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ByMary P. Campbellon July 13, 2001

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 .316" made me chuckle, but you have to find them for yourself.

To someone who has been through the rigors of math grad school, this book is a delight to read; to those who have not, they must keep in mind that this is a serious text and must be prepared to do some real work. Very bright high school students have gotten through this text with little difficulty. I want to note ahead of time - some of the questions in the book are serious research topics. They don't necessarily tell you that when they give you the problem; if you've worked on the problem for a week, you should turn to the answers in the back to check that there really is a solution.

That said, I would highly recommend this book to math-lovers who want some rigorous math outside of the usual fare. The formulas in here can actually come in handy "in real life", especially if one has to use math a lot.

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 .316" made me chuckle, but you have to find them for yourself.

To someone who has been through the rigors of math grad school, this book is a delight to read; to those who have not, they must keep in mind that this is a serious text and must be prepared to do some real work. Very bright high school students have gotten through this text with little difficulty. I want to note ahead of time - some of the questions in the book are serious research topics. They don't necessarily tell you that when they give you the problem; if you've worked on the problem for a week, you should turn to the answers in the back to check that there really is a solution.

That said, I would highly recommend this book to math-lovers who want some rigorous math outside of the usual fare. The formulas in here can actually come in handy "in real life", especially if one has to use math a lot.

3 people found this helpful

ByA customeron April 13, 2004

Basically, I like this textbook. The material is interesting, the way the authors presented the material is inspiring, and they provided a lot of jokes to make even studying for exams not that boring. But there is one big problem which made me decided to rate this book only 3 stars instead of 5 stars: the authors like to use non-standard notations. For example: m\n means "m>0 and n=mk for some integer k". One of the worst thing in scientific world is writing things others cannot read, and the authors did this by introducing many strange notations. These things makes the good work sometimes almost unreadable. This is not computer systems in which we use "cp" for the copy command and "cd" for change directory command.

What a pity the authors did that. This textbook will be perfect without those strange notations....

What a pity the authors did that. This textbook will be perfect without those strange notations....

ByMary P. Campbellon July 13, 2001

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 .316" made me chuckle, but you have to find them for yourself.

To someone who has been through the rigors of math grad school, this book is a delight to read; to those who have not, they must keep in mind that this is a serious text and must be prepared to do some real work. Very bright high school students have gotten through this text with little difficulty. I want to note ahead of time - some of the questions in the book are serious research topics. They don't necessarily tell you that when they give you the problem; if you've worked on the problem for a week, you should turn to the answers in the back to check that there really is a solution.

That said, I would highly recommend this book to math-lovers who want some rigorous math outside of the usual fare. The formulas in here can actually come in handy "in real life", especially if one has to use math a lot.

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 .316" made me chuckle, but you have to find them for yourself.

To someone who has been through the rigors of math grad school, this book is a delight to read; to those who have not, they must keep in mind that this is a serious text and must be prepared to do some real work. Very bright high school students have gotten through this text with little difficulty. I want to note ahead of time - some of the questions in the book are serious research topics. They don't necessarily tell you that when they give you the problem; if you've worked on the problem for a week, you should turn to the answers in the back to check that there really is a solution.

That said, I would highly recommend this book to math-lovers who want some rigorous math outside of the usual fare. The formulas in here can actually come in handy "in real life", especially if one has to use math a lot.

ByWilliam Stevensonon December 12, 2002

I used this book while studying Combinatorics at the University of Warwick, a leading British institution for mathematicians. At the time, the book was a little bit overwhelming - Knuth doesn't waste any time in getting to the point of solving problems in the book. Thus, if you're the type of person who needs lots of worked examples, I would supplement this with another book, for example, Grimaldi's Discrete and Combinatorial Mathematics. But this book does belong on the bookshelf - it is a great reference, particularly because it prepares one to read The Art of Computer Programming, also by Knuth. TAOCP is the definitive series on computer science, respected by computer scientists everywhere. I guess the best way to describe Concrete Mathematics is that if you are a graduate student in CS, you should own this book. If you are a mathematically-oriented undergraduate, this book will make you really understand anything that your professors will throw at you. But, if you are not a math-lover, you will want a backup and a really nice professor :)

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ByA customeron July 29, 2002

This book is great. But many excercises are too hard for non-mathematically trained reader. I can solve almost all warm-up exercises without peeking the answer. But even few warm-up excercises are virtually research one. For example, see the exercise 2.1. The answer for this exercise is that there is no agreement about this. I think it means that there is no answer for this exercise. Sometimes even understanding an answer is very hard when you read an answer because you can't solve an exercise. This book contains answers for all exercises. But this book's exercises are MUCH HARDER than many other mathematic books which contain answers for only odd number(or even number) exercises.

You need a great inductive mathematical reasoning experience to read this book. If you finish this, you can omit the first 100 pages of TAOCP vol 1.

It would be nice if there is a solution book for this hard concrete book.

You need a great inductive mathematical reasoning experience to read this book. If you finish this, you can omit the first 100 pages of TAOCP vol 1.

It would be nice if there is a solution book for this hard concrete book.

