Pragmatic Bookshelf | 2013 | 270 páginas | pdf | 5,7 Mb
link
epub - 2 Mb - link
Mathematics is beautiful--and it can be fun and exciting as well as practical. Good Math is your guide to some of the most intriguing topics from two thousand years of mathematics: from Egyptian fractions to Turing machines; from the real meaning of numbers to proof trees, group symmetry, and mechanical computation. If you've ever wondered what lay beyond the proofs you struggled to complete in high school geometry, or what limits the capabilities of computer on your desk, this is the book for you.
Why do Roman numerals persist? How do we know that some infinities are larger than others? And how can we know for certain a program will ever finish? In this fast-paced tour of modern and not-so-modern math, computer scientist Mark Chu-Carroll explores some of the greatest breakthroughs and disappointments of more than two thousand years of mathematical thought. There is joy and beauty in mathematics, and in more than two dozen essays drawn from his popular "Good Math" blog, you'll find concepts, proofs, and examples that are often surprising, counterintuitive, or just plain weird.Mark begins his journey with the basics of numbers, with an entertaining trip through the integers and the natural, rational, irrational, and transcendental numbers. The voyage continues with a look at some of the oddest numbers in mathematics, including zero, the golden ratio, imaginary numbers, Roman numerals, and Egyptian and continuing fractions. After a deep dive into modern logic, including an introduction to linear logic and the logic-savvy Prolog language, the trip concludes with a tour of modern set theory and the advances and paradoxes of modern mechanical computing.If your high school or college math courses left you grasping for the inner meaning behind the numbers, Mark's book will both entertain and enlighten you.
Contents
Preface . . . . . . . . . . . xi
Part I — Numbers
1. Natural Numbers . . . . . . . . . 3
1.1 The Naturals, Axiomatically Speaking 4
1.2 Using Peano Induction 7
2. Integers . . . . . . . . . . . 9
2.1 What’s an Integer? 9
2.2 Constructing the Integers—Naturally 11
3. Real Numbers . . . . . . . . . 15
3.1 The Reals, Informally 15
3.2 The Reals, Axiomatically 18
3.3 The Reals, Constructively 20
4. Irrational and Transcendental Numbers . . . 23
4.1 What Are Irrational Numbers? 23
4.2 The Argh! Moments of Irrational Numbers 24
4.3 What Does It Mean, and Why Does It Matter? 26
Part II — Funny Numbers
5. Zero . . . . . . . . . . . 31
5.1 The History of Zero 31
5.2 An Annoyingly Difficult Number 34
6. e: The Unnatural Natural Number . . . . . 37
6.1 The Number That’s Everywhere 37
6.2 History 39
6.3 Does e Have a Meaning? 40
7. φ: The Golden Ratio . . . . . . . . 41
7.1 What Is the Golden Ratio? 42
7.2 Legendary Nonsense 44
7.3 Where It Really Lives 45
8. i: The Imaginary Number . . . . . . . 47
8.1 The Origin of i 47
8.2 What i Does 49
8.3 What i Means 50
Part III — Writing Numbers
9. Roman Numerals . . . . . . . . 55
9.1 A Positional System 55
9.2 Where Did This Mess Come From? 57
9.3 Arithmetic Is Easy (But an Abacus Is Easier) 58
9.4 Blame Tradition 61
10. Egyptian Fractions . . . . . . . . 65
10.1 A 4000-Year-Old Math Exam 65
10.2 Fibonacci’s Greedy Algorithm 66
10.3 Sometimes Aesthetics Trumps Practicality 68
11. Continued Fractions . . . . . . . . 69
11.1 Continued Fractions 70
11.2 Cleaner, Clearer, and Just Plain Fun 72
11.3 Doing Arithmetic 74
Part IV — Logic
12. Mr. Spock Is Not Logical . . . . . . . 79
12.1 What Is Logic, Really? 81
12.2 FOPL, Logically 82
12.3 Show Me Something New! 86
13. Proofs, Truth, and Trees: Oh My! . . . . . 91
13.1 Building a Simple Proof with a Tree 92
13.2 A Proof from Nothing 94
13.3 All in the Family 96
13.4 Branching Proofs 98
14. Programming with Logic . . . . . . . 103
14.1 Computing Family Relationships 104
14.2 Computation with Logic 108
15. Temporal Reasoning . . . . . . . . 117
15.1 Statements That Change with Time 118
15.2 What’s CTL Good For? 123
Part V — Sets
16. Cantor’s Diagonalization: Infinity Isn’t Just Infinity . . . 127
16.1 Sets, Naively 128
16.2 Cantor’s Diagonalization 131
16.3 Don’t Keep It Simple, Stupid 135
17. Axiomatic Set Theory: Keep the Good, Dump the Bad .. . 139
17.1 The Axioms of ZFC Set Theory 140
17.2 The Insanity of Choice 147
17.3 Why? 150
18. Models: Using Sets as the LEGOs of the Math World . . . 153
18.1 Building Natural Numbers 154
18.2 Models from Models: From Naturals to Integers and Beyond! 156
19. Transfinite Numbers: Counting and Ordering Infinite Sets . . . . 161
19.1 Introducing the Transfinite Cardinals 161
19.2 The Continuum Hypothesis 163
19.3 Where in Infinity? 164
20. Group Theory: Finding Symmetries with Sets . . 167
20.1 Puzzling Symmetry 167
20.2 Different Kinds of Symmetry 171
20.3 Stepping into History 173
20.4 The Roots of Symmetry 176
Part VI — Mechanical Math
21. Finite State Machines: Simplicity Goes Far . . . 183
21.1 The Simplest Machine 183
21.2 Finite State Machines Get Real 187
21.3 Bridging the Gap: From Regular Expressions to Machines 189
22. The Turing Machine . . . . . . . . 197
22.1 Adding a Tape Makes All the Difference 198
22.2 Going Meta: The Machine That Imitates Machines 203
23. Pathology and the Heart of Computing . . . 209
23.1 Introducing BF: The Great, the Glorious, and the Completely Silly 211
23.2 Turing Complete, or Completely Pointless? 214
23.3 From the Sublime to the Ridiculous 215
24. Calculus: No, Not That Calculus—λ Calculus . . 219
24.1 Writing λ Calculus: It’s Almost Programming! 220
24.2 Evaluation: Run It! 224
24.3 Programming Languages and Lambda Strategies 226
25. Numbers, Booleans, and Recursion . . . . 231
25.1 But Is It Turing Complete? 231
25.2 Numbers That Compute Themselves 232
25.3 Decisions? Back to Church 235
25.4 Recursion: Y Oh Y Oh Y? 237
26. Types, Types, Types: Modeling λ Calculus . . . 243
26.1 Playing to Type 244
26.2 Prove It! 249
26.3 What’s It Good For? 250
27. The Halting Problem . . . . . . . 253
27.1 A Brilliant Failure 254
27.2 To Halt or Not To Halt? 256
Bibliography . . . . . . . . . 261
Sem comentários:
Enviar um comentário