Numbers: Their Tales, Types, and Treasures.

Chapter 1: Numbers and Counting

1.1.A MENTAL NETWORK

1.2.WHAT IS A NUMBER?

1.3.COUNTING

1.4.THE COUNTING PRINCIPLES

1.5.COLLECTIONS OF OBJECTS, ELEMENTS IN A SET

1.6.THE BIJECTION PRINCIPLE AND THE COMPARISON OF SETS

1.7.UNUSUAL COUNTING TAGS

1.8.CARDINAL AND ORDINAL NUMBERS

1.9.ABSTRACTION

1.10.COUNTING BY THE ORDINAL PRINCIPLE

1.11.SYSTEMATIC ENUMERATION

1.12.THE NUMERAL SYSTEM IN WRITING

1.13.BASE-10

1.14.MEASURING MAGNITUDES

Chapter 2: Numbers and Psychology

2.1.CORE KNOWLEDGE OF THE WORLD

2.2.OUR BUILT-IN OBJECT-TRACKING SYSTEM

2.3.THE APPROXIMATE-NUMBER SYSTEM

2.4.GOING BEYOND THE CORE SYSTEMS

2.5.HOW WE LEARN TO COUNT

2.6.LOGICAL FOUNDATIONS FIRST?

2.7.THE NUMBER LINE

2.8.EVOLUTION OF NUMERALS

2.9.IDIOSYNCRASIES OF LANGUAGES

Chapter 3: Numbers in History

3.1.NUMBERS IN BABYLON—THE FIRST PLACE-VALUE SYSTEM IN HISTORY

3.2.NUMBERS IN EGYPT—THE FIRST DECIMAL SYSTEM IN HISTORY

3.3.ARITHMETIC IN EGYPT

3.4.NUMBERS IN CHINA

3.5.THE CHINESE PLACE-VALUE NOTATION

3.6.NUMBERS IN INDIA

3.7.SYMBOLIC NUMBER NOTATION AND THE ABACUS IN INDIA

3.8.THE SLOW RECEPTION OF THE HINDU-ARABIC SYSTEM IN EUROPE

Chapter 4: Discovering Properties of Numbers

4.1.THE SEARCH FOR MEANING

4.2.PYTHAGOREAN PHILOSOPHY OF NUMBERS

4.3.EVEN AND ODD NUMBERS

4.4.RECTANGULAR AND SQUARE NUMBERS

4.5.TRIANGULAR NUMBERS

4.6.TRIANGULAR AND RECTANGULAR

4.7.POLYGONAL NUMBERS

4.8.TETRAHEDRAL NUMBERS

Chapter 5: Counting for Poets

5.1.VERSE METRIC—THE ROLE OF RHYTHM

5.2.THE ORIGIN OF LINGUISTICS

5.3.THE PROBLEM OF COUNTING METRIC PATTERNS

5.4.SOLVING PINGALA'S FIRST COUNTING PROBLEM IN SPECIAL CASES

5.5.A GENERAL SOLUTION TO PINGALA'S FIRST PROBLEM

5.6.DISCOVERING COMMON TRAITS OF COUNTING PROBLEMS

5.7.THE ART OF COUNTING SYLLABLES

5.8.THE ART OF COMBINATION

5.9.SOLVING PINGALA'S THIRD PROBLEM

5.10.THE PASCAL TRIANGLE AND PINGALA'S PROBLEMS

5.11.LOTTERY GAMES AND OTHER AMUSEMENTS

Chapter 6: Number Explorations

6.1.THE FIBONACCI NUMBERS IN EUROPE

6.2.GENERATIONS OF RABBITS

6.3.MORE ABOUT THE PASCAL TRIANGLE

6.4.COMBINATORIAL GEOMETRY

6.5.THE BINOMIAL EXPANSION

Chapter 7: Placement of Numbers

7.1.MAGIC SQUARES

7.2.GENERAL PROPERTIES OF MAGIC SQUARES

