IRRATIONALITY AND TRANSCENDENCE OF e AND Pi (Part A)
Dave L. Renfro
These are some notes on the irrationality and transcendence
of e and Pi that I thought would be useful to archive as a
sci.math post. I have not attempted to be very complete
regarding printed literature--if I come across something
relevant (especially something like an Amer. Math. Monthly
paper), I'll include it. On the other hand, I have tried to be
a bit more thorough regarding what's available on the internet.
[Incidentally, I do know about Baker's and Shidlovskii's books.
But I haven't been to a well-stocked library to look at those
(or other books) yet. The times I have been to a university
library lately were for other things, and expanding this list of
references was far down on my list of things to do.]
Because I was unable to copy and paste the entire essay into
one "message window" at the Math Forum sci.math web page, I'm
sending this as two posts. Sections 0 to 4 appear below and
Sections 5-6 appear in the following post.
Dave L. Renfro
TABLE OF CONTENTS
INTRODUCTION
0. SOME GENERAL REFERENCES
1. e IS IRRATIONAL
2. Pi IS IRRATIONAL
3. e IS TRANSCENDENTAL
4. Pi IS TRANSCENDENTAL
5. ALGEBRAIC, LIOUVILLE, AND TRANSCENDENTAL NUMBERS
6. QUESTIONS
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INTRODUCTION
There are four related issues which in order of difficulty
are: (1) e is irrational, (2) Pi is irrational, (3) e is
transcendental, and (4) Pi is transcendental. After listing
some general references I'll address each of these in turn.
Then some historical and mathematical issues involving algebraic,
Liouville, and transcendental numbers that have not already
appeared are discussed. In the last section I list some questions
I don't know the answers to. These are questions whose answers
would have been incorporated in this essay, and so if anyone
can help, I will incorporate the answers in later versions.
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0. SOME GENERAL REFERENCES
0-book.1 Ivan Niven, IRRATIONAL NUMBERS, Carus Mathematical
Monograph 11, The Mathematical Association of America,
1956.
0-book.2 Ivan Niven and Hrbert S. Zuckerman, AN INTRODUCTION
TO THE THEORY OF NUMBERS, John Wiley and Sons, latest
edition.
0-book.3 G. H. Hardy and E. M. Wright, AN INTRODUCTION TO THE
THEORY OF NUMBERS, 4'th edition, Clarendon Press, 1960.
{The transcendence of e and Pi are proved on pp. 170-173
and pp. 173-176, respectively.}
0-book.4 J. W. A. Young, editor, MONOGRAPHS ON TOPICS OF MODERN
MATHEMATICS, Dover Publications, 1911/1955. {The essay
"The History and Transcendence of Pi" by David Eugene
Smith on pages 388-416 is an excellent introductory
historical survey.}
0-book.5 Jesper Lutzen, "JOSEPH LIOUVILLE (1809-1882): MASTER
OF PURE AND APPLIED MATHEMATICS", Studies in the History
of Mathematics and Physical Sciences 15, Springer-Verlag,
1990. [QA 29 .L62 J67 1990] {Chapter XII: "Transcendental
Numbers" (pp. 513-526) is a useful source of historical
information.}
0-net.1 sci.math thread "Transcendence Overview (corrected
version)" March 22, 1995 post by Thomas O. Womack
<http://www.math.niu.edu/~rusin/known-math/95/transcend>
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1. e IS IRRATIONAL
It is relatively easy to prove e is irrational using the
expansion e = 2 + 1/2! + 1/3! + ..., and can often be found in
real analysis texts. This proof that e is irrational is due to
Joseph Fourier (1768-1830). In 1737, Leonhard Euler (1707-1783)
proved that e and e^2 are irrational. [Note that "e^2 is
irrational" implies "e is irrational" (see the "Note" in
Section 2).] In 1766, Johann Heinrich Lambert (1728-1777) proved
x rational and x not equal to 0 ==> e^x irrational.
Lambert also suspected that Pi and e are transcendental, but he
was unable to prove this.
Here are some texts where a proof that e is irrational can
be found:
1-book.1 Alan F. Beardon, LIMITS: A NEW APPROACH TO REAL ANALYSIS,
Springer-Verlag, 1997. [QA 300 .B416 1997]
{See pages 170-171.}
1-book.2 Jerrold E. Marsden, ELEMENTARY CLASSICAL ANALYSIS,
W. H. Freeman and Company, 1974. [QA 300 .M2868]
{See page 27.}
1-book.3 Michael Spivak, CALCULUS, W. A. Benjamin, 1967.
