Table Of ContentH
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Diane L. Herrmann & Paul J. Sally, Jr. u
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Through a careful treatment of number theory and geometry, Number, Shape, and
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Symmetry: An Introduction to Number Theory, Geometry, and Group Theory helps r
,
readers understand serious mathematical ideas and proofs. Classroom-tested, the
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book draws on the authors’ successful work with undergraduate students at the
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University of Chicago, seventh to tenth grade mathematically talented students in
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the University of Chicago’s Young Scholars Program, and elementary public school p
teachers in the Seminars for Endorsement in Science and Mathematics Education e
(SESAME). ,
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The first half of the book focuses on number theory, beginning with the rules of
arithmetic (axioms for the integers). The authors then present all the basic ideas
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and applications of divisibility, primes, and modular arithmetic. They also introduce y
the abstract notion of a group and include numerous examples. The final topics on m
number theory consist of rational numbers, real numbers, and ideas about infinity.
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Moving on to geometry, the text covers polygons and polyhedra, including the con-
e
struction of regular polygons and regular polyhedra. It studies tessellation by look- t
ing at patterns in the plane, especially those made by regular polygons or sets of r
y
regular polygons. The text also determines the symmetry groups of these figures and
patterns, demonstrating how groups arise in both geometry and number theory.
Mathematics
Herrmann-finalx.indd 1 9/14/12 11:27 AM
Number, Shape, and Symmetry
Number, Shape, and Symmetry
An Introduction to Number Theory,
Geometry, and Group Theory
Diane L. Herrmann
Paul J. Sally, Jr.
Cover photograph copyright © George W. Hart, http://georgehart.com
Eights by George W. Hart, 6” x 6” x 6”, paper. The underlying structure is based on five intersecting tetrahedra, with each color representing the six edges of one
tetrahedron.
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Contents
Preface xiii
0 TheTriangleGame 1
Practice Problem Solutions and Hints . . . . . . . . . . . . . 6
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1 TheBeginningsofNumberTheory 11
1.1 Setting the Table: Numbers, Sets, and Functions . . . . 11
Numbers and Number Systems . . . . . . . . . . . . . . 11
Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Math Words . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Rules of Arithmetic. . . . . . . . . . . . . . . . . . . . . 20
1.3 A New System . . . . . . . . . . . . . . . . . . . . . . . 22
1.4 One’s Digit Arithmetic . . . . . . . . . . . . . . . . . . . 23
Practice Problem Solutions and Hints . . . . . . . . . . . . . 32
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 AxiomsinNumberTheory 45
2.1 Consequences of the Rules of Arithmetic . . . . . . . . . 46
Cancelation for Addition . . . . . . . . . . . . . . . . . . 46
Properties of –1 and 0 . . . . . . . . . . . . . . . . . . . 47
Cancelation for Multiplication . . . . . . . . . . . . . . . 51
Subtraction and Division. . . . . . . . . . . . . . . . . . 53
2.2 Inequalities and Order . . . . . . . . . . . . . . . . . . . 54
Order and Other Number Systems . . . . . . . . . . . . 58
Well-Ordering . . . . . . . . . . . . . . . . . . . . . . . . 58
Practice Problem Solutions and Hints . . . . . . . . . . . . . 60
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
vii
viii Contents
3 DivisibilityandPrimes 67
3.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Greatest Common Divisor . . . . . . . . . . . . . . . . . 71
3.3 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Formulas for Primes . . . . . . . . . . . . . . . . . . . . 77
Twin Primes and Triple Primes . . . . . . . . . . . . . . 79
Other Conjectures about Primes . . . . . . . . . . . . . 80
Practice Problem Solutions and Hints . . . . . . . . . . . . . 81
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 TheDivisionandEuclideanAlgorithms 87
4.1 The Division Algorithm . . . . . . . . . . . . . . . . . . 87
The Division Algorithm with a Negative Dividend . . . 91
4.2 TheEuclideanAlgorithmandtheGreatestCommonDi-
visor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3 The Fundamental Theorem of Arithmetic . . . . . . . . 96
Why We Don’t Call 1 a Prime . . . . . . . . . . . . . . 98
Prime Factorization and the GCD . . . . . . . . . . . . 98
Practice Problem Solutions and Hints . . . . . . . . . . . . . 99
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 VariationsonaTheme 107
5.1 Applications of Divisibility. . . . . . . . . . . . . . . . . 107
Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . 107
Sum and Number of Divisors . . . . . . . . . . . . . . . 108
Perfect Numbers . . . . . . . . . . . . . . . . . . . . . . 113
5.2 More Algorithms . . . . . . . . . . . . . . . . . . . . . . 116
Rational Arithmetic and Least Common Multiples . . . 116
Egyptian Fractions . . . . . . . . . . . . . . . . . . . . . 119
Practice Problem Solutions and Hints . . . . . . . . . . . . . 122
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 CongruencesandGroups 129
6.1 Congruences and Arithmetic of Residue Classes . . . . . 129
6.2 Groups and Other Structures . . . . . . . . . . . . . . . 137
Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . 142
Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Zero Divisors and Fields . . . . . . . . . . . . . . . . . . 144
Practice Problem Solutions and Hints . . . . . . . . . . . . . 146
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Contents ix
7 ApplicationsofCongruences 155
7.1 Divisibility Tests . . . . . . . . . . . . . . . . . . . . . . 155
Divisibility by Powers of 2 . . . . . . . . . . . . . . . . . 155
Divisibility by Powers of 5 . . . . . . . . . . . . . . . . . 157
Divisibility by 3 and 9 . . . . . . . . . . . . . . . . . . . 157
Divisibility by 11 . . . . . . . . . . . . . . . . . . . . . . 158
Divisibility by 7, 11, and 13 . . . . . . . . . . . . . . . . 159
7.2 Days of the Week . . . . . . . . . . . . . . . . . . . . . . 159
Calculating from the First Date of Any Year . . . . . . 160
How to Find the Day of the Week . . . . . . . . . . . . 161
7.3 Check Digits . . . . . . . . . . . . . . . . . . . . . . . . 163
ISBNs and UPC Numbers . . . . . . . . . . . . . . . . . 164
Practice Problem Solutions and Hints . . . . . . . . . . . . . 165
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8 RationalNumbersandRealNumbers 171
8.1 Fractions to Decimals . . . . . . . . . . . . . . . . . . . 171
8.2 Decimals to Fractions . . . . . . . . . . . . . . . . . . . 174
8.3 Infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.4 Rational Numbers . . . . . . . . . . . . . . . . . . . . . 177
8.5 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . 179
8.6 How Many Real Numbers? . . . . . . . . . . . . . . . . 181
Practice Problem Solutions and Hints . . . . . . . . . . . . . 183
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9 IntroductiontoGeometryandSymmetry 187
Practice Problem Solutions and Hints . . . . . . . . . . . . . 191
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10 PolygonsandTheirConstruction 195
10.1 Polygons and Their Angles . . . . . . . . . . . . . . . . 195
Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . 196
n-gons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.2 Constructions . . . . . . . . . . . . . . . . . . . . . . . . 200
Practice Problem Solutions and Hints . . . . . . . . . . . . . 205
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