7.3.HOW TO CONSTRUCT A DOUBLY-EVEN MAGIC SQUARE

7.4.CONSTRUCTION OF A MAGIC SQUARE OF ORDER 3

7.5.CONSTRUCTING ODD-ORDER MAGIC SQUARES

7.6.CREATING SINGLY-EVEN MAGIC SQUARES

7.7.PALINDROMIC NUMBERS

7.8.NAPIER'S RODS

Chapter 8: Special Numbers

8.1.PRIME NUMBERS

8.2.SEARCHING FOR PRIMES

8.3.PLAYING WITH PRIMES

8.4.UNSOLVED QUESTIONS

8.5.PERFECT NUMBERS

8.6.KAPREKAR NUMBERS

8.7.THE KAPREKAR CONSTANT

8.8.THE MYSTICAL NUMBER 1,089

8.9.SOME NUMBER PECULIARITIES

Chapter 9: Number Relationships

9.1.BEAUTIFUL NUMBER RELATIONSHIPS

9.2.AMICABLE NUMBERS

9.3.OTHER TYPES OF AMICABILITY

9.4.PYTHAGOREAN TRIPLES AND THEIR PROPERTIES

9.5.FIBONACCI'S METHOD FOR FINDING PYTHAGOREAN TRIPLES

9.6.STIFEL'S METHOD FOR GENERATING PRIMITIVE PYTHAGOREAN TRIPLES

9.7.EUCLID'S METHOD FOR FINDING PYTHAGOREAN TRIPLES

9.8.EXPLORING PYTHAGOREAN TRIPLES

9.9.CONSECUTIVE MEMBERS OF A PYTHAGOREAN TRIPLE

9.10.SOME OTHER PYTHAGOREAN CURIOSITIES

9.11.DIVISIBILITY OF NUMBERS

Chapter 10: Numbers and Proportions

10.1.COMPARING QUANTITIES

10.2.PROPORTIONS OF LENGTHS

10.3.EUCLID'S ALGORITHM AND CONTINUED FRACTIONS

10.4.CREATING RECTANGLES FROM SQUARES

10.5.THE GOLDEN RATIO

10.6.INCOMMENSURABILITY

10.7.THE FAMOUS NUMBER π

10.8.THE AMAZING HISTORY OF π

10.9.FAMOUS NUMBERS IN THE GREAT PYRAMID

10.10.THE “PI-RAMID”

10.11.THE HISTORICAL EXPLANATION

Chapter 11: Numbers and Philosophy

11.1.NUMBERS—INVENTED OR DISCOVERED?

11.2.THE PLATONIC POINT OF VIEW

11.3.AN ONGOING DISCUSSION

11.4.PHILOSOPHY OF MATHEMATICS

11.5.THE LOGICIST DEFINITION OF A CARDINAL NUMBER

11.6.A FORMALIST'S DEFINITION OF NUMBER

11.7.THE STRUCTURALIST'S POINT OF VIEW

11.8.THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS

11.9.MATHEMATICAL MODELS

11.10.LIMITS OF THE MODEL OF NATURAL NUMBERS

11.11.THE PROBLEM WITH REALLY HUGE NUMBERS

11.12.SHUT UP AND CALCULATE!

Appendix: Tables

A.1. TABLE OF FIBONACCI NUMBERS

A.2. TABLE OF THE FIRST PRIME NUMBERS UNDER 10,000

A.3. TABLE OF ALL KNOWN MERSENNE PRIMES

A.4. TABLE OF ALL KNOWN PERFECT NUMBERS

A.5. TABLE OF KAPREKAR NUMBERS

A.6. TABLE OF ARMSTRONG NUMBERS

A.7. TABLE OF AMICABLE NUMBERS

A.8. PYTHAGOREAN TRIPLES WITH A PAIR OF PALINDROMIC NUMBERS

 

Notes