Perkins D. φ, π, e & i

Washington: Mathematical Association of America, 2018. — 192 p.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.φ
Of what is everything made?
The golden rectangle
The Eye, and the arithmetic of φ
The Fibonacci (Hemachandra) sequence
A continued fraction for
φ is irrational
The arithmetic geometric mean inequality
Further content
π
Liu Hui approximates π using polygons
Nilakantha’s arctangent series
Machin’s arctangent formula
Wallis’s formula for π/2 (via calculus)
A connection to probability
Wallis’s formula for π/2 via (sin x)/x
The generalized binomial theorem
Euler’s (1/2)! = √π/2
The Basel problem: ∑1/k² = π²/6
π is irrational
Further content
e
The money puzzle
Euler’s e = ∑1/k!
The maximum of x{exp(1/x)}
The limit of (1+1/n)ⁿ
A modern proof that e = ∑1/k!
e is irrational
Stirling’s formula
Turning a series into a continued fraction
Further content
i
Proportions
Negatives
Chimeras
Cubics
A truly curious thing
The complex plane
ln(i)
iθ = ln(cosθ + i sinθ)
e{exp(iθ)} = cosθ + i sinθ
The shortest path
φ = e{exp(iπ/5)} + e{exp(-iπ/5)}
Further content
Wallis’s original derivation of his formula for π
Newton’s original generalized binomial theoremExtra help