published by the American Mathematical Association of Two-Year Colleges volume 30 • number 1 • fall 2008
The AMAT YC Review
How to Design Your Own π to e Converter Harlan J. Brothers Harlan Brothers is Director of Technology at The Country School in Madison, CT where he teaches programming, fractal geometry, and guitar. Having worked for six years with Michael Frame and Benoit Mandelbrot at the Yale Fractal Geometry Workshops, he now lectures on the subject of fractal music. Harlan is also an inventor with five US patents. E-mail:
[email protected]
The base of the natural logarithm, e, has been studied since the early seventeenth century (Maor, 1994, pp. 26–27). Its limit definition formula n (1) e = lim 1 + n1 n→∞
is a variant of the compound-interest formula used in finance (Anton, 1980, p. 558; 2; Bradley & Smith, 1999, pp. 63–64). This article demonstrates how a simple restatement of (1) can lead to the derivation of a fascinating family of functions that can be used to convert the digits of π to those of e.
Generalizing the Limit Definition of e Starting with the right hand side (RHS) of (1), we can treat it as a real function of two variables, r and s. Replacing the integer n with a rational number in the form r/s we have r s e = rlim r+s |r| > |s|, r, s 6= 0 (2) r | s |→∞ Thus, the greater the ratio between r and s, the better (2) approximates e. The value of (2) is that it can be redefined more broadly as a compound function of one variable by replacing r with subfunction f (x) and s with subfunction g(x). The result is a formula we refer to as the Generalized Classical Method (GCM). Given: f (x) and g(x) are monotone over x, |f (x)| > |g(x)|, and f (x), g(x) are divergent, then GCM (x) =
f (x) + g(x) f (x)
f (x) g(x)
f (x), g(x) 6= 0
(3)
30 How to Design Your Own π to e Converter and e = lim
|x|→∞
f (x) + g(x) f (x)
f (x) g(x)
(4)
New connections between π and e When we assign f (x) and g(x) using the appropriate trigonometric subfunctions, the functions that result, in their limits, define new relationships between π and e. For example, we can set f (x) = cos(θ) and g(x) = sin(θ). As θ approaches π, cos(θ) → −1, sin(θ) → 0. Although the function is not defined at θ = π, we find that cos(θ) cos(θ) + sin(θ) sin(θ) 3π for 3π (5) e = lim 4
