"Blast Into Math! A Fun and Rigorous Introduction to Pure Mathematics" by Julie Rowlett
JuRo, BoBoCoAE | 2013 | ISBN: 8740303308 9788740303308 | 215 pages | PDF | 10 MB
This book is suitable for both students and a general audience interested in learning what pure mathematics is all about. Pure mathematics is presented in a friendly, accessible, and nonetheless rigorous style. Definitions, theorems, and proofs are accompanied by creative analogies and illustrations to convey the meaning and intuition behind the abstract math.
You don't need much background for the first few chapters, but the material builds upon itself, and if you don't do the exercises, eventually you'll have trouble understanding.
If you read this book and do all the exercises, you will not only learn how to prove theorems, you'll also experience what mathematics research is like: exciting, challenging, and fun!
Contents Preface
1. To the reader
2. Pure mathematics: the proof of the pudding is in the eating
2.1. A universal language
2.2. Theorems, propositions, and lemmas
2.3. Logic
2.4. Ready? Set? Prove!
2.5. Exercises
2.6. Examples and hints
3. Sets of numbers: mathematical playgrounds
3.1. Set theory
3.2. Numbers
3.3. The least upper bound property
3.4. Proof by induction
3.5. Exercises
3.6. Examples and hints
4. The Euclidean algorithm: a computational recipe
4.1. Division
4.2. Greatest common divisors
4.3. Proof of the Euclidean Algorithm
4.4. Greatest common divisors in disguise
4.5. Exercises
4.6. Examples and hints
5. Prime numbers: indestructible building blocks
5.1. Ingredients in the proof of the Fundamental Theorem of Arithmetic
5.2. Unique prime factorization: the Fundamental Theorem of Arithmetic
5.3. How many primes are there?
5.4. Counting infinity
5.5. Exercises
5.6. Examples and hints
6. Mathematical perspectives: all your base are belong to us
6.1. Number bases: infinitely many mathematical perspectives
6.2. Fractions in bases
6.3. Exercises
6.4. Examples and hints
7. Analytic number theory: ants, ghosts and giants
7.1. Sequences: mathematical ants
7.2. Real numbers and friendly rational numbers
7.3. Series: a tower of mathematical ants
7.4. Decimal expansions
7.5. The Prime Number Theorem
7.6. Exercises
7.7. Examples and hints
8. Afterword
9. Bibliography
with TOC BookMarkLinks
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