Mathematical Thinking: Why Everyone Should Study Math [1 ed.] 3031332024, 9783031332029

Table of contents :
Preface
Contents
1 Primes
1.1 Introduction
1.2 Primes and Cryptography
1.3 Back to the Math
1.4 Algorithms for Testing Primality
1.5 e and the Prime Number Theorem
1.5.1 e
1.5.2 The Prime Number Theorem
1.6 Two Fascinating Questions
1.6.1 Goldbach's Conjecture
1.6.2 Twin Primes
1.7 Puzzle
1.8 Exercises
1.8.1 Hints
2 The Euclidean Algorithm
2.1 The Extended Euclidean Algorithm
2.2 Uniqueness of Factorization into Primes
2.3 Puzzle
2.4 Exercises
2.4.1 Hint
3 Modular Arithmetic
3.1 Modular Arithmetic
3.2 Discrete Logarithms
3.3 Fermat's Little Theorem
3.4 The Chinese Remainder Theorem
3.5 Puzzle
3.6 Exercises
3.6.1 Hints
4 Irrationals
4.1 Rationals and Irrationals
4.2 The Rationals Are the Reals with Repeating Decimal Representations
4.3 Square Roots
4.3.1 The Pythagorean Theorem
4.3.2 Square Roots
4.4 Puzzle
4.5 Exercises
4.5.1 Hints
5 i
5.1 Roots of Polynomials
5.2 The Geometric Representation and Normof a Complex Number
5.3 e to a Complex Power
5.4 Infinite Series
5.5 Puzzle
5.6 Exercises
5.6.1 Hint
6 Infinities and Diagonalization
6.1 Finite Sets
6.2 Infinite Sets
6.3 The Set of Rationals Is Countable
6.4 The Set of Reals Is Uncountable
6.5 The Halting Problem Is Undecidable
6.6 Puzzle
6.7 Exercises
6.7.1 Hints
7 Binary Search
7.1 Binary Search
7.2 Computers, Calculators, and Finite Precision
7.3 Square Roots via Binary Search
7.3.1 Time Analysis
7.3.2 But Floating-Point Calculations Are Inexact!
7.4 Puzzle
7.5 Exercises
7.5.1 Hints
8 Newton's Method
8.1 Introduction
8.2 How to Improve a Solution
8.3 Extension to Other Functions
8.4 Tangent Lines
8.5 How Good Is Newton's Method?
8.6 A Personal Note
8.7 Puzzle
8.8 Exercises
8.8.1 Hint
9 Graph Theory
9.1 Introduction
9.2 Planarity
9.3 Graph Coloring
9.3.1 Vertex Coloring
9.3.2 Edge Coloring
9.4 Eulerian Multigraphs
9.5 Minimum Spanning Trees
9.5.1 Basic Facts
9.5.2 The Minimum Spanning Tree Algorithm
9.6 Puzzle
9.7 Exercises
9.7.1 Hints
10 Probability
10.1 Examples
10.2 Independence
10.3 Birthday Parties
10.3.1 Hitting a Specific Date
10.3.2 Getting Two Identical Birthdays
10.4 Gambling
10.5 Let's Make a Deal
10.6 The (Magical) Probabilistic Method
10.7 The Central Limit Theorem
10.8 Puzzle
10.9 Exercises
10.9.1 Hints
11 Fractals
11.1 The Cantor Set and the Sierpiński Triangle
11.2 The Mandelbrot Set
11.2.1 Background
11.2.2 Definition of the Mandelbrot Set
11.2.3 Drawing the Mandelbrot Set
11.3 The Newton Fractal
11.4 Box-Counting Dimension
11.4.1 Introduction
11.4.2 The Cantor Set
11.4.3 The Sierpiński Triangle
11.5 Puzzle
11.6 Exercises
11.6.1 Hint
12 Solutions to Puzzles
12.1 Chapter 1: Light Bulb Switches
12.2 Chapter 2: Belt Around the Equator
12.3 Chapter 3: Two Trains and a Fly
12.4 Chapter 4: Writing the Year as a Sum of Positive Integers with Maximum Product
12.5 Chapter 5: Two Towers
12.6 Chapter 6: A 1010 Board
12.7 Chapter 7: Distribution of Genders
12.8 Chapter 8: A Hunter and a Bear
12.9 Chapter 9: A Mathematician's Children
12.10 Chapter 10: Cutting Up Chocolate
12.11 Chapter 11: Uptown and Downtown Girlfriends
Acknowledgments
References
Index