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### 基本信息

- 原书名：Concrete Mathematics A Foundation for Computer Science(Second Edition)
- 原出版社： Addison Wesley

- 作者：
**（美）Ronald L.Graham,Donald E.Knuth,Oren Patashnik** - 丛书名：
**经典原版书库** - 出版社：机械工业出版社
- ISBN：
**9787111105763** - 上架时间：2002-9-5
- 出版日期：2002 年8月
- 开本：32开
- 页码：680
- 版次：1-1
- 所属分类：计算机 > 计算机科学理论与基础知识 > 基础知识 > 综合

教材

### 【插图】

### 编辑推荐

《具体数学：计算机科学基础》（英文版第2版）是一本全英文版的具体数学计算机科学基础的参考书。

### 作译者

### 目录

1.l The Tower of Hanoi 1

1.2 Lines in the P1ane 4

1.3 The Josephus Problem 8

Exercises 17

2 Sums

2.1 Notation 21

2.2 Sums and Recurrences 25

2.3 Mainpulation of Sums 30

2.4 Mu1tip1e Sums 34

2.5 General Methods 4l

2.6 Finite and Infinite Calcu1us 47

2.7 Infinite Sums 56

Exercises 62

3 Integer Functions

3.1 Floors and Ceilings 67

3.2 Floor/Ceiling Applications 70

3.3 Floor/Ceiling Recurrences 78

3.4 'mod" The Binary Operation 81

3.5 F1oor/Cei1ing Sums 86

### 前言

It was dark and stormy decade when Concrete Mathematics was born. Long-held values were constantly being questioned during those turbulent years; college campuses were hotbeds of controversy. The college curriculum itself was challenged, and mathematics did not escape scrutiny. John Hammersley had just written a thought-provoking article "On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities" 176 ; other worried mathematicians 332 even asked, "Can mathematics be saved?" One of the present authors had embarked on a series of books called The Art of Computer Programming, and in writing the first volume he (DEK) had found that there were mathematical tools missing from his repertoire; the mathematics he needed for a thorough, well-grounded understanding of computer programs was quite different from what he'd learned as a mathematics major in college. So he introduced a new course, teaching what he wished somebody had taught him.

The course title "Concrete Mathematics" was originally intended as an antidote to "Abstract Mathematics," since concrete classical results were rapidly being swept out of the modern mathematical curriculum by a new wave of abstract ideas popularly called the "New Math." Abstract mathematicsis a wonderful subject, and there's nothing wrong with it: It's beautiful, general, and useful. But its adherents had become deluded that the rest of mathematics was inferior and no longer worthy of attention. The goal of generalization had become so fashionable that a generation of mathematicians had become unable to relish beauty in the particular, to enjoy the challenge of solving quantitative problems, or to appreciate the value of technique. Abstract mathematics was becoming inbred and losing touch with reality; mathematical education needed a concrete counterweight in order to restore a healthy balance.

When DEK taught Concrete Mathematics at Stanford for the first time he explained the somewhat strange title by saying that it was his attempt to teach a math course that was hard instead of soft. He announced that, contrary to the expectations of some of his colleagues, he was not going to teach the Theory of Aggregates, not Stone's Embedding Theorem, nor even the Stone-Cech compactification. (Several students from the civil engineering department got up and quietly left the room.)

Although Concrete Mathematics began as a reaction against other trends, the main reasons for its existence were positive instead of negative. And as the course continued its popular place in the curriculum, its subject matter "solidified" and proved to be valuable in a variety of new applications. Meanwhile, independent confirmation for the appropriateness of the name came from another direction, when Z.A. Melzak published two volumes entitled Companion to Concrete Mathematics 267.

The material of concrete mathematics may seem at first to be a disparate bag of tricks, but practice makes it into a disciplined set of tools. Indeed, the techniques have an underlying unity and a strong appeal for many people. When another one of the authors (RLG) first taught the course in 1979, the students had such fun that they decided to hold a class reunion a year later.

But what exactly is Concrete Mathematics? It is a blend of continuous and discrete mathematics. More concretely, it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems. Once you, the reader, have learned the material in this book, all you will need is a cool head, a large sheet of paper, and fairly decent handwriting in order to evaluate horrendous-looking sums, to solve complex recurrence relations, and to discover subtle patterns in data. You will be so fluent in algebraic techniques that you will often find it easier to obtain exact results than to settle for approximate answers that are valid only in a limiting sense.

The major topics treated in this book include sums, recurrences, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods. The emphasis is on manipulative techniques rather than on existence theorems or combinatorial reasoning; the goal is for each reader to become as familiar with discrete operation (like the greatest integer function and finite summation) as a student of calculus is familiar with continuous operations (like the absolute-value function and infinite integration)

Notice that this list of topics is quite different from what is usually taught nowadays in undergraduate course entitled "Discrete Mathematics." Therefore the subject needs a distinctive name, and "Concrete Mathematics" has proved to be as suitable as another

The original textbook for Stanford's course on concrete mathematics was the "Mathematical Preliminaries" section in The Art of Computer Programming 207. But the presentation in those 110 pages is quite terse, so another author (OP) was inspired to draft a lengthy set of supplementary notes. The present book is an outgrowth of those notes; it is an expansion of, and a more leisurely introduction to, the material if Mathematical Preliminaries. Some of the more advanced parts have been omitted; on the other hand, several topics not found there have been included here so that the story will be complete