{See page 353.} [QA 303 S78]
[A 3'rd edition (1994) exists, but I don't know if
the page reference I've given is the same.]
1-book.4 Martin Aigner and Gunter M. Ziegler, PROOFS FROM THE
BOOK, Springer-Verlag, 1998. [QA 36 .A36 1998]
{Fourier's proof that e is irrational is given on
p. 27 and the proof of (A) above is given on p. 28.}
1-book.5 Eli Maor, "e: THE STORY OF A NUMBER", Princeton Univ.
Press, 1994. [QA 247.5 .M33 1994] {pp. 202-203}
1-book.6 Richard Courant and Herbert Robbins, "WHAT IS
MATHEMATICS", Oxford University Press, 1941 (renewed
1969). [QA 37 .C675] {pp. 298-299}
ON THE INTERNET, see:
1-net.1 Bryan Cooley's Dec. 2, 1997 alt.math.undergrad post at
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/hoiselkel>.
1-net.2 Kevin Brown's proof at
<http://www.seanet.com/~ksbrown/kmath400.htm>.
1-net.3 The "Ask Dr. Math" answer at
<http://forum.swarthmore.edu/dr.math/problems/brady9.4.98.html>
1-net.4 The "Ask Dr. Math" answer at
<http://mathforum.com/dr.math/problems/johnson.05.15.99.html>.
1-net.5 The following section in Ian Craw's on-line text for MA1002
(University of Aberdeen) Advanced Calculus and Analysis at
<http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node62.html>.
1-net.6 Xavier Gourdon and Pascal Sebah's "Irrationality proofs"
(Section 3: "Irrationality of e") at
<http://xavier.gourdon.free.fr/Constants/Miscellaneous/irrationality.html>
1-net.7 sci.math thread "I Need Proof: pi is Irrational"
Feb. 3, 1999 post by Zdislav Kovarik
<http://forum.swarthmore.edu/epigone/sci.math/yunthingzherd>
<http://www.math.niu.edu/~rusin/known-math/99/e_irrat>
1-net.8 "Interactive Real Analysis", ver. 1.9.3 by Bert G.
Wachsmuth {Proof that e and Pi^2 are irrational, taken
from pp. 307-310 of Michael Spivak's CALCULUS, 1980.}
<http://www.shu.edu/projects/reals/infinity/irrat_nm.html>
By extending the idea used in the standard proof that e
is irrational, one can prove that e isn't a quadratic
irrational (i.e. a number that is the root of a quadratic
polynomial ax^2 + bx + c, where a,b,c are integers): Start
with assuming ae^2 + be + c = 0 for some integers a,b,c,
rewrite this as ae + ce^(-1) = -b, and then make use of the
series expansions for both e and e^(-1). This was first
proved by Joseph Louiville (1809-1882) in 1840.
For a proof that exp(x) is irrational for any nonzero rational
number x, see corollary 4 in 1-net.6 above and
1-paper.1 Joseph Amal Nathan, "The irrationality of e^x for
nonzero rational x", The American Mathematical
Monthly 105 (1998), 762-763.
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2. Pi IS IRRATIONAL
Aristotle claimed that Pi was irrational (by asserting that the
diameter and circumference of a circle are not commensurable--see
p. 27 of 1-book.4 = 2-book.2), but I am not aware of what argument,
if any, he gave in support of this. In 1767, Johann Heinrich Lambert
(1728-1777) proved
x rational and x not equal to 0 ==> tan(x) irrational.
This implies that Pi is irrational, since tan(Pi/4) = 1. However,
Lambert's proof was not quite complete. In his proof he exhibited
tan(x) as a certain infinite continued fraction, but he did not
actually show this implies irrationality. [A finite continued
fraction is, of course, rational. We're looking the converse.]
This defect was filled in by Adrien Marie Legendre (1752-1833)
in 1794. Legendre also proved in 1794 that Pi^2 is irrational.
