# Phi, Pi, e, and i

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*David Perkins*

MAA Press: An Imprint of the American Mathematical Society

Reprinted edition available: SPEC/96

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 (\(
\varphi, \Pi, e\), 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 that are not often seen in college
classes. These truths are described here via the work of such
luminaries as Nilakantha, Liu Hui, Hemachandra, Khayyam, Newton,
Wallis, and Euler.

#### Reviews & Endorsements

This book can be used as a refresher on these aspects of the history of mathematics, and it could also work well for someone who is interested in the inner working of past mathematical geniuses' minds and the coincidences that make math so beautiful.

-- Kevin W. Pledger, Mathematics Teacher

# Table of Contents

## Phi, Pi, e, and i

- cover cover11
- copyright page ii3
- Phi, Pi, e, and i iii4
- Preface ix10
- Contents xi12
- Chapter 1 Phi 116
- Of what is everything made? 217
- The golden rectangle 520
- The Eye, and the arithmetic of 621
- The Fibonacci (Hemachandra) sequence 924
- A continued fraction for 1126
- is irrational 1429
- The arithmetic geometric mean inequality 1631
- Further content 1934
- Constructing geometrically 1934
- Binet's formula: Fn = (n - n)/5 2035
- The harmonic-geometric-arithmetic right triangle 2136
- The continued fraction for converges (via graphing) 2237
- 5 is irrational (via algebra) 2439
- 5 is irrational (via geometry) 2439
- is irrational (via its continued fraction) 2540
- 5 is irrational (via ternary arithmetic) 2641

- Chapter 2 Pi 2944
- Liu Hui approximates using polygons 3045
- Nilakantha's arctangent series 3247
- Machin's arctangent formula 3449
- Wallis's formula for / 2 (via calculus) 3651
- A connection to probability 4055
- Wallis's formula for / 2 via (sinx)/x 4156
- The generalized binomial theorem 4257
- Euler's (1/2)! = /2 4459
- The Basel problem: 1/k2 = 2 / 6 4762
- is irrational 4964
- Further content 5267

- Chapter 3 e 6580
- Chapter 4 i 99114
- Wallis's original derivation of his formula for pi 131146
- Newton's original generalized binomial theorem 139154
- Bibliography 145160
- Extra Help 149164
- Index 175190