Good Math: A Geek's Guide to the Beauty of Numbers, Logic, and Computation

335 Pages·2013·1.85 MB·English

Good Math
A Geek’s Guide to the Beauty of Numbers, Logic,
and Computation
by Mark C. Chu-Carroll
Version: P1.0 (July 2013)
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This book is dedicated to the memory of my father, Irving Carroll (zt"l).
He set me on the road to becoming a math geek, which is why this book exists.
More importantly, he showed me, by example, how to be a mensch: by living
honestly, with compassion, humor, integrity, and hard work.
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
I. Numbers
  1. Natural Numbers
  1.1 The Naturals, Axiomatically Speaking
  1.2 Using Peano Induction
  2. Integers
  2.1What’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
  4.1What Are Irrational Numbers?
  4.2 The Argh! Moments of Irrational Numbers
  4.3What Does It Mean, and Why Does It Matter?
II. 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
  7.1What Is the Golden Ratio?
  7.2 Legendary Nonsense
  7.3Where It Really Lives
  8. i: The Imaginary Number
  8.1 The Origin of i
   8.2What i Does
  8.3What i Means
III. Writing Numbers
  9. Roman Numerals
  9.1 A Positional System
  9.2Where 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
IV. Logic
  12. Mr. Spock Is Not Logical
  12.1What 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
  15.2What’s CTL Good For?
V. 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
  17.3Why?
  18. Models: Using Sets as the LEGOs of the Math World
  18.1 Building Natural Numbers
  18.2Models 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
  19.3Where 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
VI. 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
  24.1Writing 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!
   26.3What’s It Good For?
  27. The Halting Problem
  27.1 A Brilliant Failure
  27.2 To Halt or Not To Halt?
Bibliography
Copyright © 2013, The Pragmatic Bookshelf.
Early praise for Good Math
Mark Chu-Carroll is one of the premiere math bloggers in the world, able to
guide readers through complicated concepts with delightful casualness. In Good
Math he brings that same skill to a book-length journey through math, from the
basic notion of numbers through recent developments in computer programming.
If you have ever been curious about the golden ratio or Turing machines or why
pi never runs out of numbers, this is the book for you.
→ Carl Zimmer
author of “Matter,” a weekly column about science in The New York Times (
http://bit.ly/NYTZimmer ); and “The Loom,” a National Geographic
Magazine blog ( http://phenomena.nationalgeographic.com/blog/the-loom )
Fans of Mark Chu-Carroll’s lively and informative blog, Good Math/Bad Math,
will find much to savor in this mathematical guide for the “geekerati.” Chu-Carroll
covers it all, from the basics of natural, irrational, and imaginary numbers and
the golden ratio to Cantor sets, group theory, logic, proofs, programming, and
Turing machines. His love for his subject shines through every page. He’ll help
you love it, too.
→ Jennifer Ouellette
author of The Calculus Diaries
Preface