On the internet, see:
2-net.1 Kevin Brown's proof at
<http://www.seanet.com/~ksbrown/kmath313.htm>
2-net.2 Keith Lynn's (Univ. of South Alabama) proof at
<http://www.mathstat.usouthal.edu/~lynn/mathematics/pi_irr/pi_irr.html>
2-net.3 Xavier Gourdon and Pascal Sebah's "Irrationality proofs"
(Section 5: "Irrationality of $\pi$") at
<http://xavier.gourdon.free.fr/Constants/Miscellaneous/irrationality.html>
2-net.4 The Bournemouth and Poole College of Further Education
"The Transcendence of Pi" {Includes separate proofs of
the irrationality of Pi and Pi^2, and proofs of the
transcendence of e and Pi.}
<http://www.bournemouthandpoole-cfe.ac.uk/splash/maths/docs/pi.html>
<www.bournemouthandpoole-cfe.ac.uk/splash/maths/docs/pi.pdf>
2-net.5 sci.math thread "Help! irrationals drive me crazy!"
Jan. 29, 1995 post by Zdislav Kovarik
<http://www.math.niu.edu/~rusin/known-math/95/pi_irrational>
2-net.6 "Pi is Irrational" by Helmut Richter
<http://www.lrz-muenchen.de/~hr/numb/pi-irr.html>
2-net.7 Massoud Malek (California State University, Hayward)
"Pi is irrational"
<http://www.mcs.csuhayward.edu/~malek/Mathlinks/Pi.html>
2-net.8 "Interactive Real Analysis", ver. 1.9.3 by Bert G.
Wachsmuth {Proof that e and Pi^2 are irrational, taken
from pp. 307-310 of Michael Spivak's CALCULUS, 1980.}
<http://www.shu.edu/projects/reals/infinity/irrat_nm.html>
Proofs can also be found in:
2-paper.1 Ivan Niven, "A simple proof of the irrationality of Pi",
Bull. American Mathematical Association 53 (1947), 509.
[Niven's proof is given in 2-net.3 above.]
2-paper.2 Robert Breusch, "A proof of the irrationality of Pi",
The American Mathematical Monthly 61 (1954), 631-632.
Niven's 1947 proof was generalized by Parks to prove
several general trig. and logarithmic irrationalities in:
2-paper.3 Alan E. Parks, "Pi e, and other irrational numbers",
The American Mathematical Monthly 93 (1986), 722-723.
NOTE: If you know the square of a number is irrational, then
you automatically know that the number is irrational.
[Assume x^2 is irrational. If x were rational, then we
could write x = p/q for some integers p and q. (We don't
have to assume this is in lowest terms.) Then squaring
gives x^2 = (p^2)/(q^2), a quotient of integers, which
contradicts the irrationality of x^2.] But the converse is not
true, as there are many irrational numbers with a rational
square [e.g. sqrt(2)]. So a proof that Pi^2 is irrational
gives a stronger result than a proof that Pi is irrational.
[The former tells you that BOTH Pi and Pi^2 are irrational.]
Similarly, a proof that Pi^4 is irrational gives you the
irrationality of Pi^4, Pi^2, and Pi all at once.
Incidentally, the fact that all these powers of Pi are
irrational is an immediate consequence of the transcendence
of Pi, but this latter fact is much more difficult to
establish.
Proofs that Pi^2 is irrational appear in 2-net.3 above and in
the following books:
2-book.1 Michael Spivak, CALCULUS, W. A. Benjamin, 1967.
{Chapter 16, pp. 277-280.}
2-book.2 Martin Aigner and Gunter M. Ziegler, PROOFS FROM THE
BOOK, Springer-Verlag, 1998. [QA 36 .A36 1998]
{See page 29.}
Here are some journal references for proofs of the
irrationality of Pi^2 and Pi^4:
2-paper.4 Robert Breusch, "A proof of the irrationality of Pi",
The American Mathematical Monthly 61 (1954), 631-632.
{Breusch actually proves that Pi^2 is irrational.}
2-paper.5 T. Estermann, "A theorem implying the irrationality of
Pi^2", J. London Math. Soc. 41 (1966), 415-416.
2-paper.6 Jaroslav Hancl, "A simple proof of the irrationality
of Pi^4", The American Mathematical Monthly 93 (1986),
374-375.
2-paper.7 D. Desbrow, "On the irrationality of Pi^2", American
Mathematical Monthly 97 10 (1990), 903-906.
2-paper.8 K. Inkeri, "On the irrationality of certain values of
elementary transcendental functions", Nieuw Archief
Voor Wiskunde (3) 24 (1976), 226-230. {The results
proved imply that Pi^2 is irrational (p. 229).}
Walters has some interesting proofs for the irrationality
of Pi and exp(m/n) in the following book that were motivated
by a paper of his on simple continued fraction expansions for
e^{1/q} and e^{2/q}.
