φ, π, e & i | Phi, pi, e & i

✍ Scribed by David Perkins

• Publisher: American Mathematical Society
• Year: 2018
• Tongue: English
• Leaves: 192
• Series: Spectrum Ser.
• Edition: Paperback
• Category: Library

✦ Synopsis

Certain constants occupy precise balancing points in the cosmos of number, like habitable planets sprinkled throughout our galaxy at just the right distances from their suns. This book introduces and connects four of these constants (φ, π, e, and i), each of which has recently been the individual subject of historical and mathematical expositions. But here we discuss their properties, as a group, at a level appropriate for an audience armed only with the tools of elementary calculus. This material offers an excellent excuse to display the power of calculus to reveal elegant truths tha.  Read more...

✦ Table of Contents

Preface......Page 10
Contents......Page 12
1 Phi......Page 16
1.1 Of what is everything made?......Page 17
1.2 The golden rectangle......Page 20
1.3 The Eye, and the arithmetic of......Page 21
1.4 The Fibonacci (Hemachandra) sequence......Page 24
1.5 A continued fraction for......Page 26
1.6 ' is irrational......Page 29
1.7 The arithmetic geometric mean inequality......Page 31
1.8 Further content......Page 34
2 Pi......Page 44
2.1 Liu Hui approximates  using polygons......Page 45
2.2 Nilakantha’s arctangent series......Page 47
2.3 Machin’s arctangent formula......Page 49
2.4 Wallis’s formula for =2 (via calculus......Page 51
2.5 A connection to probability......Page 55
2.6 Wallis’s formula for =2 via (sin x)=x......Page 56
2.7 The generalized binomial theorem......Page 57
2.8 Euler’s......Page 59
2.9 The Basel problem:......Page 62
2.10  is irrational......Page 64
2.11 Further content......Page 67
3.1 The money puzzle......Page 80
3.2 Euler’s......Page 82
3.3 The maximum of......Page 84
3.4 The limit of......Page 88
3.5 A modern proof that......Page 92
3.7 Stirling’s formula......Page 95
3.8 Turning a series into a continued fraction......Page 101
3.9 Further content......Page 104
4 i......Page 114
4.1 Proportions......Page 115
4.2 Negatives......Page 117
4.3 Chimeras......Page 118
4.4 Cubics......Page 119
4.5 A truly curious thing......Page 122
4.6 The complex plane......Page 124
4.7 ln (i)......Page 129
4.8 i = ln(cos + i sin )......Page 131
4.9 ei = cos + i sin......Page 133
4.10 The shortest path......Page 135
4.11 ' = ei=5 + e��i=5......Page 137
4.12 Further content......Page 138
APPENDIX A Wallis’s original derivation of his formula for ......Page 146
APPENDIX B Newton’s original generalized binomial theorem......Page 154
Bibliography......Page 160
Extra help......Page 164
Index......Page 190

✦ Subjects

Mathematical constants;Pi;Golden section;Euler’s numbers