Date: Jul 22, 2005 12:19 PM Author: Dave L. Renfro Subject: Historical remarks: Irrationality of e and Liouville Timothy Murphy wrote (on July 22, in the thread

"Transcendental Dimensions"):

> It's fairly easy to prove that e is transcendental, I think.

> It might even have been proved before Liouville's theorem

> was published?

Charles Hermite (1822-1901) proved that e was

transcendental in 1873, while Liouville's work on

transcendental numbers occurred during the 1840's.

The following remarks are from a March 4, 2005

sci.math post of mine:

> For example, recently I've been working on some issues

> concerning the size of the set of Liouville numbers

> and its complement, and I can't begin to tell you how

> many times I've read that in 1844 Liouville published

> the well-known theorem concerning how well algebraic

> numbers of degree n can be approximated by rational

> numbers. No, no, no! Liouville might have known about

> this in 1844, but the result itself didn't appear in

> print until Liouville's 1851 paper on transcendental

> numbers. Liouville's 1844 transcendental proofs

> involved numbers given as continued fractions,

> and it just so happens that continued fractions

> provide very efficient rational approximations,

> but at that time (at least in print) Liouville

> had not yet identified the more general approximation

> property in our present definition of the Liouville

> numbers as the fundamental principle at work in

> his proofs.

Incidentally, Lützen (see [9] below, p. 521) has this

to say about Liouville's 1844 transcendence papers:

"Liouville did not prove [the result concerning

approximations of algebraic numbers by rationals

in his 1844 publications], but his notes [unpublished

journal notes of Liouville's that Lützen had access to]

make a reconstruction of his proof possible ..."

Here are some Liouville papers about these issues

that are on the internet. The 1844 transcendental

papers are not among these, however. [His 1844 papers

were mostly brief announcements with some proof sketches,

and the complete details didn't appear until his 1851 paper.]

[A] Joseph Liouville, "Sur l'irrationnalité du nombre

e = 2,718...", Journal de Mathématiques Pures et

Appliquées (= Liouville's Journal) (1) 5 (May 1840), 192.

Liouville's paper is on-line at

http://gallica.bnf.fr/Catalogue/noticesInd/FRBNF34348784.htm

This paper modifies Fourier's method of proving e is

irrational (see below) to prove that e is not quadratically

irrational. (This doesn't follow from the already known

fact that e^r is irrational for every nonzero rational

number, by the way.)

[B] Joseph Liouville, "Addition a la note sur

l'irrationnalité du nombre e", Journal de Mathématiques

Pures et Appliquées (= Liouville's Journal) (1) 5

(June 1840), 193-194.

Liouville's paper is on-line at

http://gallica.bnf.fr/Catalogue/noticesInd/FRBNF34348784.htm

This paper extends the proof of Liouville [A] to prove

that e^2 is not quadratically irrational.

[C] Joseph Liouville, "Sur la limite de (1 + 1/m)^m, m étant

un entier positif qui croft indéfiniment", Journal de

Mathématiques Pures et Appliquées (= Liouville's Journal)

(1) 5 (1840), 280.

Liouville's paper is on-line at

http://gallica.bnf.fr/Catalogue/noticesInd/FRBNF34348784.htm

This paper gives a proof that (1 + 1/m)^m approaches e

as the real variable m approaches infinity. Cauchy had

'proved' this in his 1821 book "Cours d'Analyse", but

Cauchy's proof involved an unsupported interchange of

the operation "limit as m --> infinity" with the

operation corresponding to the infinite summation

associated with the binomial expansion of (1 + 1/m)^m

(they were proving this for positive real number values

of m, not just positive integer values of m).

[D] Joseph Liouville, "Sur des classes très-étendues de

quantités dont la valeur n'est ni algébrique, ni meme

réductible á des irrationnelles algébriques", Journal de

Mathématiques Pures et Appliquées (= Liouville's Journal)

(1) 16 (1851), 133-142.

