Proof and the Art of Mathematics

Name: Proof and the Art of Mathematics
Brand: MIT Press
Price: 30.99 EUR
Availability: OnlineOnly

Joel David Hamkins(Autor*in)

MIT Press

Erschienen am 29. September 2020

240 Seiten

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978-0-262-36093-7 (ISBN)

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Inhalt

Intro
Title Page
Copyright
Dedication
Table of Contents
Preface
A Note to the Instructor
A Note to the Student
About the Author
1. A Classical Beginning
1.1. The number V2 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. n^2 - 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 least-number 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 pigeon-hole 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 five-room 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 Shröder-Cantor-Bernstein 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 least-upper-bound principle
15.5. The intermediate-value theorem
15.6. The Heine-Borel theorem
15.7. The Bolzano-Weierstrass 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

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