Download CLASS: MATH 301 TEACHER: D. MEREDITH IS PI+E IRRATIONAL

CLASS: MATH 301
TEACHER: D. MEREDITH
IS PI+E IRRATIONAL?
KYRA REDENBAUGH
Abstract. e, sometimes called ”Eulers number” is an irrational mathematical constant
1
and can be expressed as lim (1 + )n . Π on the other hand is also an irrational number
n→∞
n
∞
X
4
1
2
1
1
that can be expressed as a limit; π =
−
−
−
, so it
16n 8n + 1 8n + 4 8n + 5 8n + 6
n=0
would seem possible that these numbers added together would create an irrational number
as well. If put into a calculator, their sum seems irrational, rounded to about 5.86, but so
far a proof of it’s rationality (or irrationality) is unable to be proven.
The numbers π and e are the most commonly known irrational numbers used in mathematics. They are both sums of infinite series’, and can be applied many areas of math from
simple geometry to chaos theory. These numbers, according to Wolfram Alpha’s definition
p
of irrational numbers, are ”. . number(s) that cannot be expressed as a fraction for any
q
integers p and q. . .(and) have decimal expansions that neither terminate nor become periodic.” For centuries mathematicians from around the world have been trying to find patterns
in these numbers, and new numbers to add to the decimal expansions of them even though
the numbers never end. Because the sum of two irrational numbers resulting in a rational
number is rare but possible, a current open problem concerning these numbers is whether
their sum is irrational or not. To begin to get an idea of the rationality of their sum, we
need to first look at the cases for their irrationality.
According to Ivan Niven’s book, ”Irrational Numbers”, mathematician Johann Lambert
proved the irrationality of both π and e. However, although he was known for the first proof
Date: May 3, 2011.
1
of the irrationality of pi through the continued fraction expansion of tan x, other proofs
have come up through the centuries. Most significantly, in 1946 Niven proved that π 2 was
irrational through contradiction stating ”If π were rational, then cos π would be irrational,
whereas cos π = −1”(Niven, 19).
In 1997, David Bailey, P.B. Borein, and S. Plouffe, created what they called the ”BBP
(for Bailey-Borwein-Plouffe) formula of π in 1997. This formula was the most up to date
and accurate description of the number, stating that
∞
X
1
2
1
1
4
π=
−
−
−
.
n
16
8n
+
1
8n
+
4
8n
+
5
8n
+
6
n=0
.
e, also known as Eulers number, is the sum of the exponential series.The proof of e as a
sum of a series was first discovered by Newton and then assigned the particular letter by
Euler. It can approximated through a comparison of the Geometric Series and (e-1) Series
1
1
1 1 1 1 1
1 1 1 1 1
which states that + + + + + + +· · · = 2 and + + + + · · · = (e−1).
1 2 4 8 16 32 64
1! 2! 3! 4! 5!
Because the summation of (e-1) is less than 3, but e is larger than the Geometric Series, e
must be be between 2 and 3, and by simply calculating the terms of it, approximations like
2.17 are formed. (Adrian, 181)
Niven’s book states that er can be proven irrational as long as its exponent (r) is rational and not equal to 0. This is also done by contradiction because if r is rational, so
(er + e−r )
would er , and e−r , and
. However, this expression is equal to cosh(r), a hyperbolic
2
function which is always irrational for non-zero rational values of the arguments. (Niven, 22).
At this point we’ve proven π and e to be irrational, and have neatly derived them to be summations. So, to I took it upon myself to attempt to add these two numbers by evaluating the
2
addition of their summations. Since the formula for π has its base case at 0, to add these two
4
2
1
1
1
−
−
−
=
we would have to evaluate π0 , getting 0
16 8(0) + 1 8(0) + 4 8(0) + 5 8(0) + 6
106
.
819
With that cleared, we are left with
X
∞
∞
106 X 1
2
1
1
1
4
π+e =
+
−
−
−
+
n
819 n=1 16
8n + 1 8n + 4 8n + 5 8n + 6
(n − 1)!
n=1
∞
4
2
1
1
1
106 X 1
+
−
−
−
+
=
n
819 n=1 16
8n + 1 8n + 4 8n + 5 8n + 6
(n − 1)!
Although simplification of this would be immensely time consuming and might not get us
to a distinct answer anyway, Wolfram alpha displays it as a continued fraction,
and can calculate it to over a thousand digits
3
Although David H. Bailey addresses both the topic of Arbitrary Digit Calculation Formulas to find specific values throughout the sum of π + e, and Ramanujans AGM Continued
Fraction to perhaps asses the fraction shown above in the article, ”Experimental mathematics: examples, methods and implications.”, none of this really implies that this sum is
rational. Nevertheless, the answer may not be too far out of grasp.
In Bailey and Borwein’s article ”Computer Assisted Discovery and Proof” it is described
how computer assisted mathematics is emerging and helping computational aspects of math.
As seen above, it is shown how software packages and websites like Wolfram make calculation
and function manipulation easily accessible. Publications and journals have started up in the
recent years dedicated to these calculations in hopes that computer assistance can help in
the advancement in mathematics as it stands today. With advances like this, and being able
to find high decimal expansions in π and e, we can get a slight grasp on the patterns seen
in their sum. In the article, Bailey and Borwein specifically state that ”Integer detection
methods which are used to discover new mathematical identities, require very high precision
numerical imputs to obtain numerically meaningful results”(23).
4
Through research into irrationals alone, we can take into account that a rational formed
by two irrationals is often done if one is a negative of the other, and that there are many
more irrationals than irrationals. Therefore, although this continues to be an open problem
because nothing can be really proven yet, the thought of this sum seems to lean in favor of
being irrational.
Research into irrational numbers must take into account two facts: two irrationals sum to
a rational only when one is essentially the negative of the other; and the cardinality of the
set of irrationals is strictly larger than the cardinality of the set of rationals.
5
Bibliography
Bailey, David. ”Experimental mathematics: examples, methods and implications.” No-
tices of the American Mathematical Society 52.5 (2005):502-514. http://9003-sfx.calstate.edu.opac.sfsu.edu/
Bailey, David H., Jonathan M. Borwein, Vishaal Kapoor, and Eric W. Weisstein. ”Ten
Problems in Experimental Mathematics.” The American Mathematical Monthly 113.6 (2006):
481-509. Web. 5 Mar. 2011.
Borwein, Jonathan M., and David H. Bailey. Mathematics by Experiment. Wellesley
(Mass.): A. K. Peters, 2008. Print.
Dudley, Underwood. Is Mathematics Inevitable?: a Miscellany. Washington, DC: Mathematical Association of America, 2008. Print.
Eymard, Pierre, and J. P. Lafon. The Number [pi]. Providence, RI: American Mathematical Society, 2004. Print.
”Irrational Number - Wolfram—Alpha.” Wolfram—Alpha: Computational Knowledge Engine. Web. 12 Apr. 2011. <http://www.mathworld.wolfram.com/IrrationalNumber.html>.
Maor, Eli. E: the Story of a Number. Princeton, NJ: Princeton UP, 1994. Print.
Niven, Ivan. Irrational Numbers. [Buffalo]: Mathematical Association of America; Distributed by J. Wiley [New York, 1956. Print.
6
”Pi+ e - Wolfram—Alpha.” Wolfram—Alpha: Computational Knowledge Engine. Web.
12 Apr. 2011. <http://www.wolframalpha.com/input/?i=pi+e>.
Yeo, Adrian. The Pleasures of Pi,e and Other Interesting Numbers. Singapore: World
Scientific, 2007. Print.
7