Customer Reviews (28)

A difficult but worthwhile mathematics text
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.

Great book... some reviewers simply don't get it.
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.

smug math book
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.

Concrete Math is fun
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.

I wish every book were written like this!
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". ... Read more

Book Description
This completely updated second edition illustrates the mathematical concepts that a game programmer would need to develop a professional-quality 3D engine. Although the book is geared toward applications in game development, many of the topics appeal to general interests in 3D graphics. It starts at a fairly basic level in areas such as vector geometry and linear algebra, and then progresses to more advanced topics in 3D game programming such as illumination and visibility determination. Particular attention is given to derivations of key results, ensuring that the reader is not forced to endure gaps in the theory. The book assumes a working knowledge of trigonometry and calculus, but also includes sections that review the important tools used from these disciplines, such as trigonometric identities, differential equations, and Taylor series. ... Read more

Customer Reviews (13)

Little more than an outline
I was disappointed in how little explanations were included in this math book.It seemed more like a dictionary.It was as if someone listed all the math concepts that could be included and then put the list into a hard cover binder.
I did read some of the other reviews thinking I had missed something.One suggested spending a year on the book.But there is so little there.My college book on linear algebra did a much better job of explaining the material.Theorems were explained, samples worked out and 10 to 20 problems were given at the end of each section.This book as 50 problems for the entire book.This book spends eight pages on quaternions.Studying these pages over an extended period of time would not give the understanding you could get from a book like Kuipers' Quaternions and Rotation Sequences.It took 75 pages but one is able to understand the system and how it relates to game programming.
With all the colleges now offering game programming degrees, their must be a book written by someone who is use to explaining the material.That person would know the best way to present the material and could anticipate questions and include the answers in his or her explanations.

An excellent book
I have personally used the C4 game engine (which is the commercial game engine developed by the author of this book), and can attest to its high quality. The engine is top notch, and so is this book.

This book is exactly what you'd expect - a description of the math involved in game (and especially graphics) development. It also includes a fair amount of additional content (not necessarily math-related, such as a description of the stencil shadowing algorithm).

The book is well-written, and does a good job of presenting fairly complex subject matter in a straight-forward manner. This is currently my favourite text book.

a disappointment
the second edition brings a new chapter on the graphics pipeline. Well done, except that it is extremely obscure, for those who already understood the arcanes of the graphics pipeline.
Chapter 5 on "ray tracing": 6 pages on root finding of quadratic, cubic and quartic polynomials;
to compare with Chapter 7: "Visibility determination": The spatial partitioning deserves solely 4 pages.
What's the point on dwelling so much on something that is trivial and botching something that is hard to grasp and of paramount importance ?

One of a kind

As a professional 3D graphics programmer, I can not stress enough the quality of this book. This book covers 3D math fundamentals, algorithms, and it is complete with easy to understand (!) proofs. The math is difficult because there is so many problems to be solved in 3D (and they draw from many different branches of mathematics), but it is written in such a clear way that every topic is made approachable. Unlike esoteric Ph. D papers, you aren't assumed to have any specific knowledge of math idioms or jargon. You simply need a decent grasp of college calculus and trigonometry to make the most of it. There are a few samples too to test your knowledge.

Expect to spend at least a solid year to really make the use of this book. Treat it as you would a two - three semester course in college. During this process, you'll find yourself occasionally wanting to get more practice and referring to a respective book on it.

If you could only own two books for 3D programming, buy this first and buy Ericson's book on collision detection next.

In summary, Eric Lengyel's attention to detail and mastery of 3D math / algorithms really shines and this book is an example of it.

Great book
This book is great for anyone interested in computer graphics. Even for people who do not have a lot of math/graphics experience, this book starts you off with the basics of vectors and matrices and has exercises/solutions for each chapter. It saves you the time of looking through your old linear algebra and differential equation math books and contains the must know information you will use as a graphics programmer. ... Read more

Book Description
This is a new edition of a successful introduction to discrete mathematics for computer scientists, updated and reorganised to be more appropriate for the modern day undergraduate audience. Discrete mathematics forms the theoretical basis for computer science and this text combines a rigorous approach to mathematical concepts with strong motivation of these techniques via examples.
Key Features
Thorough coverage of all area of discrete mathematics, including logic, natural numbers, coding theory, combinatorics, sets, algebraic functions, partially ordered structures, graphs, formal & complexity theory
Special emphasis on the central role of propositional & predicate logic
Full chapters on algorithm analysis & complexity theory
Introductory coverage of formal machines & coding theory
Over 700 exercises
Flexible structure so that the material can be easily adapted for different teaching styles.
New to this Edition
Improved treatment of induction
Coverage of more 'basic' algebra
List of symbols including page references for definition/explantion
Modern text design and new exercises to aid student comprehension ... Read more

Customer Reviews (2)

Packed, pithy, and clear!
I will have to agree with the other reviewer that it seems as if Truss is trying to cram lots and lots of discrete mathematics into an indiscrete stack of 500pp but i think he has done very well. At least in addressing those who need to skim over the material quickly, though they would love instead to relish on Grimaldi's.

