How to Design Your Own pi to e Converter

Abstract

This article demonstrates how a simple restatement of the limit definition of e can lead to the derivation of a fascinating family of functions that can be used to convert the digits of pi to those of e.

Full text

The AMAT
YC Review
published by the
American Mathematical Association of Two-Year Colleges
published by the
American Mathematical Association of Two-Year Colleges
volume 30
•
number 1
•
fall 2008

How to Design Your Own πto eConverter
Harlan J. Brothers
Harlan Brothers is Director of Technology at The Country School in
Madison, CT where he teaches programming, fractal geometry, and gui-
tar. Having worked for six years with Michael Frame and Benoit Man-
delbrot 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: harlan@thecountryschool.org
The base of the natural logarithm, e, has been studied since the early seven-
teenth century (Maor, 1994, pp. 26–27). Its limit definition formula
e= lim
n→∞ 1 + 1
nn(1)
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 πto 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, rand s. Replacing the integer nwith a rational number in the form
r/s we have
e= lim
|r
s|→∞ r+s
rr
s|r|>|s|, r, s 6= 0 (2)
Thus, the greater the ratio between rand 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 rwith subfunction f(x)and swith 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 πto eConverter
and
e= lim
|x|→∞ f(x) + g(x)
f(x)f(x)
g(x)(4)
New connections between πand e
When we assign f(x)and g(x)using the appropriate trigonometric subfunc-
tions, 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 θapproaches π,
cos(θ)→ −1,sin(θ)→0. Although the function is not defined at θ=π, we find
that
e= lim
θ→πcos(θ) + sin(θ)
cos(θ)cos(θ)
sin(θ)for 3π
4< θ < 3π
2(5)
or equivalently,
e= lim
θ→π(1 + tan(θ))cot(θ)for 3π
4< θ < 3π
2.(6)
In the same manner, we can set the functions f(x) = cot(θ)and g(x) = tan(θ).
As θapproaches πfrom below, cot(θ)→ −∞ and tan(θ)→0. As θapproaches π
from above, cot(θ)→ ∞ while tan(θ)→0. Substituting these functions in (4) and
using trigonometric identities we arrive at
e= lim
θ→πsec2(θ)cot2(θ)for π
2< θ < 3π
2.(7)
It should be noted that it is not necessary to begin with GCM. The same
method of derivation can be applied to other closed-form approximations (Brothers
& Knox, 1998; Knox & Brothers, 1999; Richardson & Gaunt, 1927).
To assess the relative rates of convergence for these expressions it is possible to
generate their respective power series (see Brothers & Knox; Knox & Brothers) by
replacing θwith π±1/x and then evaluating the expression as x→ ∞. For (6) and
(7) the associated power series when approaching πfrom below are, respectively,
(1 + tan (π−(1/x)))cot(π−(1/x)) =e1 + 1
2x+11
24x2+29
48x3
+4207
5760x4+O1
x5, x > 4
π
(8)
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Harlan J. Brothers 31
a first-order approximation, and in alternative form,
sec2(π−(1/x))(sec2(π−(1/x))−1)−1
=e1−1
2x2+1
8x4−11
728x6
+19
13440x8−O1
x10 , x > 2
π
(9)
which, as confirmed by the RHS, is a second-order approximation.
Dozens of trigonometric approximations to ecan be derived in the manner
described. Many of these functions map to Cover at least some portion of the
interval [0,2π]. For example, the following form has the interesting property that in
addition to converging to eas θ→π, it also converges to √2at 5π/4and to 1 at
3π/2(see Figure 1 below and Figure 2 on page 32):
e= lim
θ→π(−sin(θ)−cos(θ))cot(θ)for 3π
4< θ < 7π
4(10)
Figure 1: Graph of the Real output of the function f(θ) =
(−sin (θ)−cos (θ))cot(θ)for 3π/4< θ < 11π/4with removable sin-
gularities at πand 2π. This function maps to Rfor 3π/4< θ < 7π/4
and to Cfor 7π/4< θ < 11π/4. It is undefined at θ= 3π/4and
θ= 7π/4and Real at (9π/4,−√2) and (5π/2,1).
Similar approximations can also be derived so as to converge for π/2. Higher-
order approximations can also be achieved in the same fashion. A more extensive
list of trigonometric expressions for ecan be found at (Brothers).
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32 How to Design Your Own πto eConverter
Figure 2: 3D scatter plot of the function in Figure 1 revealing the
Imaginary component for 7π/4< θ < 11π/4. The modulus of the
complex region approaches eas θ→2π.
The πto econverter
Because the above functions provide a direct link between the numerical values
of these two constants, using a computer algebra program or arbitrary precision
library one can write a procedure that takes a given number of the digits of πand,
from them, derives digits of e. Here is sample Mathematica®code based on the
first-order approximation in (6):
$MaxExtraPrecision 10000; power 1;
func _ : 1 Tan power Cot power
numDigits 12; extraPlaces 0; acc power numDigits extraPlaces;
piDigits Flatten Drop RealDigits , 10, numDigits , 1 ;
piNum FromDigits piDigits ;
eValue N func piNum 10numDigits 1, acc ;
Print "Input digits ", numDigits, " digit accuracy ", piDigits
If extraPlaces 0, Print " ", numDigits extraPlaces, " digit accuracy
", N , numDigits extraPlaces
Print " ", acc, " digit accuracy ", actualE N E, acc
Print "Output ", acc, " digit accuracy ", eValue
Print "Error ", N FromDigits RealDigits actualE , acc
N FromDigits RealDigits eValue , acc
The AMATYC Review Volume 30, Number 1

