Proof and the Art of Mathematics (9780262360937) by Hamkins, Joel David -- Read -- Imperial Library of Trantor

Index

Cover Title Page Dedication Contents Preface A Note to the Instructor A Note to the Student About the Author 1 A Classical Beginning

1.1 The number √2 is irrational 1.2 Lowest terms 1.3 A geometric proof 1.4 Generalizations to other roots Mathematical Habits Exercises

2 Multiple Proofs

2.1 n2 − n is even 2.2 One theorem, seven proofs 2.3 Different proofs suggest different generalizations Mathematical Habits Exercises Credits

3 Number Theory

3.1 Prime numbers 3.2 The fundamental theorem of arithmetic 3.3 Euclidean division algorithm 3.4 Fundamental theorem of arithmetic, uniqueness 3.5 Infinitely many primes Mathematical Habits Exercises

4 Mathematical Induction

4.1 The leastnumber principle 4.2 Common induction 4.3 Several proofs using induction 4.4 Proving the induction principle 4.5 Strong induction 4.6 Buckets of Fish via nested induction 4.7 Every number is interesting Mathematical Habits Exercises Credits

5 Discrete Mathematics

5.1 More pointed at than pointing 5.2 Chocolate bar problem 5.3 Tiling problems 5.4 Escape 5.5 Representing integers as a sum 5.6 Permutations and combinations 5.7 The pigeonhole principle 5.8 The zigzag theorem Mathematical Habits Exercises Credits

6 Proofs without Words

6.1 A geometric sum 6.2 Binomial square 6.3 Criticism of the “without words” aspect 6.4 Triangular choices 6.5 Further identities 6.6 Sum of odd numbers 6.7 A Fibonacci identity 6.8 A sum of cubes 6.9 Another infinite series 6.10 Area of a circle 6.11 Tiling with dominoes 6.12 How to lie with pictures Mathematical Habits Exercises Credits

7 Theory of Games

7.1 Twenty-One 7.2 Buckets of Fish 7.3 The game of Nim 7.4 The Gold Coin game 7.5 Chomp 7.6 Games of perfect information 7.7 The fundamental theorem of finite games Mathematical Habits Exercises Credits

8 Pick’s Theorem

8.1 Figures in the integer lattice 8.2 Pick’s theorem for rectangles 8.3 Pick’s theorem for triangles 8.4 Amalgamation 8.5 Triangulations 8.6 Proof of Pick’s theorem, general case Mathematical Habits Exercises Credits

9 Lattice-Point Polygons

9.1 Regular polygons in the integer lattice 9.2 Hexagonal and triangular lattices 9.3 Generalizing to arbitrary lattices Mathematical Habits Exercises Credits

10 Polygonal Dissection Congruence Theorem

10.1 The polygonal dissection congruence theorem 10.2 Triangles to parallelograms 10.3 Parallelograms to rectangles 10.4 Rectangles to squares 10.5 Combining squares 10.6 Full proof of the dissection congruence theorem 10.7 Scissors congruence Mathematical Habits Exercises Credits

11 Functions and Relations

11.1 Relations 11.2 Equivalence relations 11.3 Equivalence classes and partitions 11.4 Closures of a relation 11.5 Functions Mathematical Habits Exercises

12 Graph Theory

12.1 The bridges of Königsberg 12.2 Circuits and paths in a graph 12.3 The fiveroom puzzle 12.4 The Euler characteristic Mathematical Habits Exercises Credits

13 Infinity

13.1 Hilbert’s Grand Hotel

Hilbert’s bus Hilbert’s train Hilbert’s half marathon Cantor’s cruise ship

13.2 Countability 13.3 Uncountability of the real numbers

Alternative proof of Cantor’s theorem Cranks

13.4 Transcendental numbers 13.5 Equinumerosity 13.6 The ShröderCantorBernstein theorem 13.7 The real plane and real line are equinumerous Mathematical Habits Exercises Credits

14 Order Theory

14.1 Partial orders 14.2 Minimal versus least elements 14.3 Linear orders 14.4 Isomorphisms of orders 14.5 The rational line is universal 14.6 The eventual domination order Mathematical Habits Exercises

15 Real Analysis

15.1 Definition of continuity 15.2 Sums and products of continuous functions 15.3 Continuous at exactly one point 15.4 The leastupperbound principle 15.5 The intermediatevalue theorem 15.6 The HeineBorel theorem 15.7 The BolzanoWeierstrass theorem 15.8 The principle of continuous induction Mathematical Habits Exercises Credits

Answers to Selected Exercises Bibliography Index of Mathematical Habits Notation Index Subject Index