:)

Skims the Surface
I am a teacher's assistant for an undergraduate computer science course that uses this book. I have to say that it really is a terrible book for students to learn from who have never had much exposure to non-calculus math and the concept of the "mathematical proof". It skims over topics without providing enough exposition on the topics to allow students to have a fair grasp on the subject. This may just be the nature of teaching discrete math, but there seem to be far too many topics that Truss is trying to squeeze into too small of a space. He tries to throw in some more advanced topics such as formal machines and complexity theory, but only at the cost of having the overall quality of the material be watered down. Sadly, though, from what I have heard, this is the best current intro discrete math book out there. ... Read more

Book Description
Master the fundamentals of discrete mathematics with DISCRETE MATHEMATICS FOR COMPUTER SCIENCE with Student Solutions Manual CD-ROM! An increasing number of computer scientists from diverse areas are using discrete mathematical structures to explain concepts and problems and this mathematics text shows you how to express precise ideas in clear mathematical language. Through a wealth of exercises and examples, you will learn how mastering discrete mathematics will help you develop important reasoning skills that will continue to be useful throughout your career. ... Read more

Book Description
Combinatorial Methods with Computer Applications provides in-depth coverage of recurrences, generating functions, partitions, and permutations, along with some of the most interesting graph and network topics, design constructions, and finite geometries. Requiring only a foundation in discrete mathematics, it can serve as the textbook in a combinatorial methods course or in a combined graph theory and combinatorics course.After an introduction to combinatorics, the book explores six systematic approaches within a comprehensive framework: sequences, solving recurrences, evaluating summation expressions, binomial coefficients, partitions and permutations, and integer methods. The author then focuses on graph theory, covering topics such as trees, isomorphism, automorphism, planarity, coloring, and network flows. The final chapters discuss automorphism groups in algebraic counting methods and describe combinatorial designs, including Latin squares, block designs, projective planes, and affine planes. In addition, the appendix supplies background material on relations, functions, algebraic systems, finite fields, and vector spaces.Paving the way for students to understand and perform combinatorial calculations, this accessible text presents the discrete methods necessary for applications to algorithmic analysis, performance evaluation, and statistics as well as for the solution of combinatorial problems in engineering and the social sciences. ... Read more

Customer Reviews (1)

Combinatorics
Good book overall. Easy to read and informative. Good at teaching you how to solve problems. Example problems are sometimes only the easiest problems, and extending the method to harder problems is not always trivially apparent. Also sometimes skips somewhat important steps in solving example problems. ... Read more

Book Description

"Discrete Mathematics for Computer Science" is the perfect text to combine the fields of mathematics and computer science. Written by leading academics in the field of computer science, readers will gain the skills needed to write and understand the concept of proof. This text teaches all the math, with the exception of linear algebra, that is needed to succeed in computer science. The book explores the topics of basic combinatorics, number and graph theory, logic and proof techniques, and many more. Appropriate for large or small class sizes or self study for the motivated professional reader. Assumes familiarity with data structures. Early treatment of number theory and combinatorics allow readers to explore RSA encryption early and also to encourage them to use their knowledge of hashing and trees (from CS2) before those topics are covered in this course.

Customer Reviews (1)

I did not find it suitable for my discrete math course
I evaluated this book for possible adoption in a course in introductory discrete mathematics. My decision was that I would not use it in the course. One primary reason is that there are no sections devoted to set theory and functions. Most of the introductory material in these areas is included in the book, but only in conjunction with other topics, such as counting, solving recurrences and computing probabilities. In my experience, students need to be exposed to the material as a point of emphasis, rather than embedded inside other topics.
The first chapter introduces the basic principles of counting, permutations, combinations, binomial coefficients and a section on equivalence relations that is considered optional. This is because it is not used again in later chapters, something I don't agree with. Chapter two deals with cryptography and number theory. While I have no objection to this material in a discrete mathematics course, I prefer that it be put off to the latter part of the course. In chapter three, the logic of propositions and predicates as well as the laws of inference are examined. I generally prefer more coverage of these areas. Chapter four is 84 pages and covers induction, recursion and recurrence relations. Taking up approximately one fourth of the book, the coverage is complete. Probability is covered in chapter 5 and graph theory in chapter 6. The coverage in both is fairly typical, so I have no positive or negative comments on either one. Relations are covered in depth in an appendix. Solutions to the odd exercises are included in an appendix.
Since I prefer to start my discrete mathematics course by covering set theory, functions and logic, I have removed this book for adoption consideration.
... Read more

