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.
Buy Now
About this Book
 282 pages
 Published:
 Release: P1.0 (20130807)
 ISBN: 9781937785338
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 fastpaced tour of modern and notsomodern math, computer scientist Mark ChuCarroll 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 logicsavvy 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.
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 uptodate Apple, Windows, or Linux computer.
Contents and 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 4000Year 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 λCalculus: 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
Comments and Reviews

—Carl Zimmer Author of "Matter" and "The Loom" New York Times and National Geographic Magazine
Mark ChuCarroll 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 booklength 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.

—Jennifer Ouellette author of "The Calculus Diaries"
Fans of Mark ChuCarroll’s lively and informative blog, Good Math/Bad Math, will find much to savor in this mathematical guide for the “geekerati.” ChuCarroll 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.