This textbook invites readers to explore mathematical thinking by finding the beauty in the subject. With an accessible tone and stimulating puzzles, the author will convince curious non-mathematicians to continue their studies in the area. It has an expansive scope, covering everything from probability and graph theory to infinities and Newton’s method. Many examples of proofs appear as well, offering readers the opportunity to explore these topics with the amount of rigor that suits them. Programming exercises in Python are also included to show how math behaves in action.
Mathematical Thinking is an ideal textbook for transition courses aimed at undergraduates moving from lower level to more advanced topics, as well as for math recruitment and invitational courses at the freshman or sophomore level. It may also be of interest in computer science departments and can be used as a supplemental text for courses in discrete mathematics and graph theory.
Author(s): Howard Karloff
Series: Compact Textbooks in Mathematics
Edition: 1
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 208
City: New York
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