2-book.3 R.F.C. Walters, NUMBER THEORY: AN INTRODUCTION, Carslaw
Publications (University of Sydney), 1987.
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3. e IS TRANSCENDENTAL
In 1873, Charles Hermite (1822-1901) gave two proofs that e is
transcendental and the approximation
e approx.= 58291 / 21444.
It actually doesn't take much tweaking of Hermite's proof to prove
that Pi is transcendental, but Hermite didn't realize this and
consequently never made a serious attempt. [In a letter to a former
student Hermite expressed the opinion that proving Pi is
transcendental would be much harder. See p. 193 of 1-book.5.]
3-book.1 G. H. Hardy and E. M. Wright, AN INTRODUCTION TO THE
THEORY OF NUMBERS, 4'th edition, Clarendon Press, 1960.
{See pp. 170-173.}
3-book.2 I. N. Herstein, TOPICS IN ALGEBRA, 2'nd edition,
Xerox, 1975. {See pages 217-219.}
3-book.3 Michael Spivak, CALCULUS, W. A. Benjamin, 1967.
{See Chapter 20, pp. 362-367.}
3-book.4 Nathan Jacobson, BASIC ALGEBRA I, W. H. Freeman, 1974.
[QA 154.2 .J32] {On pp. 268-277 it is proved that if
x1, x2, ..., xn are algebraic complex numbers that are
linearly independent over the rationals, then exp(x1),
exp(x2), ..., exp(xn) are algebraically independent over
the algebraic complex numbers (i.e. denoting exp(x1) by
y1, etc., then (y1, y2, ..., yn) is not the root of any
n-variable algebraic polynomial having complex-algebraic
number coefficients).}
3-net.1 You can find a proof (in French) that e is
transcendental in Édouard Lebeau's on-line paper
[Le Journal de Maths des élèves de l'école normale
supérieure de lyon, Volume 1, Numéro 3] at
<http://www.ens-lyon.fr/JME/Vol1Num3/LebeauJME3/LebeauJME3.html>.
3-net.2 The Bournemouth and Poole College of Further Education
"The Transcendence of Pi" {Includes separate proofs of
the irrationality of Pi and Pi^2, and proofs of the
transcendence of e and Pi.}
<http://www.bournemouthandpoole-cfe.ac.uk/splash/maths/docs/pi.html>
<www.bournemouthandpoole-cfe.ac.uk/splash/maths/docs/pi.pdf>
3-net.3 Michael Filaseta (Univ. of South Carolina)
"6: The Transcendence of e and Pi"
3 pages, 51 K .pdf file
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4. Pi IS TRANSCENDENTAL
This is the most difficult and was first proved in 1882
by Carl Louis Ferdinand Lindemann (1852-1939), from whom the term
"transcendental" originated. Lindemann actually proved a stronger
result, namely that whenever you have
a_0 + (a_1)*exp(b_1) + ... + (a_n)*exp(b_n) = 0
for some complex numbers a_0, a_1, ..., a_n and
b_1, b_2, ..., b_n, then these coefficients and
exponents cannot all be algebraic. [Note: A complex number is
said to be algebraic if and only if both its real and imaginary
parts are algebraic.] Using Euler's formula 1 + exp(i*Pi) = 0,
Lindemann's result implies that Pi is transcendental. [The two
coefficients are 1, hence algebraic, and so the single exponential
exponent i*Pi must be non-algebraic. It is easy to see that
i*Pi non-algebraic implies that Pi is non-algebraic.]
Proofs that Pi is transcendental can be found in the following:
4-book.1 G. H. Hardy and E. M. Wright, AN INTRODUCTION TO THE
THEORY OF NUMBERS, 4'th edition, Clarendon Press, 1960.
{See pp. 173-176.}
4-book.2 Nathan Jacobson, BASIC ALGEBRA I, W. H. Freeman, 1974.
[QA 154.2 .J32] {On pp. 268-277 a result stronger than
what Lindemann proved, due to Weierstrass I believe,
is proved. (See my comments in 3-book.4 above.)}
4-paper.1 Ivan Niven, "The transcendence of Pi", The American
Mathematical Monthly 46 (1939), 469-471.