Liouville's paper is on-line at

http://gallica.bnf.fr/Catalogue/noticesInd/FRBNF34348784.htm

Below is an excerpt from a much longer essay on types

of numbers [rational, Pythagorean/Hilbert field,

constructible, constructible with marked rulers,

expressible in terms of real radicals, expressible

in terms of radicals, algebraic, elementary in various

ways (Joseph Ritt, Timothy Chow, etc.), provably

recursive in various ways, recursive, various levels of

montonically recursive classes, arithmetical hierarchy,

hyper-arithmetical, etc.] that I've been working on, off

and on, for the past year or so. I don't know when it'll

eventually get finished (the Usenet version), but I'm

simultaneously working on a much more extensive LaTeX

manuscript version that eventually I'll make available

either through publication or in the "mathematics arXiv" at

http://front.math.ucdavis.edu/

*** 2. LAMBERT AND LEGENDRE: THE IRRATIONALITY OF PI AND e ***

In 1714 Roger Cotes obtained an infinite continued fraction

expansion for e-1. Although this can be used to establish

the irrationality of e-1, and hence the irrationality of e,

it doesn't appear as if Cotes ever attempted to draw any

irrationality conclusions from his result. In 1737 Leonhard

Euler obtained the same infinite continued fraction expansion

for e-1 that Cotes did, as well as an infinite continued

fraction expansion for (e+1)/(e-1). In addition, Euler

argued that these expansions showed they were irrational,

and Euler was then able to argue fairly convincingly,

although perhaps not entirely rigorously, that e and e^2

were irrational. Note that the irrationality of e^2 implies

the irrationality of e, but not conversely (since there

exist irrational numbers whose squares are rational).

However, the irrationality of (e+1)/(e-1) is equivalent

to the irrationality of e, since it is clear that

"e rational implies (e+1)/(e-1) is rational", and

the identity (x+1)/(x-1) = e when x = (e+1)/(e-1)

gives the other direction.

My source for these remarks about Cotes and Euler is

Kline [7], pp. 459-460. Kline [7] does not indicate how

Euler proved e^2 is irrational, and so I do not know

if this can be deduced purely from the irrationality

of e and of (e+1)/(e-1) (which I haven't been able to do),

or whether something about the specific nature of the

continued fraction representations of these numbers

(or something else) was used by Euler. However, according

to Pringsheim [11] (p. 327), it appears that Euler also

obtained continued fraction expansions for (e^2 - 1)/2

and [e^(1/3) - 1]/2. Thus, perhaps when Kline [7] commented

that Euler essentially obtained the irrationality of e^2,

Kline should have additionally mentioned that Euler had

obtained an expansion for (e^2 - 1)/2.

In 1761 Johann Heinrich Lambert continued the continued

fraction investigations of Euler and proved that e^x and

tan(x) (radian measure) were each irrational for every

nonzero rational number x. Thus, Lambert proved:

* e^x is irrational for all nonzero rational values of x.

[But note this leaves unresolved the irrationality of

numbers such as e^2 + e, 4e^5 + 3e^2 - e, etc.]

* ln(x) is irrational for all positive rational numbers x.

[If ln(x) were rational for x different from 1, then

exp(ln x) would be the exponential of a nonzero rational

number, and hence irrational.]

* Pi is irrational. [Consider tan(Pi/4).]

For Lambert's proof, see Hobson [5] (Sections #302-303,

pp. 374-375), Laczkovich [8], Stevens [15], Struik [16]

(Chapter V.17: "Lambert. Irrationality of Pi", pp. 369-374),

and Wallisser [17].

In 1794 Adrien-Marie Legendre proved that Pi^2 is irrational,

which also implies the irrationality of Pi. Legendre

conjectured in Note 4 of his 1794 paper that Pi is not the

root of any polynomial with rational coefficients (i.e. that

Pi is a transcendental number).

It is very easy to prove "x irrational ==> 1/x irrational"

and "x irrational ==> x^(1/n) irrational for each n = 1, 2,

3, ...". Thus, by 1794 it was known that Pi^r is irrational

for each nonzero rational number r such that, when expressed

in lowest terms m/n, m is one of -2, -1, 1, 2 and n is any

positive integer.

Incidentally, the ideas above can be used to give a simple

proof of the following theorem. This result is probably

well known but I don't recall having seen it in print

anywhere.

Theorem: If x^n is irrational for each n = 1, 2, 3, ...,

then x^r is irrational for each nonzero rational

number r.

As an application, it is easy to prove that every positive

integer power of 1 + sqrt(2) is irrational. Hence, it follows

that every nonzero rational power of 1 + sqrt(2) is irrational.

It is instructive to note that irrationality results about e

are much easier to prove than corresponding irrationality results

about Pi. This is due to the fact that the continued fraction

expansion for e is much simpler than the continued fraction

expansion for Pi and the nice differentiability properties of

e^x (which imply, among other things, that all the coefficients

in the series expansion of e^x about x=0 are rational). Indeed,

the transcendence of Pi was proved as a consequence of having

sufficiently strong results involving e so that the identity

e^(Pi*i) = -1 could be used to draw conclusions about Pi.

