Softcover ISBN:  9781470461287 
Product Code:  DOL/53.S 
List Price:  $52.00 
MAA Member Price:  $39.00 
AMS Member Price:  $39.00 
Electronic ISBN:  9781470451561 
Product Code:  DOL/53.E 
List Price:  $52.00 
MAA Member Price:  $39.00 
AMS Member Price:  $39.00 

Book DetailsDolciani Mathematical ExpositionsVolume: 53; 2019; 480 ppMSC: Primary 11; 00; 70;
There is a nineteenyear 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 SternBrocot tree, and a number of combinatorial sequences.
The book features a pleasantly discursive style with excursions into music (The WellTempered 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.ReadershipUndergraduate students interested in number theory.

Table of Contents

Chapters

Patterns

Tally bones to the integers

Leibniz and the binary revolution

Mathematical induction

AlMaghribî meets Sodoku

GCD’s and diophantine equations

Fractions in the Pythagorean scale

A tree of fractions

Bach and the welltempered clavier

The harmonic series

A clay tablet

Families of numbers

Planetary conjunctions

Simple and strange harmonic motion

The size and shape of Utopia Island

Classic elliptical fractions

The Cantor set

Continued fractions

The longevity of the 17year cicada

Transits of Venus

Meton of Athens

Lunar rhythms

Eclipse lore and legends

Diophantine eclipses

List of symbols used in the test

An introduction to vectors and matrices

Computer algebra system codes

Comments on selected exercises


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There is a nineteenyear 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 SternBrocot tree, and a number of combinatorial sequences.
The book features a pleasantly discursive style with excursions into music (The WellTempered 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.
Undergraduate students interested in number theory.

Chapters

Patterns

Tally bones to the integers

Leibniz and the binary revolution

Mathematical induction

AlMaghribî meets Sodoku

GCD’s and diophantine equations

Fractions in the Pythagorean scale

A tree of fractions

Bach and the welltempered clavier

The harmonic series

A clay tablet

Families of numbers

Planetary conjunctions

Simple and strange harmonic motion

The size and shape of Utopia Island

Classic elliptical fractions

The Cantor set

Continued fractions

The longevity of the 17year cicada

Transits of Venus

Meton of Athens

Lunar rhythms

Eclipse lore and legends

Diophantine eclipses

List of symbols used in the test

An introduction to vectors and matrices

Computer algebra system codes

Comments on selected exercises