Blast into Math - A Fun and Rigorous Introduction to Pure Mathematics

Cover......Page 1
Blast Into Math!: A fun and rigorous introduction to pure mathematics......Page 2
©......Page 3
A fun and rigorous introduction to pure mathematics......Page 4
Contents......Page 5
Preface......Page 8
Acknowledgments......Page 9
1 To the reader......Page 11
2 Pure mathematics: the proof of the pudding is in the eating......Page 13
2.1 A universal language......Page 14
2.3 Logic......Page 15
2.4 Ready? Set? Prove!......Page 22
2.5 Exercises......Page 24
2.6 Examples and hints......Page 25
3.1 Set theory......Page 28
3.2 Numbers......Page 34
3.3 The least upper bound property......Page 46
3.4 Proof by induction......Page 51
3.5 Exercises......Page 58
3.6 Examples and hints......Page 63
4.1 Division......Page 67
4.2 Greatest common divisors......Page 74
4.3 Proof of the Euclidean Algorithm......Page 79
4.4 Greatest common divisors in disguise......Page 81
4.5 Exercises......Page 85
4.6 Examples and hints......Page 87
5.1 Ingredients in the proof of the Fundamental Theorem of Arithmetic......Page 94
5.2 Unique prime factorization: the Fundamental Theorem of Arithmetic......Page 98
5.3 How many primes are there?......Page 102
5.4 Counting infinity......Page 104
5.5 Exercises......Page 119
5.6 Examples and hints......Page 120
6.1 Number bases: infinitely many mathematical perspectives......Page 123
6.2 Fractions in bases......Page 134
6.3 Exercises......Page 137
6.4 Examples and hints......Page 140
7.1 Sequences: mathematical ants......Page 146
7.2 Real numbers and friendly rational numbers......Page 162
7.3 Series: a tower of mathematical ants......Page 171
7.4 Decimal expansions......Page 184
7.5 The Prime Number Theorem......Page 194
7.6 Exercises......Page 202
7.7 Examples and hints......Page 205
8 Afterword......Page 213
9 Bibliography......Page 215