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Phi, Pi, e, and i
 
David Perkins
Phi, Pi, e, and i
MAA Press: An Imprint of the American Mathematical Society
This title is not currently available
Phi, Pi, e, and i
Click above image for expanded view
Phi, Pi, e, and i
David Perkins
MAA Press: An Imprint of the American Mathematical Society
This title is not currently available
  • Book Details
     
     
    Spectrum
    Volume: 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 harmonic-geometric-arithmetic 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 -e-x2 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 + e-i / 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
Spectrum
Volume: 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.
  • 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 harmonic-geometric-arithmetic 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 -e-x2 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 + e-i / 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
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