Book contents
- Frontmatter
- Preface
- Contents
- Introduction
- 1 A Garden of Integers
- 2 Distinguished Numbers
- 3 Points in the Plane
- 4 The Polygonal Playground
- 5 A Treasury of Triangle Theorems
- 6 The Enchantment of the Equilateral Triangle
- 7 The Quadrilaterals' Corner
- 8 Squares Everywhere
- 9 Curves Ahead
- 10 Adventures in Tiling and Coloring
- 11 Geometry in Three Dimensions
- 12 Additional Theorems, Problems, and Proofs
- Solutions to the Challenges
- References
- Index
- About the Authors
12 - Additional Theorems, Problems, and Proofs
- Frontmatter
- Preface
- Contents
- Introduction
- 1 A Garden of Integers
- 2 Distinguished Numbers
- 3 Points in the Plane
- 4 The Polygonal Playground
- 5 A Treasury of Triangle Theorems
- 6 The Enchantment of the Equilateral Triangle
- 7 The Quadrilaterals' Corner
- 8 Squares Everywhere
- 9 Curves Ahead
- 10 Adventures in Tiling and Coloring
- 11 Geometry in Three Dimensions
- 12 Additional Theorems, Problems, and Proofs
- Solutions to the Challenges
- References
- Index
- About the Authors
Summary
In our final chapter, we present a collection of theorems and problems from various branches of mathematics and their proofs and solutions. We begin by discussing some set theoretic results concerning infinite sets, including the Cantor-Schröder-Bernstein theorem. In the next two sections we present proofs of the Cauchy-Schwarz inequality and the AM-GM inequality for sets of size n. We then use origami to solve the classical problems of trisecting angles and doubling cubes, followed by a proof that the Peaucellier-Lipkin linkage draws a straight line. We then look at several gems from the theory of functional equations and inequalities. In the final sections we conclude with an infinite series and an infinite product for simple expressions involving π, and illustrate each with an application.
Denumerable and nondenumerble sets
The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.
David HilbertTo infinity, and beyond!
Buzz Lightyear, Toy Story (1995)Two sets have the same cardinality if there exists a one-to-one function from one set onto the other, i.e., a one-to-one correspondence between the sets. An infinite set of numbers is denumerable (or countably infinite) if it has the same cardinality as the set ℕ = {1, 2, 3, …} of natural (or counting) numbers.
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- Information
- Charming ProofsA Journey into Elegant Mathematics, pp. 209 - 238Publisher: Mathematical Association of AmericaPrint publication year: 2010
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