Book Description

Baffled by maths? Then don't give up hope.

Customer Reviews (2)

Broad but cursory
This give a very broad overview of many topics from matrix transformations to analytic geometry, but in a paperback less than a half inch thick, the topics are only given a cursory introduction.

not a book to learn new material from
Vince takes you through what is hopefully a revision course on maths that you will need if you are going to program graphics. The topics include matrix algebra. (Of course.) This gives the essence of translation and scaling in a three dimensional space. Then analytic geometry and B splines are covered.

The book has no problem sets. Though it does have one chapter at the end devoted to sample problems with their answers. It is not a book to learn new material from. ... Read more

Book Description

The two-volume textbook Comprehensive Mathematics for Computer Scientists, of which this is the first volume, is a self-contained comprehensive presentation of mathematics including sets, numbers, graphs, algebra, logic, grammars, machines, linear geometry, calculus, ODEs, and special themes such as neural networks, Fourier theory, wavelets, numerical issues, statistics, categories, and manifolds. The concept framework is streamlined but defining and proving virtually everything. The style implicitly follows the spirit of recent topos-oriented theoretical computer science. Despite the theoretical soundness, the material stresses a large number of core computer science subjects, such as, for example, a discussion of floating point arithmetic, Backus-Naur normal forms, L-systems, Chomsky hierarchies, algorithms for data encoding, e.g., the Reed-Solomon code. The numerous course examples are motivated by computer science and bear a generic scientific meaning.

Customer Reviews (2)

for computer scientists, not programmers
If you are a computer professional and wondering how much maths you need, this book might be a good answer. Much of its content is discrete maths. Like factorising primes. A background in this leads into the theory of public key encryption. Something well worth knowing. Note that the book's discussion of factorising primes is brief. You'd need another text for the public key material.

Also potentially useful is a foray into graph theory. Rings and modules also make their appearance. Predicate logic is another topic explained.

The text also discusses matrices from a viewpoint more formal than that of a typical first course in linear algebra. It generalises to an exposition of vector spaces. This portion of the text is continuum maths.

Do note that the average computer programmer actually uses very little of the material. Emphasis on programmer. But as the title says, the book is directed towards the computer scientist, who needs a more formal grounding.

Set high standards
This book is very exciting and motivating, as it sets high standards both with respect to the contents and the style.

It contains the main topics of modern math from the beginning, with applications to Computer Science: Sets, Relations, Logic, Numbers, Groups, Rings, Modules, Linear Algebra, Geometry, Quaternions.

The style is extremely compact, precise and readable, with a professional notation. The book is optimal designed (modularized)
which results in a moderate size (360 p.)

The objective of his approach is to provide a better formal competence for computer scientists by a sound mathematical education:

"Improved formal competence is urged by the object-oriented paradigm which progressively requires a programming style and a design strategy of high abstraction level in conceptual engineering."

The accompanying online course contains illustrations and animations, which at the moment don't go far beyond the book's contents.
Nevertheless, the online part has a great potential for further applications, e.g. programming projects (Moebius-Strip, Raytracing, 3D Stereo, ...)

We hope, this book will establish itself as a standard for ambitious schools and students. ... Read more

Book Description
Possibly the most comprehensive overview of computer graphics as seen in the context of geometric modelling, this two volume work covers implementation and theory in a thorough and systematic fashion. Computer Graphics and Geometric Modelling: Mathematics, contains the mathematical background needed for the geometric modeling topics in computer graphics covered in the first volume. This volume begins with material from linear algebra and a discussion of the transformations in affine & projective geometry, followed by topics from advanced calculus & chapters on general topology, combinatorial topology, algebraic topology, differential topology, differential geometry, and finally algebraic geometry. Two important goals throughout were to explain the material thoroughly, and to make it self-contained. This volume by itself would make a good mathematics reference book, in particular for practitioners in the field of geometric modelling. Due to its broad coverage and emphasis on explanation it could be used as a text for introductory mathematics courses on some of the covered topics, such as topology (general, combinatorial, algebraic, and differential) and geometry (differential & algebraic). ... Read more

