Good Math

Good Math: A Geek’s Guide to the Beauty of Numbers, Logic, and Computation (Pragmatic Programmers)

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.

Table of Contents

Part I: Numbers
Chapter 1. Natural Numbers
Chapter 2. Integers
Chapter 3. Real Numbers
Chapter 4. Irrational and Transcendental Numbers

Part II: Funny Numbers
Chapter 5. Zero
Chapter 6. e: The Unnatural Natural Number
Chapter 7. φ: The Golden Ratio
Chapter 8. i: The Imaginary Number
Chapter 9. Roman Numerals
Chapter 10. Egyptian Fractions
Chapter 11. Continued Fractions

Part IV: Logic
Chapter 12. Mr. Spock Is Not Logical
Chapter 13. Proofs, Truth, and Trees: Oh My!
Chapter 14. Programming with Logic
Chapter 15. Temporal Reasoning

Part V: Sets
Chapter 16. Cantor’s Diagonalization: Infinity Isn’t Just Infinity
Chapter 17. Axiomatic Set Theory: Keep the Good, Dump the Bad
Chapter 18. Models: Using Sets as the LEGOs of the Math World
Chapter 19. Transfinite Numbers: Counting and Ordering Infinite Sets
Chapter 20. Group Theory: Finding Symmetries with Sets

Part VI: Mechanical Math
Chapter 21. Finite State Machines: Simplicity Goes Far
Chapter 22. The Turing Machine
Chapter 23. Pathology and the Heart of Computing
Chapter 24. Calculus: No, Not That Calculus—λ Calculus
Chapter 25. Numbers, Booleans, and Recursion
Chapter 26. Types, Types, Types: Modeling λ Calculus
Chapter 27. The Halting Problem