It is sometimes claimed that Lambert's proofs were not entirely

rigorous. Thus, besides the additional results that Legendre

proved, Legendre is sometimes credited with giving the first

rigorous proofs that e and Pi are irrational. I said this in

Renfro [12], for instance, having seen it mentioned on p. 401

of Smith [14], among other places. However, these claims that

Lambert's proofs were incomplete (in a footnote on p. 521,

Wallisser [17] lists 8 such references besides the two I mention

in this paragraph) appear to have been based on an incorrect

assessment of Lambert made on p. 67 of Rudio's influential

historical survey [13]. According to Archibald (see Archibald [1],

pp. 253-254, or what is essentially the same thing, Archibald's

notes to Part II of Chapter II in Klein [7], pp. 88-90),

Klein's comments on p. 59 of [7], which seem to imply that

the irrationality of Pi was not rigorously established until

Legendre's work, were likely based on Rudio's [13] comments

on the matter.

Incidentally, Lambert's rigor was supported in at least one

paper before Rudio [13], which adds to the mystery of Rudio's

criticism of Lambert's proofs.

"Although Legendre's method is quite as rigorous as that on

which it is founded, still, on the whole, the demonstration

of Lambert seems to afford a more striking and convincing

proof of the truth of the proposition; his investigation,

however, is given in such detail, and so many properties

of continued fractions, no well known, are proved, that

it is not very easy to follow his reasoning, which extends

over more than thirty pages. The object of the present paper

is to exhibit Lambert's demonstration of this important

theorem concisely, and in a form free from unnecessary

details, and to apply his method to deduce some results

with regard to the irrationality of certain circular and

other functions." (p. 12 of Glaisher [4])

"That Lambert's proof is perfectly rigorous and places

the fact of the irrationality of Pi beyond all doubt,

is evident to every one who examines it carefully;

and considering the small attention that had been paid

to continued fractions previously to the time at which

it was written, it cannot but be regarded as a very

admirable work." (p. 14 of Glaisher [4])

Pringsheim [11], who does not mention Glaisher's paper [4],

investigated Lambert's proof and found it to be perfectly

adequate, and then Pringsheim [11] went on to speculate as

to why Rudio might have claimed Lambert's proofs were lacking.

"In 1892 F. Rudio stated that Lambert's proof was not

sufficient and that Legendre had supplied the deficiency.

This statement is an error, as has been shown by

Pringsheim's careful study. He found Lambert's work

was more rigorous than Legendre's."

(p. 494 of Mitchell/Strain [10])

"At the turn of the century the University of Munich had

several professors, such as Pringsheim and Tietze, whom

were interested in continued fractions. In 1898 Pringsheim

wrote a paper on the first proofs of the irrationality of

e and Pi. A large part of the paper is devoted to the

question of how Rudio had arrived at his conclusion about

the gap in Lambert's proof. Pringsheim states: Lambert

has written two papers on the quadrature of the circle;

a popular one entitled: "Vorläufige Kenntnisse für die,

so die Quadratur und die Rectification des Circuls suchen";

and a scientific one entitled: "Mémoire sur quelques

proprietés remarquables des quantités transcendantes

circulaires et logarithmiques". The former serves more

as an orientation to the problem and gives a good but

very general description of his results. Obviously Rudio

only considered this paper, which is completely reprinted

in his monograph on Archimedes, Huygens, Lambert, Legendre."

(p. 522 of Wallisser [17])

Brezinski [2] (pp. 110-111) also discusses some of these issues.

[1] Raymond Clare Archibald, "Remarks on Klein's 'Famous Problems

of Elementary Geometry'", The American Mathematical Monthly

21 (1914), 247-259. [JFM 45.0742.03]

http://www.emis.de/cgi-bin/JFM-item?45.0742.03

[2] Claude Brezinski, HISTORY OF CONTINUED FRACTIONS AND PADÉ

APPROXIMANTS, Springer-Verlag, 1991.

[MR 92c:01002; Zbl 714.01001]

http://www.emis.de/cgi-bin/MATH-item?0714.01001

Reviewed by John H. McCabe in SIAM Review 35 (1993), 159-160.

Reviewed by Wolfgang J. Thron in Mathematics of Computation

62 (1994), 432-433.

[3] George Chrystal, ALGEBRA: AN ELEMENTARY TEXTBOOK FOR THE

HIGHER CLASSES OF SECONDARY SCHOOLS AND FOR COLLEGES,

7'th Edition, two parts, American Mathematical Society, 1999.

[MR 22 #12066; Zbl 91.01402; JFM 18.0051.01; JFM 21.0073.01]

http://www.emis.de/cgi-bin/Zarchive?an=0091.01402

http://www.emis.de/cgi-bin/JFM-item?18.0051.01

http://www.emis.de/cgi-bin/JFM-item?21.0073.01

Chapter XXXII, Section 17 (pp. 512-514) gives a couple

of irrationality results on continued fractions that he

attributes to Legendre. Chrystal doesn't say whether

the proofs are Legendre's original proofs.