Book Description
This book introduces readers to the mathematics of computer science and prepares them for the math they will encounter in other college courses. It includes applications that are specific to computer science, helps learners to develop reasoning skills, and provides the fundamental mathematics necessary for computer scientists.Chapter topics include sets, functions and relations, Boolean algebra, natural numbers and induction, number theory, recursion, solving recurrences, counting, matrices, and graphs.For computer scientists and the enhancement of programming skills. ... Read more

Customer Reviews (1)

This book sucks
If you are a math professor, considering using this book for a discrete math class, please, spare your students the misery and the money. My professor, with good intentions, assigned this book because it is cheeper than a traditional text book (still extremely expensive considering the book is very short and does not go into detail.)Most students in the class ended up buying an additional discrete mathematics textbook to supplement this one because this book did not explain most of the concepts in enough detail. I am studying for a test right now and the book is not helping me at all, especially since for a comprehensive discrete math course, my professor had to include in his lecture a lot of content not mentioned in this book.
Also, there was no CS prerequirement for this course, so the content of this book that deals with computer programing is off limits for the class and irrelevent and impossible to understand for me, a student with no background in computer science.
I would have prefered to spend the additional 40-60$ of the bat for a comprehensive well written discrete mathematics book, rather than waste $40-$50 on this one and have to spend $100 on another textbook to supplement the content. ... Read more

Book Description

The mathematical knowledge needed for computer and information sciences­­including, particularly, the binary number system, logic circuits, graph theory, linear systems, probability and statistics­­get clear and concise coverage in this invaluable study guide. Basic high school math is all that's needed to follow the explanations and learn from hundreds of practical problems solved step-by-step. Hundreds of review questions with answers help reinforce learning and increase skills.

Customer Reviews (4)

Well written but all over the map as far as subject matter
I found this Schaum's outline to be a very well written book on a variety of elementary mathematical subjects probably most suitable for someone who has just graduated high school and is a freshman starting some kind of technical degree. However, beyond that, I am not sure exactly who the target audience is supposed to be. The book contains some information on the very basic computer math that a college freshman engineering or ITstudent would need to know such as binary number systems and computer codes and math. However, some of the other material is very much oriented towards the aspiring computer engineer but is not really mathematical such as the sections on logic, flowcharts, sets and relations, and logic gates. Other sections are mathematical but don't necessarily have anything to do with computers and could be just as helpful to a student of applied mathematics such as the sections on linear algebra, linear equations, statistics, and graph theory.

Thus my recommendation would be that if you need a good tutorial on a variety of mathematical and pseudo-mathematical topics that tie in well with a technical degree of any kind, the low price of admission on this odd little outline is probably worth it. Just don't expect it to rescue you in any one course in particular. It will more likely rescue you in particular sections of several courses instead.

Great book which helped me tremendously
This book tremendously helped me understand computers and their relationship to mathematics and helped clear about doubts all through the semesters in college that I studied computer science. I strongly recommend this book to anyone majoring in computer science, information technologies, computer programming, etc ... as this book serves as a pillar to understanding the fundamentals of computer hardware and software.

A cheap and good book
The book is pretty much suitable for students who has finishedhigh school maths but requires revision on maths for computing. A few chapters are simple enough for students to run over quickly. There is asupplementary questions and solved questions with working for student tounderstand and practice on. Overall the book is good enough to refresh yourmemory on computer maths.

very poorly structured
For a start this book is never intended to be used by itself but as a supplement,so any attempt to use it as a textbook will only reslut in frustartion. Furthermore its not geared to teach as many VERY importantconcepts are simply ommited and the author makes far too many assumptionswhich all results in an incomplete book. The book attempts to cut studytime but also ends up cutting too much in clear and sufficent explanations.Use this book once you have completly mastered whathever topic you areinvolved in,and even then its only good for the practise problems at theend of each chapter. All in all DONT buy this book if you are trying tolearn something...use it ONLY for reviwing and just as a supplement. ... Read more

Book Description
This is a mathematics book, not a programming book, although it explains Pascal to beginners. It is aimed at high school students and undergraduates with a strong interest in mathematics, and teachers looking for fresh ideas. It is full of diverse mathematical ideas requiring little background. It includes a large number of challenging problems, many of which illustrate how numerical computation leads to conjectures which can then be proved by mathematical reasoning. It is assumed that readers have a PC at their disposal. ... Read more

