**Dolciani Mathematical Expositions**

Volume: 53;
2019;
480 pp;
Softcover

MSC: Primary 11; 00; 70;

**Print ISBN: 978-1-4704-6128-7
Product Code: DOL/53.S**

List Price: $52.00

AMS Member Price: $39.00

MAA Member Price: $39.00

**Electronic ISBN: 978-1-4704-5156-1
Product Code: DOL/53.E**

List Price: $52.00

AMS Member Price: $39.00

MAA Member Price: $39.00

#### Supplemental Materials

# Exploring Continued Fractions: From the Integers to Solar Eclipses

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*Andrew J. Simoson*

MAA Press: An Imprint of the American Mathematical Society

There is a nineteen-year recurrence in the
apparent position of the sun and moon against the background of the
stars, a pattern observed long ago by the Babylonians. In the course
of those nineteen years the Earth experiences 235 lunar cycles.
Suppose we calculate the ratio of Earth's period about the sun to the
moon's period about Earth. That ratio has 235/19 as one of its early
continued fraction convergents, which explains the apparent
periodicity.

Exploring Continued Fractions explains this and other recurrent
phenomena—astronomical transits and conjunctions, lifecycles of
cicadas, eclipses—by way of continued fraction expansions. The deeper
purpose is to find patterns, solve puzzles, and discover some
appealing number theory. The reader will explore several algorithms
for computing continued fractions, including some new to the
literature. He or she will also explore the surprisingly large
portion of number theory connected to continued fractions: Pythagorean
triples, Diophantine equations, the Stern-Brocot tree, and a number of
combinatorial sequences.

The book features a pleasantly discursive style with excursions
into music (The Well-Tempered Clavier), history (the Ishango bone and
Plimpton 322), classics (the shape of More's Utopia) and whimsy
(dropping a black hole on Earth's surface). Andy Simoson has won both
the Chauvenet Prize and Pólya Award for expository writing from the
MAA and his Voltaire's Riddle was a Choice magazine
Outstanding Academic Title. This book is an enjoyable ramble through some
beautiful mathematics. For most of the journey the only necessary
prerequisites are a minimal familiarity with mathematical reasoning
and a sense of fun.

#### Readership

Undergraduate students interested in number theory.

#### Table of Contents

# Table of Contents

## Exploring Continued Fractions: From the Integers to Solar Eclipses

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents vii9
- Introduction xiii15
- Strand I: Patterns 123
- Chapter I: Tally Bones to the Integers 931
- Strand II: Leibniz and the Binary Revolution 2951
- Chapter II: Mathematical Induction 3759
- Strand III: Al-Maghribî meets Sudoku 6991
- Chapter III: GCDs and Diophantine Equations 7395
- The greatest common divisor 7496
- An ancient algorithm for the greatest common divisor 78100
- The Diophantine solution 85107
- A litmus test for Euclid’s solution 88110
- Clock arithmetic 89111
- Systems of Diophantine equations 92114
- The totient is multiplicative 93115
- A problem from Diophantus’s Arithmetica 93115
- Exercises 94116

- Strand IV: Fractions in the Pythagorean Scale 99121
- Chapter IV: A Tree of Fractions 107129
- Strand V: Bach and The Well-Tempered Clavier 139161
- Chapter V: The Harmonic Series 147169
- Strand VI: A Clay Tablet 169191
- Chapter VI: Families of Numbers 185207
- Strand VII: Planetary Conjunctions 221243
- Chapter VII: Simple and Strange Harmonic Motion 229251
- Strand VIII: The Size and Shape of Utopia Island 261283
- Chapter VIII: Classic Elliptical Fractions 271293
- Strand IX: The Cantor Set 303325
- Chapter IX: Continued Fractions 311333
- A local approach to continued fractions 311333
- A global approach to continued fractions 318340
- A plethora of continued fractions 322344
- Why the ugly duckling 𝐺 is really a swan 328350
- An interlude delineating Algorithm 𝑂* 330352
- Dominance domains 331353
- The harmonic algorithm is a chameleon 332354
- Applying continued fractions to factoring integers 335357
- The first infinite continued fraction 336358
- Black holes and the receding Moon 340362
- Exercises 345367

- Strand X: The Longevity of the 17-year Cicada 351373
- Chapter X: Transits of Venus 357379
- Strand XI: Meton of Athens 379401
- Chapter XI: Lunar Rhythms 383405
- Strand XII: Eclipse Lore and Legends 399421
- Chapter XII: Diophantine Eclipses 405427
- Adapting the Earth-Moon-Sun model 405427
- Eclipse duration 408430
- A sufficient condition for eclipses 408430
- Finding 𝐻 at any lunation 410432
- Using Condition 1 to find the lapse between successive eclipses 412434
- Continued fraction insight 412434
- Some Diophantine magic 415437
- Lunar eclipses 418440
- A reality check 419441
- A final note 420442
- Exercises 421443

- Appendix I: List of Symbols Used in the Text 425447
- Appendix II: An Introduction to Vectors and Matrices 429451
- Appendix III: Computer Algebra System Codes 437459
- Appendix IV: Comments on Selected Exercises 453475
- Bibliography 465487
- Index 473495
- Back Cover Back Cover1503