[4] James Whitbread Lee Glaisher, "On Lambert's proof of the

irrationality of Pi and on the irrationality of certain

other quantities", Report of the British Association for

the Advancement of Sciences, 1871, 12-16. [JFM 3.0198.04]

http://www.emis.de/cgi-bin/JFM-item?03.0198.04

[5] Ernest W. Hobson, A TREATISE ON PLANE AND ADVANCED

TRIGONOMETRY, 7'th edition, Dover Publications, 1928/1957.

[MR 19,876h; Zbl 78.13205]

http://www.emis.de/cgi-bin/Zarchive?an=0078.13205

[6] Ernest W. Hobson, "Squaring the circle: A history of

the problem", pp. 1-57 in SQUARING THE CIRCLE AND OTHER

MONOGRAPHS, Chelsea Publishing Company, 1913/1957.

[JFM 44.0050.02; Zbl 52.16301; MR 14,1114a]

http://www.emis.de/cgi-bin/JFM-item?44.0050.02

http://www.emis.de/cgi-bin/Zarchive?an=0052.16301

Hobson's essay is on-line at

http://name.umdl.umich.edu/ABN2635

[7] Felix Klein, FAMOUS PROBLEMS OF ELEMENTARY GEOMETRY,

Dover Publications, 1930/2003. [The original German edition

appeared in 1895 and the first English translation in 1897.

This is a reprint of the 1930 English translation which

included some notes by Raymond Archibald.]

[JFM 28.0438.05; Zbl 70.38003; MR 17,883e]

http://www.emis.de/cgi-bin/JFM-item?28.0438.05

http://www.emis.de/cgi-bin/Zarchive?an=0070.38003

The 1897 English translation of Klien's book is on-line at

http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABN2381

[8] Miklós Laczkovich, "On Lambert's proof of the irrationality

of Pi", The American Mathematical Monthly 104 (1997),

439-444. [MR 98a:11090; Zbl 876.11032]

http://www.emis.de/cgi-bin/MATH-item?0876.11032

http://www.maa.org/pubs/monthly_may97_toc.html

[9] Jesper Lützen, JOSEPH LIOUVILLE (1809-1882): MASTER OF PURE

AND APPLIED MATHEMATICS, Springer-Verlag, 1990.

[MR 91h:01067; Zbl 701.01015]

http://www.emis.de/cgi-bin/MATH-item?0701.01015

Reviewed by David E. Zitarelli in Historia Mathematica

20 (1993), 205-210.

[10] Ulysses Grant Mitchell and Mary Strain, "The number e",

Osiris 1 (1936), 476-496. [Zbl 14.24401; JFM 62.0003.06]

http://www.emis.de/cgi-bin/Zarchive?an=0014.24401

http://www.emis.de/cgi-bin/JFM-item?62.0003.06

[11] Alfred Pringsheim, "Ueber die ersten Beweise der Irrationalität

von e und Pi", Sitzungsberichte der Bayerischen Akademie der

Wissenschaften Mathematisch-Physikalische Klasse 28 (1898),

325-337. [JFM 29.0373.08]

http://www.emis.de/cgi-bin/JFM-item?29.0373.08

[12] Dave L. Renfro, sci.math posts, 25 March 2001.

http://groups-beta.google.com/group/sci.math/msg/ffcc71614fcfdcac

http://groups-beta.google.com/group/sci.math/msg/2e6b0d8a073302fe

Extensive list of references (print and internet) for proofs

that (1) e is irrational, (2) Pi is irrational, (3) e is

transcendental, and (4) Pi is transcendental.

[13] Ferdinand Rudio, ARCHIMEDES, HUYGENS, LAMBERT, LEGENDRE. VIER

ABHANDLUNGEN ÜBER DIE KREISMESSUNG, versehen, Leipzig.

B. G. Teubner (Leipzig), 1892. [JFM 24.0050.02]

http://www.emis.de/cgi-bin/JFM-item?24.0050.02

[14] David Eugene Smith, "The history and transcendence of Pi",

pp. 388-416 in Jacob W. Young (editor), MONOGRAPHS ON TOPICS

OF MODERN MATHEMATICS, Dover Publications, 1911/1955.

