
Book DetailsSpectrumVolume: 86; 2017; 176 pp
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.

Table of Contents

cover

copyright page

Phi, Pi, e, and i

Preface

Contents

Chapter 1 Phi

Of what is everything made?

The golden rectangle

The Eye, and the arithmetic of

The Fibonacci (Hemachandra) sequence

A continued fraction for

is irrational

The arithmetic geometric mean inequality

Further content

Constructing geometrically

Binet's formula: Fn = (n  n)/5

The harmonicgeometricarithmetic right triangle

The continued fraction for converges (via graphing)

5 is irrational (via algebra)

5 is irrational (via geometry)

is irrational (via its continued fraction)

5 is irrational (via ternary arithmetic)

Chapter 2 Pi

Liu Hui approximates using polygons

Nilakantha's arctangent series

Machin's arctangent formula

Wallis's formula for / 2 (via calculus)

A connection to probability

Wallis's formula for / 2 via (sinx)/x

The generalized binomial theorem

Euler's (1/2)! = /2

The Basel problem: 1/k2 = 2 / 6

is irrational

Further content

A geometric derivation of a formula for

Wallis's formula for / 2 (via algebra)

Converting an infinite product to a continued fraction

A harmonic continued fraction for

Finding the nth digit of

Chapter 3 e

The money puzzle

Euler's e = 1/k!

The maximum of x1/x

The limit of ( 1 + 1n )n

A modern proof that e = 1/k!

e is irrational

Stirling's formula

Turning a series into a continued fraction

Further content

The probability of derangement.

Deranged matchings.

Euler's constant

Euler's constant via e and

The probability integral ex2 dx =

Chapter 4 i

Proportions

Negatives

Chimeras

Cubics

A truly curious thing

The complex plane

ln(i)

i = ln(cos + i sin)

ei = cos + i sin

The shortest path

= ei / 5 + ei / 5

Further content

Khayyám's geometric solution to a cubic

Viète's trigonometric approach to cubics

A complex approach to the Basel problem

Hamilton discovers the quaternions

Wallis's original derivation of his formula for pi

Newton's original generalized binomial theorem

Bibliography

Extra Help

Index


Additional Material

Reviews

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

 Book Details
 Table of Contents
 Additional Material
 Reviews
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.

cover

copyright page

Phi, Pi, e, and i

Preface

Contents

Chapter 1 Phi

Of what is everything made?

The golden rectangle

The Eye, and the arithmetic of

The Fibonacci (Hemachandra) sequence

A continued fraction for

is irrational

The arithmetic geometric mean inequality

Further content

Constructing geometrically

Binet's formula: Fn = (n  n)/5

The harmonicgeometricarithmetic right triangle

The continued fraction for converges (via graphing)

5 is irrational (via algebra)

5 is irrational (via geometry)

is irrational (via its continued fraction)

5 is irrational (via ternary arithmetic)

Chapter 2 Pi

Liu Hui approximates using polygons

Nilakantha's arctangent series

Machin's arctangent formula

Wallis's formula for / 2 (via calculus)

A connection to probability

Wallis's formula for / 2 via (sinx)/x

The generalized binomial theorem

Euler's (1/2)! = /2

The Basel problem: 1/k2 = 2 / 6

is irrational

Further content

A geometric derivation of a formula for

Wallis's formula for / 2 (via algebra)

Converting an infinite product to a continued fraction

A harmonic continued fraction for

Finding the nth digit of

Chapter 3 e

The money puzzle

Euler's e = 1/k!

The maximum of x1/x

The limit of ( 1 + 1n )n

A modern proof that e = 1/k!

e is irrational

Stirling's formula

Turning a series into a continued fraction

Further content

The probability of derangement.

Deranged matchings.

Euler's constant

Euler's constant via e and

The probability integral ex2 dx =

Chapter 4 i

Proportions

Negatives

Chimeras

Cubics

A truly curious thing

The complex plane

ln(i)

i = ln(cos + i sin)

ei = cos + i sin

The shortest path

= ei / 5 + ei / 5

Further content

Khayyám's geometric solution to a cubic

Viète's trigonometric approach to cubics

A complex approach to the Basel problem

Hamilton discovers the quaternions

Wallis's original derivation of his formula for pi

Newton's original generalized binomial theorem

Bibliography

Extra Help

Index

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