Good Math (For Jan S Morrison) by Chu-Carroll, Mark C. -- Read -- Imperial Library of Trantor

Index

Good Math Table of Contents Early praise for Good Math Preface

Where’d This Book Come From? Who This Book Is For How to Read This Book What Do You Need? Acknowledgments

Part 1: Numbers 1 Natural Numbers

1.1 The Naturals, Axiomatically Speaking 1.2 Using Peano Induction

2 Integers

2.1 What’s an Integer? 2.2 Constructing the Integers—Naturally

3 Real Numbers

3.1 The Reals, Informally 3.2 The Reals, Axiomatically 3.3 The Reals, Constructively

4 Irrational and Transcendental Numbers

4.1 What Are Irrational Numbers? 4.2 The Argh! Moments of Irrational Numbers 4.3 What Does It Mean, and Why Does It Matter?

Part 2: Funny Numbers 5 Zero

5.1 The History of Zero 5.2 An Annoyingly Difficult Number

6 e : The Unnatural Natural Number

6.1 The Number That’s Everywhere 6.2 History 6.3 Does e Have a Meaning?

7 φ : The Golden Ratio

7.1 What Is the Golden Ratio? 7.2 Legendary Nonsense 7.3 Where It Really Lives

8 i : The Imaginary Number

8.1 The Origin of i 8.2 What i Does 8.3 What i Means

Part 3: Writing Numbers 9 Roman Numerals

9.1 A Positional System 9.2 Where Did This Mess Come From? 9.3 Arithmetic Is Easy (But an Abacus Is Easier) 9.4 Blame Tradition

10 Egyptian Fractions

10.1 A 4000-Year-Old Math Exam 10.2 Fibonacci’s Greedy Algorithm 10.3 Sometimes Aesthetics Trumps Practicality

11 Continued Fractions

11.1 Continued Fractions 11.2 Cleaner, Clearer, and Just Plain Fun 11.3 Doing Arithmetic

Part 4: Logic 12 Mr. Spock Is Not Logical

12.1 What Is Logic, Really? 12.2 FOPL, Logically 12.3 Show Me Something New!

13 Proofs, Truth, and Trees: Oh My!

13.1 Building a Simple Proof with a Tree 13.2 A Proof from Nothing 13.3 All in the Family 13.4 Branching Proofs

14 Programming with Logic

14.1 Computing Family Relationships 14.2 Computation with Logic

15 Temporal Reasoning

15.1 Statements That Change with Time 15.2 What’s CTL Good For?

Part 5: Sets 16 Cantor’s Diagonalization: Infinity Isn’t Just Infinity

16.1 Sets, Naively 16.2 Cantor’s Diagonalization 16.3 Don’t Keep It Simple, Stupid

17 Axiomatic Set Theory: Keep the Good, Dump the Bad

17.1 The Axioms of ZFC Set Theory 17.2 The Insanity of Choice 17.3 Why?

18 Models: Using Sets as the LEGOs of the Math World

18.1 Building Natural Numbers 18.2 Models from Models: From Naturals to Integers and Beyond!

19 Transfinite Numbers: Counting and Ordering Infinite Sets

19.1 Introducing the Transfinite Cardinals 19.2 The Continuum Hypothesis 19.3 Where in Infinity?

20 Group Theory: Finding Symmetries with Sets

20.1 Puzzling Symmetry 20.2 Different Kinds of Symmetry 20.3 Stepping into History 20.4 The Roots of Symmetry

Part 6: Mechanical Math 21 Finite State Machines: Simplicity Goes Far

21.1 The Simplest Machine 21.2 Finite State Machines Get Real 21.3 Bridging the Gap: From Regular Expressions to Machines

22 The Turing Machine

22.1 Adding a Tape Makes All the Difference 22.2 Going Meta: The Machine That Imitates Machines

23 Pathology and the Heart of Computing

23.1 Introducing BF: The Great, the Glorious, and the Completely Silly 23.2 Turing Complete, or Completely Pointless? 23.3 From the Sublime to the Ridiculous

24 Calculus: No, Not That Calculus— λ Calculus

24.1 Writing λ Calculus: It’s Almost Programming! 24.2 Evaluation: Run It! 24.3 Programming Languages and Lambda Strategies

25 Numbers, Booleans, and Recursion

25.1 But Is It Turing Complete? 25.2 Numbers That Compute Themselves 25.3 Decisions? Back to Church 25.4 Recursion: Y Oh Y Oh Y?

26 Types, Types, Types: Modeling λ Calculus

26.1 Playing to Type 26.2 Prove It! 26.3 What’s It Good For?

27 The Halting Problem

27.1 A Brilliant Failure 27.2 To Halt or Not To Halt?

Bibliography