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Good Math

Book Description

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.

Table of Contents

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