Customer Reviews (1)

Great book when published, now very much out of date
When this book was published in 1993, it was a great book, suitable for use in many different math classes. Engel uses programs written in the language Pascal to solve many different math problems to the extent that Pascal could handle them. At that time, the size of integers was rather limited, so solutions could be found only as long as they were small. The problems and solutions are split into the following categories:

a) Number theory.
*) Probability
*) Statistics
*) Combinatorial algorithms
*) Numerical algorithms

There is also a section of miscellaneous problems. Each problem is explained in detail and there is a functioning Pascal program that will identify any solutions.
Time and the several swimlanes of advancement in technology have rendered this book obsolete. Pascal is no longer the primary language of computer science instruction and the limits on the size of integers are now almost laughable. The power of symbolic mathematics packages such as Mathematica has rendered the writing of a program in any other way an act of foolishness. With their ability to store numbers almost unlimited in size and the ease and simplicity of writing programs inside them, symbolic mathematics packages give you almost unlimited power to perform mathematical tests. The associated ease and quality of display render the simple dot diagrams in this book almost useless.
A significant achievement in its time, this book is now largely of historical interest. Engel was a trailblazer with an axe and a shovel, only now even the programming novice can use bulldozers and chainsaws. ... Read more

Book Description
This book is the final version of a course on algorithmic information theory and the epistemology of mathematics and physics. It discusses Einstein and Goedel's views on the nature of mathematics in the light of information theory, and sustains the thesis that mathematics is quasi-empirical. There is a foreword by Cris Calude of the University of Auckland, and supplementary material is available at the author's web site. The special feature of this book is that it presents a new "hands on" didatic approach using LISP and Mathematica software. The reader will be able to derive an understanding of the close relationship between mathematics and physics.
"The Limits of Mathematics is a very personal and idiosyncratic account of Greg Chaitin's entire career in developing algorithmic information theory. The combination of the edited transcripts of his three introductory lectures maintains all the energy and content of the oral presentations, while the material on AIT itself gives a full explanation of how to implement Greg's ideas on real computers for those who want to try their hand at furthering the theory."
(John Casti, Santa Fe Institute) ... Read more

Customer Reviews (1)

A subjective estimation
I wish that I could give this book a higher rating than I have, for the subject matter is one that I find of enormous intrinsic interest. Moreover, Dr. Chaitin is one of the most important contributors to this field of the last 30+ years.

My reasons for being disappointed in this book may well be the reasons others enthusiastically endorse it. Dr. Chaitin himself, in his preface, places this volume as the one thing he would most wish to save were a disaster to wipe out the rest of his oeuvre.

The sub-title of the book is "A Course on Information Theory and the Limits of Formal Reasoning." This sub-heading I find to be quite misleading. The book is not a "course" on any thing -- rather, it is a collection of a very small number of informal papers that Dr. Chaitin has given in recent years, and a very large number of pages devoted to LISP programs that can be used to demonstrate aspects of his extensions to the results of Turing and Goedel. The collection of articles seem largely redundant to me; any one of the articles by itself would be sufficient to summarize the rest of the book's contents. As for the programming, that should either have been provided in the form of a CD-ROM (as only someone of a genuinely "special" nature would actually sit down and manually type in all those instructions) or a functioning URL (such URL's as do appear in the book do not seem to be working as of this writing, Mar. 2007).

I was hoping to get something more comprehenive, and that could function as a stand-alone text. This book seems to be neither. The technical details are all to be found elsewhere, and the functional aspects that might translate into an actual course of study are simply not to be found at all. Dr. Chaitin notes that the original technical work of his, published in the 60's, had a formal error that has since been corrected. Quite frankly, I would rather have that work plus a footnote regarding the later developments, than this volume which (sadly) I find of no real help. (I have since ordered and received a used, 1987 imprint of his "Algorithmic Information Theory" as printed by Cambridge.)

Alternatively, and perhaps more importantly, I would very much liked to have seen this "course" developed as a *COURSE*, rather than as three more or less popular, and largely independent, lectures. These lectures seem, at best, only to minimally build upon one another. A more integrated and coherent work that developed its subject in a step-wise manner, rather than repeating itself with only slightly different glosses, is something that I would have liked much more.

