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 the computer on your desk, this is the book for you.

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

## by Mark C. Chu-Carroll

# Good Math

## A Geek's Guide to the Beauty of Numbers, Logic, and Computation

### by Mark C. Chu-Carroll

# Customer Reviews

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" and "The Loom", New York Times and National Geographic Magazine

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"

# What You Need

No special equipment or software is required. Although the book contains brief code examples, they can all be run with open source software on any up-to-date Apple, Windows, or Linux computer.

# Resources

# Contents & Extracts

**Preface****Numbers**- Natural Numbers
- The Naturals, Axiomatically Speaking
- Using Peano Induction

- Integers
- What’s an Integer?
- Constructing the Integers—Naturally

- Real Numbers
- The Reals, Informally
- The Reals, Axiomatically
- The Reals, Constructively

- Irrational and Transcendental Numbers
- What are Irrational Numbers?
- The Argh! Moments of Irrational Numbers
- What Does It Mean, and Why Does It Matter?

- Natural Numbers
**Funny Numbers**- Zero
- The History of Zero
- An Annoyingly Difficult Number

- e: The Unnatural Natural Number
- The Number That’s Everywhere
- History
- Does e Have a Meaning?

- The Golden Ratio
- What Is the Golden Ratio?
- Legendary Nonsense
- Where it Really Lives

- i: The Imaginary Number
- The Origin of i
- What i Does
- What i Means

- Zero
**Writing Numbers**- Roman Numerals
- A Positional System
- Where Did This Mess Come From?
- Arithmetic is Easy (But an Abacus is Easier)
- Blame Tradition

- Egyptian Fractions
- A 4000-Year Old Math Exam
- Fibonacci’s Greedy Algorithm
- Sometimes Aesthetics Trumps Practicality

- Continued Fractions
- Continued Fractions
- Cleaner, Clearer and Just Plain Fun
- Doing Arithmetic

- Roman Numerals
**Logic**- Mr. Spock is Not Logical
- What is Logic, Really?
- FOPL, Logically
- Show Me Something New!

- Proofs, Truth, and Trees: Oh My!
- Building a Simple Proof with a Tree
- A Proof From Nothing
- All in the Family
- Branching Proofs

- Programming with Logic
- Computing Family Relationships
- Computation with Logic

- Temporal Reasoning
- Statements that Change with Time
- What’s CTL good for?

- Mr. Spock is Not Logical
**Sets**- Cantor’s Diagonalization: Infinity Isn’t Just Infinity
- Sets, Naively
- Cantor’s Diagonalization
- Don’t Keep it Simple Stupid

- Axiomatic Set Theory: Keep the Good, Dump the Bad
- The Axioms of ZFC Set Theory
- The Insanity of Choice
- Why?

- Models: Using Sets as the Legos of the Math World
- Building Natural Numbers
- Models from Models: From Naturals to Integers and Beyond!

- Transfinite Numbers: Counting and Ordering Infinite Sets
- Introducing the Transfinite Cardinals
- The Continuum Hypothesis
- Where in Infinity?

- Group Theory: Finding Symmetries with Sets
- Puzzling Symmetry
- Different Kinds of Symmetry
- Stepping into History
- The Roots of Symmetry

- Cantor’s Diagonalization: Infinity Isn’t Just Infinity
**Mechanical Math**- Finite State Machines: Simplicity Goes Far
- The Simplest Machine
- Finite State Machines Get Real
- Bridging the Gap: From Regular Expressions to Machines

- The Turing Machine
- Adding a Tape Makes All the Difference
- Going Meta: The Machine That Imitates Machines

- Pathology and the Heart of Computing
- Introducing BF: the Great, the Glorious, the Completely Silly
- Turing Complete, or Completely Pointless?
- From the Sublime to the Ridiculous

- Calculus. No, Not That Calculus: λ Calculus
- Writing λ-Calculus: It’s Almost Programming!
- Evaluation: Run It!
- Programming Languages and Lambda Strategies

- Numbers, Booleans, and Recursion
- But Is It Turing Complete?
- Numbers That Compute Themselves
- Decisions? Back to Church
- Recursion: Y Oh Y Oh Y?

- Types, Types, Types! Modeling λ Calculus
- Playing to Type
- Prove it!
- What’s It Good For?

- The Halting Problem
- A Brilliant Failure
- To Halt, or Not To Halt?

- Finite State Machines: Simplicity Goes Far

# Author

**Mark Chu-Carroll** is a PhD computer scientist and professional software engineer. His professional interests include collaborative software development, programming languages and tools, and how to improve the daily lives of software developers. Aside from general geekery and blogging, he plays classical music on the clarinet, traditional Irish music on the wooden flute, and folds elaborate structures out of paper.