. I show the table of contents next. Note that there are exercises at the conclusion of each chapter with solutions in the back of the book. However, most of the exercises are not so simple that you can just glimpse at the solution and figure out how to get from A to B. I recommend it if you have the time. It can really bring out thoughts and the beauty of mathematics that you may not have considered before.

1. Recurrent Problems.
The Tower of Hanoi.
Lines in the Plane.
The Josephus Problem.
Exercises.

2. Sums.
Notation.
Sums and Recurrences.
Manipulation of Sums.
Multiple Sums.
General Methods.
Finite and Infinite Calculus.
Infinite Sums.
Exercises.

3. Integer Functions.
Floors and Ceilings.
Floor/Ceiling Applications.
Floor/Ceiling Recurrences.
'mod': The Binary Operation.
Floor/Ceiling Sums.
Exercises.

4. Number Theory.
Divisibility.
Factorial Factors.
Relative Primality.
'mod': The Congruence Relation.
Independent Residues.
Additional Applications.
Phi and Mu.
Exercises.

5. Binomial Coefficients.
Basic Identities.
Basic Practice.
Tricks of the Trade.
Generating Functions.
Hypergeometric Functions.
Hypergeometric Transformations.
Partial Hypergeometric Sums.
Mechanical Summation.
Exercises.

6. Special Numbers.
Stirling Numbers.
Eulerian Numbers.
Harmonic Numbers.
Harmonic Summation.
Bernoulli Numbers.
Fibonacci Numbers.
Continuants.
Exercises.

7. Generating Functions.
Domino Theory and Change.
Basic Maneuvers.
Solving Recurrences.
Special Generating Functions.
Convolutions.
Exponential Generating Functions.
Dirichlet Generating Functions.
Exercises.

8. Discrete Probability.
Definitions.
Mean and Variance.
Probability Generating Functions.
Flipping Coins.
Hashing.
Exercises.

9. Asymptotics.
A Hierarchy.
O Notation.
O Manipulation.
Two Asymptotic Tricks.
Euler's Summation Formula.
Final Summations.
Exercises.

A. Answers to Exercises.
B. Bibliography.

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. They could throw entire chapters into Appendices, but things build on each other in such a way that you'd simply have to skip around from the main chapters to the Appendix anyhow.

Fourth, what other books cover this material? I'm not well-qualified to talk about the entire universe of books, but I must say that the three Analysis of Algorithm books I have for my current class definitely give only the very basics of this material and really only present two possibilities: 1) fiddle around with the equation, possibly using a graphic representation, until you see a pattern and make a guess, then prove it by induction, or 2) if your algorithm is one specific class, plug some numbers into this 3-part formula and if one of the parts applies an answer will pop out for you. Concrete Math is gives you many powerful tools to solve such problems.

Fifth, what is the flavor of the book? The authors have an informal writing style -- outside of the very formal math and proofs -- and the book has marginal notes that were contributed by the "beta-tester students" as the book was being written.

Some reviewers have criticized the marginal notes, and I simply have to shake my head and be glad I don't have to work alongside them. Yes, many of the notes are puns or other humor, but those are a nice break from the heavy math. And many of the notes provide great hints and perspectives from students who are also learning the material. I wish all technical books had such notes, but only a Knuth could get a publisher to go to the trouble.

So that's my review. An excellent book that's very intense and covers a hard, very technical topic. It's like learning math from algebra to differential equations all over again in a different language, and perhaps the negative reviewers simply never understood this. If you're not a programmer who needs to rigorously analyze algorithms, skip the book unless you simply want to learn for the joy of learning.

This is one of those math books where the authors enjoy making inexplicable leaps between equations that really don't follow in a smooth logical fashion one from another. They are terrible at explaining things. They like to boast that they are from Stanford, (subtext: implied: if you are not from Stanford you probably wouldn't understand it anyway, you poor pitiful low-income commoner). I absolutely hate the tone of this book. And the side comments in the margins are inane, weak and mostly irritating. I threw it aside in disgust and went back to Warren Weaver.

This book is great. It is the funnest math book I have worked with, and I appreciate the intensity of the mathematics -- something that is falling out of the norm in computer science. The book is also a great source of fantastic combinatorics.