My background in logic and computer science, while not trivial, remains that of a studious and committed autodidact. It is possible that someone with less of a background in topics of formal reasoning than myself would find this book of enormous value. For me, however, it lacked both the technical details to make it a worthy struggle, and the pedagogical depth to make it of significant value. ... Read more

Book Description

This book covers all the traditional topicsof discrete mathematics—logic, sets, relations,functions, and graphs—and reflects recent trends incomputer science.Shows how to use discretemathematics and logic for specifying new computerapplications, and how to reason about programs in asystematic way. Describes Prolog, a programminglanguage based on logic, and a section on Miranda,language bad on functions. Features numerous exampleswhich relate the mathematical concepts to problems incomputer science.

Customer Reviews (2)

University Level material
I have used this textbook when delivering a second year University class on this topic and found that the thorough coverage of the topics was appreciated by my students. Even more important to them was the large number of examples that are presented, in detail, throughout the text.

This is a University Level textbook, not a Study Guide, and respects the reader's intellectual maturity by preparing them for subsequent classes. The perception of "density" implies that it is best taken with a liberal dose of classroom instruction - not many students seem to intuitively grasp discrete mathematics and learn the material wholly on their own. I know that I certainly did not when I was an undergrad!

For students who feel that the material is difficult, I always suggest using the library for another point of view. I also recommend the Schaum's Outline for Discrete Mathematics as a companion if the student is having significant difficulty with the concepts.

Obviously, I like the book, so why not a 5? Unfortunately, I don't know of any books that I would grant a 5 - the authors can always do something better :-)

Not for first-time students..
I'm sure this book covers all the stuff that it's meant to and I'm sure that if I was a post-grad in maths this would be a good book to use. However for anyone else this book is way too heavy reading. The authors have made no attempt to keep the material easy to study and understand. The whole book is just a continuous stream of information with the density of a neutron star and where every 5th or so word is a mathematical formula. Then again maybe I'm just biased because I hate the subject matter - I think it's unnecesarily obscure and difficult for a general computer science course.
And look at the price - that's nearly $150 for us Aussies, (although our uni co-op sells it for about A$90) and that doesn't include shipping fees. Don't you hate it the way they jack up the price on these text books because they know that you have to buy it to have any chance of passing the course. ... Read more

Book Description

• Includes new chapters on symmetry, limit and continuity, constructive solid geometry, and the Bezier curve.
• Provides many new figures and exercises.
• Contains an annotated suggested reading list with exercises and answers in each chapter.
• Appeals to both academics and professionals.
• Offers a new solutions manual for instructors.

Customer Reviews (8)

Good geometry and few computer stuff
This 2nd edition of a 1989 book maintains its mathematic approach with few or no relation to computers. The long title suggests a book with practical examples, but what it contains is pure math. Just to give an idea, the word "pixel" appears just in one page. However, the basics on hypernumbers, introduction on matrix methods, limits, points, lines, CSG spend half of book.

Computer Graphics today is much more than geometry, by the way solved by powerful graphics cards and APIs like OpenGL. Scientific visualization uses intensively colors, transparencies, movements, animation and none of these subjects are even touched. Nowadays mathematics apply also to fog, antialianing, specular, caustics, LOD, transparencies, reflexes, lightning. Unfortunately these math applications aren't tackled in this book. So, don't expect to find computer graphics in this book, as we understand it today. This is still a 1989 book.

If the title was just "Computer Graphics: An Introduction to the Mathematics and Geometry", as in its first edition, it was more related to its content. Unfortunately, just extending the book's title doesn't make it up-to-date.

Don't. No, no, no.
I can't understand the raving reviews. It is weak on math, and has no practical example on CG.
Let me give you an example: Chapter 2 (allegedly on "Matrix Methods"), exercise 2.20 (there are 25): Compute the determinant of the following matrix (I'll use Matlab/Scilab notation): M = [ 2 0; -3 2] Are you serious? What about learning Limits and Continuity in one short chapter? No comments...Seriously, if you don't already know this stuff, should you be looking at CG in the first place?
After I got this book, I got F.S. Hill's Computer graphics Using OpenGL, which is much thicker, mathematically oriented, and has practical examples in C++, and Parberry's and Dunn's 3D Math Primer for Graphics and Game Development which doesn't attempt so much as Hill's and Mortenson's, but has nice code in C++. There might be better options. Not to mention that one has to buy and study real math books.
I give it 1 star as the book falls short of its stated goals of preparing the reader for more advanced study.

