
Good Math
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Content
- Cover
- Table of Contents
- Preface
- Where'd This Book Come From?
- Who This Book Is For
- How to Read This Book
- What Do You Need?
- Acknowledgments
- Part I-Numbers
- 1. Natural Numbers
- The Naturals, Axiomatically Speaking
- Using Peano Induction
- 2. Integers
- What's an Integer?
- Constructing the Integers-Naturally
- 3. Real Numbers
- The Reals, Informally
- The Reals, Axiomatically
- The Reals, Constructively
- 4. Irrational and Transcendental Numbers
- What Are Irrational Numbers?
- The Argh! Moments of Irrational Numbers
- What Does It Mean, and Why Does It Matter?
- Part II-Funny Numbers
- 5. Zero
- The History of Zero
- An Annoyingly Difficult Number
- 6. e: The Unnatural Natural Number
- The Number That's Everywhere
- History
- Does e Have a Meaning?
- 7. f: The Golden Ratio
- What Is the Golden Ratio?
- Legendary Nonsense
- Where It Really Lives
- 8. i: The Imaginary Number
- The Origin of i
- What i Does
- What i Means
- Part III-Writing Numbers
- 9. Roman Numerals
- A Positional System
- Where Did This Mess Come From?
- Arithmetic Is Easy (But an Abacus Is Easier)
- Blame Tradition
- 10. Egyptian Fractions
- A 4000-Year-Old Math Exam
- Fibonacci's Greedy Algorithm
- Sometimes Aesthetics Trumps Practicality
- 11. Continued Fractions
- Continued Fractions
- Cleaner, Clearer, and Just Plain Fun
- Doing Arithmetic
- Part IV-Logic
- 12. Mr. Spock Is Not Logical
- What Is Logic, Really?
- FOPL, Logically
- Show Me Something New!
- 13. Proofs, Truth, and Trees: Oh My!
- Building a Simple Proof with a Tree
- A Proof from Nothing
- All in the Family
- Branching Proofs
- 14. Programming with Logic
- Computing Family Relationships
- Computation with Logic
- 15. Temporal Reasoning
- Statements That Change with Time
- What's CTL Good For?
- Part V-Sets
- 16. Cantor's Diagonalization: Infinity Isn't Just Infinity
- Sets, Naively
- Cantor's Diagonalization
- Don't Keep It Simple, Stupid
- 17. Axiomatic Set Theory: Keep the Good, Dump the Bad
- The Axioms of ZFC Set Theory
- The Insanity of Choice
- Why?
- 18. Models: Using Sets as the LEGOs of the Math World
- Building Natural Numbers
- Models from Models: From Naturals to Integers and Beyond!
- 19. Transfinite Numbers: Counting and Ordering Infinite Sets
- Introducing the Transfinite Cardinals
- The Continuum Hypothesis
- Where in Infinity?
- 20. Group Theory: Finding Symmetries with Sets
- Puzzling Symmetry
- Different Kinds of Symmetry
- Stepping into History
- The Roots of Symmetry
- Part VI-Mechanical Math
- 21. Finite State Machines: Simplicity Goes Far
- The Simplest Machine
- Finite State Machines Get Real
- Bridging the Gap: From Regular Expressions to Machines
- 22. The Turing Machine
- Adding a Tape Makes All the Difference
- Going Meta: The Machine That Imitates Machines
- 23. Pathology and the Heart of Computing
- Introducing BF: The Great, the Glorious, and the Completely Silly
- Turing Complete, or Completely Pointless?
- From the Sublime to the Ridiculous
- 24. Calculus: No, Not That Calculus-? Calculus
- Writing ? Calculus: It's Almost Programming!
- Evaluation: Run It!
- Programming Languages and Lambda Strategies
- 25. Numbers, Booleans, and Recursion
- But Is It Turing Complete?
- Numbers That Compute Themselves
- Decisions? Back to Church
- Recursion: Y Oh Y Oh Y?
- 26. Types, Types, Types: Modeling ? Calculus
- Playing to Type
- Prove It!
- What's It Good For?
- 27. The Halting Problem
- A Brilliant Failure
- To Halt or Not To Halt?
- Bibliography
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