[Zbl 67.29202]

http://www.emis.de/cgi-bin/Zarchive?an=0067.29202

The book this essay is in can be found on-line by searching

for 'Young' using "Identify works by author and title" at

http://www.hti.umich.edu/u/umhistmath/

[15] Jan Stevens, "Zur irrationalität von Pi", Mitteilungen der

Mathematischen Gesellschaft in Hamburg 18 (1999), 151-158.

[MR 2000i:11114; Zbl 1045.11049]

http://www.emis.de/cgi-bin/MATH-item?1045.11049

[16] Dirk Jan Struik, A SOURCE BOOK IN MATHEMATICS, 1200-1800,

Harvard University Press, 1969. [MR 39 #11; Zbl 205.29202]

http://www.emis.de/cgi-bin/Zarchive?an=0205.29202

[17] Rolf Wallisser, "On Lambert's proof of the irrationality of Pi",

pp. 521-530 in Franz Halter-Koch and Robert F. Tichy (editors),

ALGEBRAIC NUMBER THEORY AND DIOPHANTINE ANALYSIS, Walter de

Gruyter & Co., 2000. [MR 2001h:01022; Zbl 973.11005]

http://www.emis.de/cgi-bin/MATH-item?0973.11005

************** 3. FOURIER'S PROOF THAT e IS IRRATIONAL ***************

Although Euler is credited with proving the irrationality of e

in 1737, Euler's proof is not the well known proof (see [18])

that makes use of the series expansion

e = 1/0! + 1/1! + 1/2! + 1/3! + ...

The well known proof of the irrationality of e that appears in

most textbooks is due to Joseph Fourier, although this doesn't

seem to be very widely known. For example, there is no mention of

Fourier in this regard in Maor's semi-historical book [21] about

e. At least Maor doesn't claim that this proof is due to Euler,

which Gourdon/Sebah [20] do: "Euler gave in 1737 a very elementary

proof of the irrationality of e based on the sequence ..." ([20],

accessed 22 July 2005). Maor [21] simply says on p. 192 that

Euler first proved the irrationality of e in 1737, and then

Maor [21] cites Courant/Robbins's book WHAT IS MATHEMATICS

for the well known series proof that Maor [21] gives on

pp. 202-203. Brabenec [19] (Problem 23, p. 82) also

incorrectly attributed Fourier's proof to Euler: "Euler

used a Maclaurin series as the key element in his proof, ..."

What follows is all that I've managed to uncover so far about the

historical background of Fourier's proof that e is irrational.

Hobson [6] (p. 44) and Ribenboim [22] (pp. 285 & 301) say

this proof is due to Joseph Fourier (1768-1830). Both cite

Stainville's 1815 book MÉLANGES D'ANALYSE ALGÉBRIQUE ET DE

GÉOMETRIE, possibly implying that the proof first appeared here,

although neither explicitly says this.

Lützen [9] (p. 516) also mentions that this proof is due

to Fourier, and Lützen cites pp. 57-58 of Rudio [13],

a book that I have not yet seen a copy of. However,

Lützen has told me (e-mail dated 24 May 2004) that

Rudio [13] says the usual textbook proof for the

irrationality of e appears on p. 339 of Stainville's

book and that Rudio says this proof is due to Fourier.

Finally, Mitchell/Strain [10] (p. 495) and Ross [23]

(p. 72) state that this proof was given by Fourier in 1815.

I don't know whether Fourier came up with this proof prior

to 1815, nor does Lützen know.

[18] The Math Forum: Ask Dr. Math: "The Irrationality of e".

http://mathforum.org/library/drmath/view/53910.html

[19] Robert L. Brabenec, RESOURCES FOR THE STUDY OF REAL

ANALYSIS, Classroom Resource Materials, Mathematical

Association of America, 2004, xii + 231 pages.

[Zbl 1059.26001]

http://www.emis.de/cgi-bin/MATH-item?1059.26001

[20] Xavier Gourdon and Pascal Sebah, "Irrationality proofs",

Numbers, Constants and Computation web pages.

http://tinyurl.com/4c62w

[21] Eli Maor, e: THE STORY OF A NUMBER, Princeton University

Press, 1994. [MR 95a:01002; Zbl 805.01001]

http://www.emis.de/cgi-bin/MATH-item?0805.01001

[22] Paulo Ribenboim, MY NUMBERS, MY FRIENDS: POPULAR LECTURES

ON NUMBER THEORY, Springer--Verlag, 2000, xii + 375 pages.

[MR 2002d:11001; Zbl 947.11001]

http://www.emis.de/cgi-bin/MATH-item?0947.11001

[23] Marty Ross, "Irrational thoughts", The Mathematical Gazette

88 #511 (March 2004), 68-78.

*****************************************************************

Dave L. Renfro