Not bad, but not the best
Though the book covers a broad range of mathematics present in the field of computer graphics, I found it lacking in certain respects.The plentiful diagrams and figures greatly assisted comprehension of stated materials, yet I would like to see written proofs of methods used.In addition, such topics as quaternions are only briefly mentioned in historical references.Although this is an excellant mathematics book, I strongly recommend the purchase of Mathematics for 3D Game Programming & Computer Graphics in addition to this title.

great math book (did i really say that!)
if you want to learn the mathematics behind computer graphics buy this book.

it is well written and the explanations are first rate.i'm studying on my own with the book and hardly have need to ask others for any help with understanding topics.

i'm not sure what else to say, except I first tried Mathematics for Computer Graphics and Game Programming and it was pretty horrible compared to this book.

It's cheap, so get it.

Good resource for game developers
This is a good book that is better suited as a reference than a text book. The book seems to be set up with that in mind as the format allows for each concept covered to be easily found. For example, I have taken a course in Linear Algebra. I was still a little shakey about the rotation matricies that are commonly used in game programming. This book brought me up to speed in no time, but if I hadn't taken Linear Algebra, I'm not so sure that I would have fully understood that section. But again, overall this is a great reference to have around! ... Read more

Product Description
The advent of fast computers and the search for efficient algorithms revolutionized combinatorics and brought about the field of discrete mathematics. This book is an introduction to the main ideas and results of discrete mathematics, and with its emphasis on algorithms it should be interesting to mathematicians and computer scientists alike. The book is organized into three parts: enumeration, graphs and algorithms, and algebraic systems. There are 600 exercises with hints and solutions to about half of them. The only prerequisites for understanding everything in the book are linear algebra and calculus at the undergraduate level.Martin Aigner is a professor of mathematics at the Free University of Berlin. He received his PhD at the University of Vienna and has held a number of positions in the USA and Germany before moving to Berlin. He is the author of several books on discrete mathematics, graph theory, and the theory of search. The Monthly article Turan's graph theorem earned him a 1995 Lester R. Ford Prize of the MAA for expository writing, and his book Proofs from the BOOK with Günter M. Ziegler has been an international success with translations into 12 languages. ... Read more

Customer Reviews (2)

Solutions
I wanted to know whether there is any officially published solution book for the above book?As its certainly not possible to solve each n every problem which one(actually each n everyone) would find time consuming. So let me know about it.That's the only reason i rated the book 4 stars.Its worth 5 starsotherwise.

I need detail solution book for its exercise!
Can u tell me if this text book has a detail answer book to its exercis ?? I need to know... Please mail me about my question... Thank you so much ... Read more

Product Description
Unlike many engineering mathematics books, the new edition of this comprehensive applications-oriented book uses computer programs in almost every chapter to demonstrate the mathematical concepts under discussion. Designed for engineering students as well as practicing engineers and scientists, the book has hundreds of examples with in-text solutions. In terms of content, it covers the entire sequence of mathematical topics needed by the majority of university programs, including ODE, PDE, complex variables, probability/statistics, and numerical methods. The authors demonstrate how the mathematical concepts will be used in practical applications such as Fractals, Robotics, Circuits, Membrane simulation, Collision Detection, Ray Tracing, Signal Processing, and more. A CD-ROM with the source code for the in-text computer programs (written in C) includes calculation routines and simulations. Features„X Includes computer programs to demonstrate mathematical concepts used in Fractals, Robotics, Circuits, Membrane simulation, Collision Detection, Ray Tracing, Signal Processing and more„X Accompanied by a CD-ROM with source code, calculation programs, ¡§live¡¨ figures, and simulations „X Provides self-instruction techniques with hundreds of examples with in text, worked-out solutionsBrief Table of Contents1. Complex Numbers; 2. Theory of Equations and Curve Fitting; 3. Determinants and Matrices; 4. Analytical Solid Geometry; 5. Partial Differentiation; 6. Multiple Integrals; 7. Vector Algebra; 8. Vector Calculus; 9. Infinite Series; 10. Fourier Series; 11. Differential Equations of First Order; 12. Applications of Differential Equations of First Order; 13. Linear Differential Equations; 14. Applications of Linear Differential Equations; 15. Special Functions and Series Solution of Differential Equations; 16. Partial Differential Equations; 17. Applications of Partial Differential Equations; 18. The Laplace Transforms; 19. Functions of a Complex Variable; 20. Integral Transforms; 21. Statistics and Probability; 22. Finite Differences and Numerical Methods; 23. The Z-Transforms; 24. Numerical Solution of Ordinary Differential Equations; 25. Numerical Solution of Partial Differential Equations; 26. Curvilinear Co-ordinates; 27. Tensor Analysis; 28. Virtual Work; Appendixes ... Read more

Book Description
This volume is a collection of articles written by experiencedprimary, secondary, and collegiate educators. The book explainswhy discrete mathematics should be taught in K--12 classroomsand offers practical guidance on how to do so.

