φ, π, e & i

Description

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... Preface Contents 1 φ 1.1 Of what is everything made? 1.2 The golden rectangle 1.3 The Eye, and the arithmetic of φ 1.4 The Fibonacci (Hemachandra) sequence 1.5 A continued fraction for 1.6 φ is irrational 1.7 The arithmetic geometric mean inequality 1.8 Further content 2 π 2.1 Liu Hui approximates π using polygons 2.2 Nilakantha’s arctangent series 2.3 Machin’s arctangent formula 2.4 Wallis’s formula for π/2 (via calculus) 2.5 A connection to probability 2.6 Wallis’s formula for π/2 via (sin x)/x 2.7 The generalized binomial theorem 2.8 Euler’s (1/2)! = √π/2 2.9 The Basel problem: ∑1/k² = π²/6 2.10 π is irrational 2.11 Further content 3 e 3.1 The money puzzle 3.2 Euler’s e = ∑1/k! 3.3 The maximum of x{exp(1/x)} 3.4 The limit of (1+1/n)ⁿ 3.5 A modern proof that e = ∑1/k! 3.6 e is irrational 3.7 Stirling’s formula 3.8 Turning a series into a continued fraction 3.9 Further content 4 i 4.1 Proportions 4.2 Negatives 4.3 Chimeras 4.4 Cubics 4.5 A truly curious thing 4.6 The complex plane 4.7 ln(i) 4.8 iθ = ln(cosθ + i sinθ) 4.9 e{exp(iθ)} = cosθ + i sinθ 4.10 The shortest path 4.11 φ = e{exp(iπ/5)} + e{exp(-iπ/5)} 4.12 Further content APPENDIX A: Wallis’s original derivation of his formula for π APPENDIX B: Newton’s original generalized binomial theorem Bibliography Extra help Index