What You Can Learn from Proofs from THE BOOK
A Proofs from THE BOOK Guide by the University of Dating
Proofs from THE BOOK - Martin Aigner , Günter M. Ziegler
"THE BOOK" continues to have a profound influence on mathematicians as a book that introduces the beautiful theorems of mathematics. The book was written by mathematicians Martin Aigner and Günter M. Ziegler. The title "THE BOOK" actually originates from a mathematical legend. There is a discussion among mathematicians: "If all the beautiful theorems of mathematics were compiled into one book, what kind of book would it be?" This book was born as that "compendium of beautiful theorems."
The Origin of "THE BOOK"
The origin of "THE BOOK" is deeply tied to the legendary mathematician Paul Erdős. Erdős was a unique figure in the world of mathematics, and his contributions are immeasurable. However, the key point in this book is the kinds of theorems Erdős favored. Erdős was deeply interested in the concept of "the most beautiful theorems" and encouraged mathematicians to pursue that "beauty." Erdős himself sought after an ideal form of mathematical beauty and collaborated with many mathematicians, publishing a great number of theorems. His famous quote, "Mathematics has beautiful theorems," underpins this book.
The authors of "THE BOOK", Aigner and Ziegler, were inspired by Erdős's philosophy. They aimed to create a book that would collect the theorems Erdős idealized. They gathered what are considered the "most beautiful" theorems in mathematics and provided accessible explanations of the deep mathematical meanings behind them. The theorems included in "THE BOOK" symbolize the "simple yet deep, beautiful theorems" that Erdős desired.
"THE BOOK" and the Bible
The title "THE BOOK" also subtly alludes to the Bible, as the title suggests the meaning of a "book containing God's words." The Bible is regarded as the most important book for faith, and similarly, "THE BOOK" is positioned as a collection of the most important theorems for mathematicians. Of course, there is no direct connection to the Bible, but it is a book highly valued by mathematicians as the "ideal compendium of theorems." Like the Bible, each theorem in "THE BOOK" holds deep meaning and allows mathematicians to touch upon "truths," showing some shared aspects between the two.
Contents and Features of the Book
"THE BOOK" features a selection of theorems regarded as "beautiful" in mathematics, drawn from various mathematical fields. Specifically, it includes theorems from combinatorics, number theory, geometry, probability theory, and analysis. The hallmark of this book is that it is not just a list of theorems; it emphasizes intuitive understanding, allowing readers to appreciate the beauty of the theorems. Each theorem is accompanied by explanations of its proof, historical background, and mathematical significance.
For example, classic theorems like the Pythagorean Theorem and Fermat's Last Theorem are presented, with detailed explanations of their proofs and historical development. Furthermore, it not only introduces theorems but also shows how they relate to other areas of mathematics and how they give rise to new problems. In this way, the book explores the deeper philosophical aspects of mathematics and its development, making it enjoyable for a wide range of readers, from beginners to seasoned researchers.
Detailed Chapter Overview
"Proofs from THE BOOK (6th Edition)" is a collection of beautiful mathematical proofs that cover a wide range of fields, including number theory, geometry, analysis, combinatorics, and graph theory. Below is a detailed chapter overview, translated into Japanese:
Number Theory
1. Proof of the Infinity of Primes (Six proofs of the infinity of primes)
This chapter introduces six distinct proofs showing that there are infinitely many primes. The most famous of these is Euclid's proof that primes are infinite. Other approaches to the same result are also presented.
2. Bertrand's Postulate
This theorem, also called Bertrand's conjecture, asserts that for any natural number , there exists at least one prime number greater than . This conjecture was later definitively proven.
3. Binomial Coefficients Are (Almost) Never Powers
This chapter proves that binomial coefficients are almost never powers (e.g., in the form ), except in certain special cases.
4. Representing Numbers as Sums of Two Squares
This theorem discusses how any natural number can be represented as the sum of two squares in specific cases. This result is a significant one in number theory.
5. The Law of Quadratic Reciprocity
Quadratic reciprocity is a fundamental theorem in number theory concerning primes. It is a key tool for determining whether a prime is a quadratic residue modulo another prime.
