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Publisher:

Princeton University Press

Publication Date:

2013

Number of Pages:

228

Format:

Hardcover

Price:

24.95

ISBN:

9780691151007

Category:

General

[Reviewed by , on ]

Charles Ashbacher

11/8/2012

Nearly every teaching mathematician has heard some form of the question “What is math good for?” This book starts with the work of George Boole and his “Laws of Thought” that are quite abstract and takes the implementation of those ideas by Alan Turing and Claude Shannon to the development and structure of the modern computer. Through the treatment, which includes some basic digital logic design and probabilistic computations, the reader is taken on a journey from the development of some abstract mathematical ideas through a nearly ubiquitous application of those ideas within the modern world with so many embedded digital computers.

While most of the treatment is understandable, readers who are unfamiliar with Turing machines will likely struggle a bit when reading the section that covers them. While simple in structure, Turing machines are an abstraction that is best understood by seeing some examples, which is not done well here. The sections on digital devices such as logic gates and flip-flops may also prove challenging.

I enjoyed the discussion of Claude Shannon. In the history of the computer and development of the internet and World Wide Web, his ideas and contributions are too often overlooked. He is one of my heroes and I believe that everyone that reads this book will come to the same conclusion. If you read this book and hear the question, “What good is algebra?” you will have a ready and irrefutable answer, “Boolean algebra is the basis for describing and designing the circuits of computers.”

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing *The Journal of Recreational Mathematics*. In his spare time, he reads about these things and helps his daughter in her lawn care business.

*Preface* xi

**1 What You Need to Know to Read This Book** 1

Notes and References 5

**2 Introduction** 6

Notes and References 14

**3 George Boole and Claude Shannon: Two Mini-Biographies** 17

- 3.1 The Mathematician 17
- 3.2 The Electrical Engineer 28
- Notes and References 39

**4 Boolean Algebra** 43

- 4.1 Boole's Early Interest in Symbolic Analysis 43
- 4.2 Visualizing Sets 44
- 4.3 Boole's Algebra of Sets 45
- 4.4 Propositional Calculus 48
- 4.5 Some Examples of Boolean Analysis 52
- 4.6 Visualizing Boolean Functions 59
- Notes and References 65

**5 Logical Switching Circuits** 67

- 5.1 Digital Technology: Relays versus Electronics 67
- 5.2 Switches and the Logical Connectives 68
- 5.3 A Classic Switching Design Problem 71
- 5.4 The Electromagnetic Relay and the Logical NOT 73
- 5.5 The Ideal Diode and the Relay Logical AND and OR 76
- 5.6 The Bi-Stable Relay Latch 81
- Notes and References 84

**6 Boole, Shannon, and Probability** 88

- 6.1 A Common Mathematical Interest 88
- 6.2 Some Fundamental Probability Concepts 89
- 6.3 Boole and Conditional Probability 96
- 6.4 Shannon, Conditional Probability, and Relay Reliability 99
- 6.5 Majority Logic 106
- Notes and References 110

**7 Some Combinatorial Logic Examples** 114

- 7.1 Channel Capacity, Shannon's Theorem, and Error-Detection Theory 114
- 7.2 The Exclusive-OR Gate (XOR) 122
- 7.3 Error-Detection Logic 127
- 7.4 Error-Correction Theory 128
- 7.5 Error-Correction Logic 132
- Notes and References 137

**8 Sequential-State Digital Circuits** 139

- 8.1 Two Sequential-State Problems 139
- 8.2 The NOR Latch 142
- 8.3 The Clocked RS Flip-Flop 146
- 8.4 More Flip-Flops 154
- 8.5 A Synchronous, Sequential-State Digital Machine Design Example 158
- Notes and References 160

**9 Turing Machines** 161

- 9.1 The First Modern Computer 162
- 9.2 Two Turing Machines 164
- 9.3 Numbers We Can't Compute 168
- Notes and References 173

**10 Beyond Boole and Shannon** 176

- 10.1 Computation and Fundamental Physics 176
- 10.2 Energy and Information 178
- 10.3 Logically Reversible Gates 180
- 10.4 Thermodynamics of Logic 184
- 10.5 A Peek into the Twilight Zone: Quantum Computers 188
- 10.6 Quantum Logic--and Time Travel, Too! 197

Notes and References 205

**Epilogue**

For the Future: The Anti-Amphibological Machine 210

**Appendix**

Fundamental Electric Circuit Concepts 219

*Acknowledgments* 223

*Index* 225

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