4-net.1 The Bournemouth and Poole College of Further Education
"The Transcendence of Pi" {Includes separate proofs of
the irrationality of Pi and Pi^2, and proofs of the
transcendence of e and Pi.}
<http://www.bournemouthandpoole-cfe.ac.uk/splash/maths/docs/pi.html>
<www.bournemouthandpoole-cfe.ac.uk/splash/maths/docs/pi.pdf>
4-net.2 Michael Filaseta (Univ. of South Carolina)
"6: The Transcendence of e and Pi"
3 pages, 51 K .pdf file
<www.math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes6.pdf>
4-net.3 David Hilbert, "Ueber die Transzendenz der Zahlen e und
Pi", Math. Annalen 43 (1893), 216-219. [Digitally scanned
copy of the paper. However, the date given at this web
page is incorrectly given as 1896.]
<http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/local/hilbert.html>
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Jan Stevens
In article <zmek2c...@forum.mathforum.com>,
ren...@central.edu (Dave L. Renfro) writes:
>
> 2. Pi IS IRRATIONAL
>
> Aristotle claimed that Pi was irrational (by asserting that the
> diameter and circumference of a circle are not commensurable--see
> p. 27 of 1-book.4 = 2-book.2), but I am not aware of what argument,
> if any, he gave in support of this. In 1767, Johann Heinrich Lambert
> (1728-1777) proved
>
> x rational and x not equal to 0 ==> tan(x) irrational.
>
> This implies that Pi is irrational, since tan(Pi/4) = 1. However,
> Lambert's proof was not quite complete. In his proof he exhibited
> tan(x) as a certain infinite continued fraction, but he did not
> actually show this implies irrationality. [A finite continued
> fraction is, of course, rational. We're looking the converse.]
> This defect was filled in by Adrien Marie Legendre (1752-1833)
> in 1794. Legendre also proved in 1794 that Pi^2 is irrational.
This is an often repeated statement. Apparently it is not correct.
See:
Wallisser, Rolf. On Lambert's proof of the irrationality of $\pi$.
Algebraic number theory and Diophantine analysis (Graz, 1998),
521--530, de Gruyter, Berlin, 2000
He claims that on the contrary Lambert's proof is better than Legendre's,
who by the way cites Lambert.
> *******************************************************************
>
> 3. e IS TRANSCENDENTAL
>
> In 1873, Charles Hermite (1822-1901) gave two proofs that e is
> transcendental and the approximation
>
> e approx.= 58291 / 21444.
>
> It actually doesn't take much tweaking of Hermite's proof to prove
> that Pi is transcendental, but Hermite didn't realize this and
> consequently never made a serious attempt. [In a letter to a former
> student Hermite expressed the opinion that proving Pi is
> transcendental would be much harder. See p. 193 of 1-book.5.]
>
I suppose this refers to the letter to Borchardt. Borchardt is not a
former student, but the editor of J. Reine Angew. Math. (Crelle's Journal).
I quote from the paper
Ch. Hermite, Sur quelques approximations alg\'ebriques.
J. Reine Angew. Math. {\bf 76} (1873), 342--344. Also in:
Oeuvres III, pp. 146--149
..... Je ne me hasarderai point \`a la recherche d'une d\'emonstration
de la transcendance du nombre $\pi$. Que d'autres tentes l'entreprise,
nul ne sera plus heureux que moi de leur succ\`es, mais croyez-m'en,
mon cher ami, il ne laissera pas que de leur en co\^uter quelques efforts.
Tout ce que je puis, c'est de r\'efaire ce qu'a d\'ej\`a fait
{\it Lambert}, seulement d'une autre mani\`ere,
after which he goes on to give a proof of the irrationality of pi,
which is basically the same as Niven's new proof of 1947.
(See also my paper:
Zur Irrationalität von $\pi$. (German) [On the irrationality of $\pi$]
Mitt. Math. Ges. Hamburg 18 (1999), 151--158.)
I can only speculate what happened. After Lindemann's quite complicated
transcendence proof all attention was directed to simplifying it,
culminating in Hilbert's paper. There the idea behind the proof comes
out of the blue. A new generation only knew Hilbert's proof.
It was an appealing idea to simplify it to give the weaker result
on irrationality.
Actually, not everybody seems to be ignorant of the historical development.
Alan Baker writes in his book that the best way to understand the
ideas behind the transcendence proof is to read Hermite's papers.
In general, mathematicians treat the history of our subject quite badly
(as opposed to e.g. physicists). New ideas, concepts and proofs supersede
the old ones (newspeak).
I would be interested in other examples of this phenomenon.
Bye,
Jan.
--
email: ste...@math.chalmers.se
Matematiska institutionen
G"oteborgs universitet
Chalmers tekniska h"ogskola
SE 412 96 G"oteborg
Sweden