In this book, teachers at all levels will find a great deal ofvaluable material to help them introduce discrete mathematics intheir classrooms. One main article provides a comprehensive anddetailed view of discrete mathematics for K--12. Another surveysthe resources that are available for teachers. School anddistrict curriculum leaders will find material that addresseshow discrete mathematics can be introduced into their curricula.College faculty members will find ideas and topics that can beincorporated into a variety of courses.

Features:

Classroom activities and an annotated list of resources.

Authors who are directors of innovative programs and who are wellknown for their work.

A description of discrete mathematics providing the opportunityfor a fresh start for students who have been previouslyunsuccessful in mathematics.

Discussion on discrete mathematics as it is used to achieve thegoals of the current effort to improve mathematics education.

Guidance on topics, resources and teaching; a valuable guide forboth pre-service and in-service professional development. ... Read more

Book Description
Baffled by maths? Then don't give up hope.
John Vince will show you how to understand many of the mathematical ideas used in computer animation, virtual reality, CAD, and other areas of computer graphics.
In ten chapters you will rediscover - and hopefully discover for the first time a new way of understanding - the mathematical techniques required to solve problems and design computer programs for computer graphic applications. Each chapter explores a specific mathematical topic and takes you forward into more advanced areas until you are able to understand 3D curves and surface patches, and solve problems using vectors.
After reading the book, you should be able to refer to more challenging books with confidence and develop a greater insight into the design of computer graphics software.
Get to grips with mathematics fast ...
- Numbers
- Algebra
- Trigonometry
- Coordinate geometry
- Transforms
- Vectors
- Curves and surfaces
- Analytic geometry
Essential Mathematics for Computer Graphics fast
The book you will read once, and refer to over and over again! ... Read more

Customer Reviews (3)

Great refresher
I love that the author was able to fit so much into so few pages. If I had to carry my old math books from college, I would have to hire a chiropractor.

It's important to point out that this isn't really the text to learn math from; however, it's a great little refresher that's to the point. So if you're rusty, get this book... if this is all new to you, I'd look into more indepth books.

Terrific Book!
This book is terrific. The word "fast" in the title hints that you are only going to get a superficial overview of various topics, but this is not the case: you actually come away with an UNDERSTANDING of the topics. The chapter on interpolation is a case in point... I came away being able to derive cubic interpolants, which means that I UNDERSTOOD cubic interpolants. The discussion on quaternions paved the way for me to be able to use them, and the chapter on Bezier curves and patches also gave me an excellent foundation on which to begin using them in OpenGL. In short, I guess this is just one of those magical books in which the author knows how to explain complex subjects in a simple, direct manner. This book is excellent, and I heartily recommend it.

Concise, tight mathematics reference.
First off, here is what you should expect when buying this book: it's a mathematical quick-reference book, and as such, it contains absolutely no code.What it does contain is background information and formulas for a bunch of the most common situations you'll encounter in computer graphics.

This book covers a lot of ground in relatively few pages, making it a tight read.It moves through Algebra, Trig, and Geometry, along the way covering such topics as 2D and 3D transforms, perspective, and Beziers.Most of the topics in the book are explained in sufficient detail in a small amount of space.The author makes great and liberal use of diagrams and pictures, which I find particularly helpful to understanding the material since math isn't one of my strong points.

There are also straightforward examples that demonstrate the mathematical principles, although there is usually only one or two examples per topic.This can be difficult if, like me, you have problems "getting it" without lots of examples to chew on.

As a software engineer, I enjoy figuring out how things work.And in that respect, this book is great.I have some experience dabbling in OpenGL, and I've read several other computer graphics texts that were laden with code examples but lacking in adequate explanations of the math being used.I have also read basic math books that were not geared towards computer programming topics, which made application and transfer of the mathematical threory difficult.This book filled the gap for me, taking me from thinking things like "Hey, look what I can do with OpenGL" to "Hey, I know how OpenGL is doing that!!"Well, maybe I don't know *exactly* how OpenGL works behind the scenes, but because of this book I have a much better understanding of basic computer graphics.

Highly recommended, especially if you don't have a strong math or computer science background. ... Read more