Harlan J. Brothers 33
Using the variable numDigits, this program takes the first ndigits of πas its
input, rounded to ndigit accuracy (d.a.). It outputs the actual value of e(at nd.a.)
along with the computed value and the error, expressed as eminus the computed
value. For the first-order approximation (6), inputting the first twelve digits of π
yields the first 12 digits of e(at 12 d.a.).
Higher-order approximations can also be tested with the program. For exam-
ple, using trigonometric identities, the second-order approximation in (7) can be
rewritten as
e= lim
θ→π1 + tan2(θ)cot2(θ).(11)
Setting the variable power = 2 and using the same 12 digit input yields 24 digits
of e. In general, the number of accurate digits obtained is proportional to order of
accuracy of the approximation’s associated power series; a kth order approximation
returns approximately ktimes the number of digits that are input.
The decimal place accuracy of the output (the number of accurate digits to the
right of the decimal point) can be directly approximated using the log of the error
term in the approximation’s associated power series (see equations (8) and (9)):
d.p.a. ≈j−Log10 
a
bxnk (12)
where a,b, and nare constants and “b c” denotes the floor function. Let Paequal
the actual value of πand let Prequal the input value rounded to nd.a.. The value
of xis then calculated according to
x= 1/(Pa−Pr).(13)
It is interesting to note that certain inputs are inherently more accurate at any
order. Because the function fis defined by the limit e= limθ→πf(θ), it converges as
|Pa−Pr| → 0. Rounding error therefore changes according to the value of the digits
immediately following the specific number of digits chosen for input. Remembering
that the first digit of πoccurs to the left of the decimal point, if we denote the
maximum possible rounding error by ε, then at any given value of accuracy n,
ε= 5 ×10−n+C×10−(n+1), for C < 1as C→0. The maximum error in the
converted value therefore occurs when |Pa−Pr|=ε. Conversely, minimum error
occurs when |Pa−Pr|<1×10−n.
For any arbitrary number of input digits, the potential range in accuracy is
equal to (|Pa−Pr|max − |Pa−Pr|min). This is simply ε, which to one d.p.a. in the
nth decimal place is equal to 0.5. For the function in (8), the error term in its power
series is (1/2x). Evaluating this term with x= 1/(0.5) (see Eq. 13), we find that the
Fall 2008 The AMATYC Review

34 How to Design Your Own πto eConverter
maximum potential error ηin the estimation of the overall d.p.a. of the function is
equal to
η=−Log10 1
4≈0.6(14)
Referring back to the above code, the maximum decimal place error (m.d.p.e.)
for this function (and its higher-order cousins) can therefore be closely estimated by
m.d.p.e. ≈3
5·powerwhere “ d e” denotes the ceiling function. At maximum error,
the minimum accuracy is then given by d.p.a ≈numDigits*power −3
5·power.
Using a non-zero value for the variable extraPlaces allows one to compare the
values of Paand Pr.
Conclusion
It should be noted that using πto generate the digits eis not very “practical.”
The simple fact that one needs to first calculate trigonometric values to be used
as part of the overall calculation severely limits efficiency in a computational sense.
Clearly, if armed only with pencil and paper, there are easier ways to calculate e
using infinite series (Brothers, 2004). Nonetheless, these unexpected expressions
appear interesting in their own right and provide fertile territory for exploration and
research by first- and second-year calculus students. Indeed, students can use the
methods presented here to produce their own original results, reinforcing the intimate
link between two numbers that have fascinated mathematicians for centuries.
References
Anton, H. (1980). Calculus. New York: John Wiley & Sons.
Bradley, G. L. & Smith, K. J.. (1999). Single variable calculus. (2nd ed.). Baskin-Ridge,
NJ: Prentice-Hall.
Brothers, H. J. (2004). Improving the convergence of Newton’s series approximation for e.
The College Mathematics Journal, 35 (1), 34–39.
Brothers, H. J. http://www.brotherstechnology.com/math/converter.html.
Brothers, H. J. & Knox, J. A. (1998). New closed-form approximations to the logarithmic
constant e.The Mathematical Intelligencer, 20(4), 25–29.
Knox, J. A. & Brothers, H. J. (1999) Novel series-based approximations to e.The College
Mathematics Journal, 30 (4), 269-275.
Maor, E. (1994), e: The story of a number. Princeton, NJ: Princeton University Press.
Richardson, L. F. & Gaunt, J. A. (1927). The deferred approach to the limit. Philosophical
Transactions of the Royal Society of London, series A, 226.
The AMATYC Review Volume 30, Number 1

Harlan J. Brothers 35
Acknowledgments
This paper would not have been possible without the support and encouragement of
Professor John Knox, University of Georgia. His initial interest and subsequent devotion
to the development of series-based approximations for eprovided the foundation for this
work. The author is grateful to Professor Miguel Garcia of Gateway Community College in
Connecticut for his unflagging interest and encouragement. Thanks also to Robert Speiser at
Brigham Young University for his editorial expertise. Mathematica is a registered trademark
of Wolfram Research, Inc.
Lucky Larry #91
Solve the equation (x−1) (3x+ 2) = 4x.
Larry’s solution:
(x−1) (3x+ 2) = 4x
x−1 = 4xor 3x+ 2 = 4x
−1 = 3xor 2 = x
−1
3=xor 2 = x
Solution set =n−1
3,2oMichael W. Lanstrum
Cuyahoga Community College
Parma, OH
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who are thus ignorant are unable to perceive their own ignorance and so do not seek
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Letter to Abigail Adams, May 12, 1780
Fall 2008 The AMATYC Review