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數形與對稱性--數論幾何和群論導論(英文)/國外優秀數學著作原版系列

  • 作者:(美)黛安·L.赫爾曼//小保羅·J.薩利|責編:杜瑩雪//李蘭靜
  • 出版社:哈爾濱工業大學
  • ISBN:9787560399713
  • 出版日期:2022/05/01
  • 裝幀:平裝
  • 頁數:456
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內容大鋼
    本書的前半部分著重於數論,以算術規則(整數的規則)作為開始。作者之後呈現了所有整除性、素數和模算術的基本思想和應用。同時介紹了群的抽象記號,並且包含了許多例子。數論的最後一個話題由有理數、實數及無窮的思想組成。在幾何方面,本書涵蓋了多邊形和多面體,包括正多邊形和正多面體的構造。它通過觀察平面上的圖案來研究鑲嵌,特別是那些由正多邊形及正多邊形的集合組成的圖形。本書同時研究了這些圖形和圖案的對稱群,解釋了這些群是如何在幾何和數論中同時出現的。

作者介紹
(美)黛安·L.赫爾曼//小保羅·J.薩利|責編:杜瑩雪//李蘭靜

目錄
Preface
0 Warm-up: The Triangle Game
  Practice Problem Solutions and Hints
  Exercises
  The Beginnings of Number Theory
  1.1 Setting the Table: Numbers, Sets, and Functions
    Numbers and Number Systems
    Sets
    Functions
    Math Words
  1.2 Rules of Arithmetic
  1.3 A New System
  1.4 One's Digit Arithmetic
  Practice Problem Solutions and Hints
  Exercises
2 Axioms in Number Theory
  2.1 Consequences of the Rules of Arithmetic
    Cancelation for Addition
    Properties of-1 and 0
    Cancelation for Multiplication
    Subtraction and Division
  2.2 Inequalities and Order
    Order and Other Number Systems
    Well-Ordering
  Practice Problem Solutions and Hints
  Exercises
3 Divisibility and Primes
  3.1 Divisibility
  3.2 Greatest Common Divisor
  3.3 Primes
    Formulas for Primes
    Twin Primes and Triple Primes
    Other Conjectures about Primes
  Practice Problem Solutions and Hints
  Exercises
4 The Division and Euclidean Algorithms
  4.1 The Division Algorithm
    The Division Algorithm with a Negative Dividend
  4.2 The Euclidean Algorithm and the Greatest Common Divisor
  4.3 The Fundamental Theorem of Arithmetic
    Why We Don't Call 1 a Prime
    Prime Factorization and the GCD
  Practice Problem Solutions and Hints
  Exercises
5 Variations on a Theme
  5.1 Applications of Divisibility
    Fibonacci Numbers
    Sum and Number of Divisors
    Perfect Numbers
  5.2 More Algorithms

    Rational Arithmetic and Least Common Multiples
    Egyptian Fractions
  Practice Problem Solutions and Hints
  Exercises
6 Congruences and Groups
  6.1 Congruences and Arithmetic of Residue Classes
  6.2 Groups and Other Structures
    Cyclic Groups
    Rings
    Zero Divisors and Fields
  Practice Problem Solutions and Hints
  Exercises
7 Applications of Congruences
  7.1 Divisibility Tests
    Divisibility by Powers of 2
    Divisibility by Powers of 5
    Divisibility by 3 and 9
    Divisibility by 11
    Divisibility by 7, 11, and 13
  7.2 Days of the Week
    Calculating from the First Date of Any Year
    How to Find the Day of the Week
  7.3 Check Digits
    ISBNs and UPC Numbers
  Practice Problem Solutions and Hints
  Exercises
  Rational Numbers and Real Numbers
  8.1 Fractions to Decimals
  8.2 Decimals to Fractions
  8.3 Infinity
  8.4 Rational Numbers
  8.5 Irrational Numbers
  8.6 How Many Real Numbers?
  Practice Problem Solutions and Hints
  Exercises
9 Introduction to Geometry and Symmetry
  Practice Problem Solutions and Hints
  Exercises
IO Polygons and Their Construction
  10.1 Polygons and Their Angles
    Triangles
    Quadrilaterals
    n -gons
  10.2 Constructions
  Practice Problem Solutions and Hints
  Exercises
  Symmetry Groups
  11.1 Symmetric Motions of the Triangle
  11.2 Symmetric Motions of the Square
    Reflections and Rotations

    Impossible Motions
    Economy of Notation Revisited
  11.3 Symmetries of Regular n-gons
  Practice Problem Solutions and Hints
  Exercises
12 Permutations
  12.1 Symmetric Motions as Permutations
    Permutations and the Motions of the Square
  12.2 Counting Permutations and Symmetric Groups
  12.3 Even More Economy of Notation
    Transpositions
  Practice Problem Solutions and Hints
  Exercises
13 Polyhedra
  13.1 Regular Polyhedra
  13.2 Euler's Formula
  13.3 Symmetries of Regular Polyhedra
    Rotations of the Tetrahedron
    Tetrahedron "Flips," or Reflections
    Economy of Notation
    Rotations of the Cube
    Cube "Flips," or Reflections
  13.4 Reflections and Rotations
    Symmetries of the Octahedron
    The Dodecahedron and the Icosahedron
  13.5 Variations on a Theme: Other Polyhedra
    Prisms and Pyramids
    Other Convex and Nonconvex Polyhedra
    Diagrams (Nets) for Making Polyhedra
  Practice Problem Solutions and Hints
  Exercises
14 Graph Theory
   14.1 Introduction
   14.2 The KSnigsberg Bridge Problem
   14.3 Colorability and Planarity
   14.4 Graphs and Their Complements
   14.5 Trees
  Practice Problems Solutions and Hints
  Exercises
15 Tessellations
   15.1 Tessellating with a Single Shape
  15.2 Tessellations with Multiple Shapes
  15.3 Variations on a Theme: Polyominoes
  15.4 Frieze Patterns
    Symmetry Groups
    The Four Symmetric Motions
    Classification of Friezes
    The Seven Symmetries
  15.5 Infinite Patterns in Two and Three Dimensions .
  Practice Problem Solutions and Hints

  Exercises
16 Connections
  16.1 The Golden Ratio and Fibonacci Numbers
    The Golden Ratio and Geometry
    Constructing the Golden Ratio
    Fibonacci Numbers
  16.2 Constructible Numbers and Polygons
    Constructing √a
    Constructible Polygons
    Gauss's Construction of a Regular Pentagon
    Constructing Other Regular n-gons
  Practice Problem Solutions and Hints
  Exercises
A Appendix: Euclidean Geometry Review
  Part 1
  Part 2
Glossary
Bibliography
Index
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