Table Of ContentJulie Rowlett
Blast Into Math!
A fun and rigorous introduction to pure mathematics
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Blast into Math!
A fun and rigorous introduction to pure mathematics
First Edition
© 2013 Julie Rowlett & (Ventus Publishing ApS)
ISBN 978-87-403-0330-8
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A fun and rigorous introduction to
pure mathematics
Julie Rowlett
with illustrations by Henry Segerman
Blast into Math! Contents
Contents
Preface 8
Acknowledgments 9
1 To the reader 11
2 Pure mathematics: the proof of the pudding is in the eating 13
2.1 A universal language 14
2.2 Theorems, propositions, and lemmas 15
2.3 Logic 15
2.4 Ready? Set? Prove! 22
2.5 Exercises 24
2.6 Examples and hints 25
3 Sets of numbers: mathematical playgrounds 28
3.1 Set theory 28
3.2 Numbers 34
3.3 The least upper bound property 46
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Blast into Math! Contents
3.4 Proof by induction 51
3.5 Exercises 58
3.6 Examples and hints 63
4 The Euclidean algorithm: a computational recipe 67
4.1 Division 67
4.2 Greatest common divisors 74
4.3 Proof of the Euclidean Algorithm 79
4.4 Greatest common divisors in disguise 81
4.5 Exercises 85
4.6 Examples and hints 87
5 Prime numbers: indestructible building blocks 94
5.1 Ingredients in the proof of the Fundamental Theorem of Arithmetic 94
5.2 Unique prime factorization: the Fundamental Theorem of Arithmetic 98
5.3 How many primes are there? 102
5.4 Counting infinity 104
5.5 Exercises 119
5.6 Examples and hints 120
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Blast into Math! Contents
6 Mathematical perspectives: all your base are belong to us 123
6.1 Number bases: infinitely many mathematical perspectives 123
6.2 Fractions in bases 134
6.3 Exercises 137
6.4 Examples and hints 140
7 Analytic number theory: ants, ghosts and giants 146
7.1 Sequences: mathematical ants 146
7.2 Real numbers and friendly rational numbers 162
7.3 Series: a tower of mathematical ants 171
7.4 Decimal expansions 184
7.5 The Prime Number Theorem 194
7.6 Exercises 202
7.7 Examples and hints 205
8 Afterword 213
9 Bibliography 215
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Blast into Math! Preface
Preface
The purpose of this book is to offer readers a fun mathematical learning experience without sacrificing
or oversimplifying the mathematics. Pure, rigorous mathematics is presented with concise definitions,
theorems and proofs; accompanying the mathematics are lively descriptions, colorful exposition, and
analogies. My goal is to share with readers through an active reading experience how mathematicians
perceive and experience mathematics. It is vibrant, exciting, and dynamic, like the analogies used in
this book to describe it.
Readers will notice that each chapter has a theme color. According to psychological research [W-S-G],
people remember things better when they are in color. The theme color is used for emphasis within the
chapter and is also an associative mnemonic for each chapter’s topic. The first chapter, “To the Reader,”
explains how the reader should actively read the book. Mathematics is experiential; one must do math to
understand it. The second chapter introduces the fundamental topics in logic upon which all mathematical
proofs are based, teaching readers what constitutes a mathematical proof and guiding them to begin
writing their own proofs. The next three chapters cover set theory and basic topics in number theory.
Chapter six teaches readers to be creative by changing their mathematical perspective, by writing numbers
in different bases. The last chapter introduces analysis.
Traditional textbooks tend to focus on a specific area of mathematics without explaining how
mathematicians do research. In this book a parallel is made between the reader’s experience and the
experience of research mathematicians. The basic principles and process of mathematics research are
analogous to the reader’s process of working through the book. When the reader is asked to complete
part of the proof of a theorem as an exercise this is compared to collaboration amongst mathematicians.
This book not only presents fundamental topics in pure mathematics but also shares with readers the
basic principles of pure mathematics research.
The intended audience includes undergraduate students in “Transition to Higher Mathematics” courses
and advanced high school students. The pre-requisite knowledge has most likely been met by a standard
high-school education, and so this book could also be used in community college and continuing
education or read by a general audience. It offers readers active participation, in the spirit of Sudoku, but
unlike Sudoku it delves deeper into the world of mathematics research. By the end of the book readers
will have worked on problems which no mathematician has ever solved!
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Blast into Math! Preface
I believe that anyone can learn mathematics if it is presented in a way they can understand, and that
a positive mathematical learning experience can increase one’s overall confidence. This book aims to
challenge its readers while supporting their efforts with enthusiastic encouragement, lively examples,
entertaining mnemonics, and helpful hints. Motivating by example, a parallel is made between readers
working and struggling through the book and research mathematicians working and struggling with their
research problems; the reader is not working alone, he or she is working within the global community of
mathematicians. In the end the reader’s efforts will be rewarded with a strong sense of accomplishment,
and it is my sincere hope that this sense of accomplishment based on overcoming mathematical challenges
will help readers to face and overcome other challenges as well.
Acknowledgments
I am deeply grateful to colleagues, students, friends, and family who encouraged and supported me in
this endeavor. It began as lecture notes for the Education Program for Gifted Youth at Stanford University,
2006, and I am grateful to Rick Sommer and my students for inspiring me to turn those notes into a
book. Many thanks to colleagues and friends who read and commented on the draft: Henry Segerman,
Jeff Stopple, Antonio Iarrobino, Gerald Folland, Pieter Moree, Ravi Vakil, Michail Verskajn, and Vickie
Kern. I am also grateful to Ina Mette for her suggestion to choose a catchier and more succinct title. Sophie
Tergeist’s enthusiastic support and Bookboon have been wonderful. Finally, I gratefully acknowledge the
support of the Max Planck Institute for Mathematics in Bonn.
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To Romeo.
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