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ByM. Le Brunon May 15, 2001

Lest others find this wonderful book as disappointing as the reviewer from Osan, Korea: note that "concrete" in the title is just meant in contrast to "abstract". But both concrete and abstract are adjectives intended only to describe different apporaches to *theoretical* math, as opposed to *applied* math, which addresses examples directly relevant to the real world (and thus is probably of more interest to engineers and their ilk). This *isn't* an applied math text. The difference between the concrete and abstract styles is that concrete math generally takes a "bottom up" tack, arising from specific given "concrete" entities, such as certain special functions, sums, sequences etc and tends to involve more derivation and calculation. In contrast typical abstract math is more "top down", proceeding, say, from axioms, perhaps even non-constructively, and tends to involve more reasoning and proving. If you dig the theoretical stuff, and like the concrete approach, this book is a treasure trove.

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ByA customeron July 16, 1999

This book is an excellent introduction to the mathematics of algorithms, and besides featuring an engaging, chatty style, is for the most part accessible to people who do not have a tremendous amount of mathematical preparation. Some calculus is helpful, but the authors develop most of their proofs from first principles, so if you took "Calculus for Liberal Arts Majors" while in college you should still be able to learn a great deal from this book if you are interested.

The only problem is that in discussing the "Repertoire Method" of solving recurrences, they never really give a top-down presentation, and we're expected to figure it out by an example. This is unfortunate because later parts of the book build on this.

ANYONE OUT THERE WHO'S SEEN THIS AND IS WILLING & ABLE TO HELP PLEASE CONTACT ME.

However, all exercises are solved, most in reasonable detail, which makes the book particularly useful for those who for professional or other reasons develop an interest in mathematics later in life.

The only problem is that in discussing the "Repertoire Method" of solving recurrences, they never really give a top-down presentation, and we're expected to figure it out by an example. This is unfortunate because later parts of the book build on this.

ANYONE OUT THERE WHO'S SEEN THIS AND IS WILLING & ABLE TO HELP PLEASE CONTACT ME.

However, all exercises are solved, most in reasonable detail, which makes the book particularly useful for those who for professional or other reasons develop an interest in mathematics later in life.

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ByK. Wongon January 20, 2016

The definitive book on Computer Science - Wither or not you're seeking a job in industry or pursuing grad studies, this book will prepare you for any of the math thrown at you in full fledged Computer Science program (Summations, Proving Correctness, Asymptotic Running times, Series, etc).

The book is not written like your typical academic text book you see these days. It contains full solutions, in plain and concise English, and a lot of humorous footnotes by TA's and Students.

If you're a freshman or junior in a Computer Science program and just have done Calculus 1 and Discrete Math, you'll want to work through this if you want to masterfully analysis algorithms later on in your program.

Anyways, it's from the Legend himself - Knuth!

The book is not written like your typical academic text book you see these days. It contains full solutions, in plain and concise English, and a lot of humorous footnotes by TA's and Students.

If you're a freshman or junior in a Computer Science program and just have done Calculus 1 and Discrete Math, you'll want to work through this if you want to masterfully analysis algorithms later on in your program.

Anyways, it's from the Legend himself - Knuth!

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ByPeteris Kruminson July 9, 2015

Yet another classic. Concrete Mathematics is the second most accessible book by Knuth as one of the co-authors. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. This book focuses on practical skills rather than theory. It contains many examples, tricks of trade, and problems with solutions. This book is also fun to read. After reading this book you will be very comfortable manipulating sums, recurrences, discrete probabilities, and number theory.

I've placed this book #15 in my Top 100 Programming, Computer and Science books list:

http://www.catonmat.net/blog/top-100-books-part-three/

.

I've placed this book #15 in my Top 100 Programming, Computer and Science books list:

http://www.catonmat.net/blog/top-100-books-part-three/

.

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ByN. Novikon January 30, 2000

This is one of those books you keep forever, purely for its utility: it's packed with formulas, techniques, examples. But more than that, the authors lead you through the techniques and explain the concepts behind them, with the goal of equipping you with the mental tools to attack any mathematical problem you encounter. And to top it off, it's well-written, and the "margin notes" provide some comic relief. The material is very dense, and it's not a book I'd recommend for casual reading: this is stuff you only work through if you're going to need it. But if you *are* going to need it, this book will make it a lot more pleasant.

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ByA customeron April 30, 1999

hmm, well I myself am an engineer but also have an avid interest in maths..of course the maths average engineer uses is calculus , linear algebra et al. not pure maths..so a common engineer might be dissatisfied with this book. Neverhteless that shouldn't dissuade the maths enthusiast because this is one of the good books written on discrete maths essential for comp sci student and/or a maths major to get a sound grasp of the basics. I hope everyone benefits from this effort to popularise ancient pure maths in the world of abstract maths !

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ByRomani Chezon September 15, 2013

This book is an excellent introduction to math in the computer world. It is exactly what it describes itself to be and exactly what everyone told me it would be. My last math experience was in high school and so far it is proving difficult but it is fun and definitely a mind workout. I would say it might be better suited for people with a college math background but I have heard of high school students working through it. Overall an awesome book if you love to learn.

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