6. Every Finite Division Ring is a Field
This theorem deals with finite division rings (also known as division algebras) and proves that every such ring is a field. This result is significant in algebra.
7. The Spectral Theorem and Hadamard's Determinant Problem
This chapter covers the spectral theorem and uses it to address Hadamard’s determinant problem. It deals with the eigenvalues of matrices and related topics.
8. Some Irrational Numbers
Here, several examples of irrational numbers are introduced, and their proofs are explained.
9. Calculation of
This proof concerns an interesting formula in both analysis and number theory.
Geometry
10. Hilbert’s Third Problem: Decomposing Polyhedra
This problem asks whether a polyhedron can be decomposed into smaller pieces and reassembled into another polyhedron. This chapter discusses the decomposition and reassembly of polyhedra.
11. Lines in the Plane and Decompositions of Graphs
This chapter tackles problems related to lines in the plane and decompositions of graphs, combining geometric structures with graph theory.
12. The Slope Problem
This is a geometric problem concerning the slopes of lines in a given configuration, specifically the distribution of slopes.
13. Three Applications of Euler’s Formula
In this chapter, Euler's formula is used to prove three different geometric results related to polyhedra.
14. Cauchy’s Rigidity Theorem
Cauchy’s theorem states that the shape of a polyhedron is uniquely determined by the shapes of its faces. This is a crucial result in the study of rigidity in mathematics and physics.
15. The Borromean Rings Don’t Exist
This proof shows that the Borromean rings—three rings that are interlinked such that removing one causes the others to separate—do not exist.
16. Touching Simplices
This problem involves the conditions under which simplices (simple polyhedra) can touch one another. The conditions are formulated and proven here.
17. Every Large Point Set Has an Obtuse Angle
This theorem shows that in any sufficiently large set of points, there will always be a set of points forming an obtuse angle.
18. Borsuk’s Conjecture
This conjecture posits that any set can be divided into a small number of smaller sets. This conjecture is proved here.
Analysis
19. Sets, Functions, and the Continuum Hypothesis
This chapter addresses set theory and functions, focusing particularly on the continuum hypothesis.
20. In Praise of Inequalities
This chapter discusses important mathematical inequalities and provides proofs for several famous inequalities.
21. The Fundamental Theorem of Algebra
The fundamental theorem of algebra asserts that every non-constant polynomial has at least one root in the complex numbers. This is a central result in analysis.
22. One Square and an Odd Number of Triangles
This chapter deals with a proof involving a square and an odd number of triangles. It explores a fascinating geometric result.
23. A Theorem of Pólya on Polynomials
This chapter proves a theorem by George Pólya regarding polynomials and their properties.
24. VanderWaerden’s Permanent Conjecture
This conjecture, related to matrices, is proved here in the context of algebraic theory.
25. On a Lemma of Littlewood and Offord
This lemma, relevant to probability theory and combinatorics, is proven in this section.
26. Cotangent and the Herglotz Trick
This chapter involves a proof related to cotangent functions and the Herglotz trick.
27. Buffon’s Needle Problem
The Buffon needle problem is a famous problem in probability theory, and the proof is discussed here.
This book emphasizes the beauty and creativity of mathematical proofs, introducing the intuitive ideas and clever insights behind each one. It is an invaluable resource for math enthusiasts and students, offering fascinating and sometimes challenging problems to work through.
The 6th Edition (2018) and Its Revisions
The first edition of "THE BOOK" was published in 1998, and it has since been revised several times. The most recent edition, the 6th Edition (2018), reflects the latest developments in mathematics. New theorems and results, particularly from combinatorics and number theory, have been added, and the explanations of previously covered topics have been expanded for greater clarity. The revised edition delves even deeper into the mathematical intuition and historical development behind the theorems.
Summary
"THE BOOK" is a collection of the most
beautiful theorems in mathematics, created with the intention of pursuing the "beauty" of mathematics. Written based on the philosophical ideas of Paul Erdős, it not only explores the theorems but also examines their development and significance. This book offers profound insights and is a remarkable resource for anyone interested in understanding the richness and depth of mathematics, from beginners to experts.
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