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Historical remarks: Irrationality of e and Liouville
Posted:
Jul 22, 2005 12:19 PM


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 (18221901) 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 wellknown 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prove [the result concerning approximations of algebraic numbers by rationals in his 1844 publications], but his notes [unpublished journal notes of Liouville's that Lützen had access to] make a reconstruction of his proof possible ..."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 online 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), 193194.
Liouville's paper is online 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 online 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 trèsétendues 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), 133142.
Liouville's paper is online 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, hyperarithmetical, 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 e1. Although this can be used to establish the irrationality of e1, 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 e1 that Cotes did, as well as an infinite continued fraction expansion for (e+1)/(e1). 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)/(e1) is equivalent to the irrationality of e, since it is clear that "e rational implies (e+1)/(e1) is rational", and the identity (x+1)/(x1) = e when x = (e+1)/(e1) gives the other direction.
My source for these remarks about Cotes and Euler is Kline [7], pp. 459460. 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)/(e1) (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 #302303, pp. 374375), Laczkovich [8], Stevens [15], Struik [16] (Chapter V.17: "Lambert. Irrationality of Pi", pp. 369374), and Wallisser [17].
In 1794 AdrienMarie 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. 253254, or what is essentially the same thing, Archibald's notes to Part II of Chapter II in Klein [7], pp. 8890), 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. 110111) 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), 247259. [JFM 45.0742.03] http://www.emis.de/cgibin/JFMitem?45.0742.03
[2] Claude Brezinski, HISTORY OF CONTINUED FRACTIONS AND PADÉ APPROXIMANTS, SpringerVerlag, 1991. [MR 92c:01002; Zbl 714.01001] http://www.emis.de/cgibin/MATHitem?0714.01001
Reviewed by John H. McCabe in SIAM Review 35 (1993), 159160.
Reviewed by Wolfgang J. Thron in Mathematics of Computation 62 (1994), 432433.
[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/cgibin/Zarchive?an=0091.01402 http://www.emis.de/cgibin/JFMitem?18.0051.01 http://www.emis.de/cgibin/JFMitem?21.0073.01
Chapter XXXII, Section 17 (pp. 512514) 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, 1216. [JFM 3.0198.04] http://www.emis.de/cgibin/JFMitem?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/cgibin/Zarchive?an=0078.13205
[6] Ernest W. Hobson, "Squaring the circle: A history of the problem", pp. 157 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/cgibin/JFMitem?44.0050.02 http://www.emis.de/cgibin/Zarchive?an=0052.16301
Hobson's essay is online 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/cgibin/JFMitem?28.0438.05 http://www.emis.de/cgibin/Zarchive?an=0070.38003
The 1897 English translation of Klien's book is online at http://www.hti.umich.edu/cgi/t/text/textidx?c=umhistmath;idno=ABN2381
[8] Miklós Laczkovich, "On Lambert's proof of the irrationality of Pi", The American Mathematical Monthly 104 (1997), 439444. [MR 98a:11090; Zbl 876.11032] http://www.emis.de/cgibin/MATHitem?0876.11032 http://www.maa.org/pubs/monthly_may97_toc.html
[9] Jesper Lützen, JOSEPH LIOUVILLE (18091882): MASTER OF PURE AND APPLIED MATHEMATICS, SpringerVerlag, 1990. [MR 91h:01067; Zbl 701.01015] http://www.emis.de/cgibin/MATHitem?0701.01015
Reviewed by David E. Zitarelli in Historia Mathematica 20 (1993), 205210.
[10] Ulysses Grant Mitchell and Mary Strain, "The number e", Osiris 1 (1936), 476496. [Zbl 14.24401; JFM 62.0003.06] http://www.emis.de/cgibin/Zarchive?an=0014.24401 http://www.emis.de/cgibin/JFMitem?62.0003.06
[11] Alfred Pringsheim, "Ueber die ersten Beweise der Irrationalität von e und Pi", Sitzungsberichte der Bayerischen Akademie der Wissenschaften MathematischPhysikalische Klasse 28 (1898), 325337. [JFM 29.0373.08] http://www.emis.de/cgibin/JFMitem?29.0373.08
[12] Dave L. Renfro, sci.math posts, 25 March 2001.
http://groupsbeta.google.com/group/sci.math/msg/ffcc71614fcfdcac http://groupsbeta.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/cgibin/JFMitem?24.0050.02
[14] David Eugene Smith, "The history and transcendence of Pi", pp. 388416 in Jacob W. Young (editor), MONOGRAPHS ON TOPICS OF MODERN MATHEMATICS, Dover Publications, 1911/1955. [Zbl 67.29202] http://www.emis.de/cgibin/Zarchive?an=0067.29202
The book this essay is in can be found online 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), 151158. [MR 2000i:11114; Zbl 1045.11049] http://www.emis.de/cgibin/MATHitem?1045.11049
[16] Dirk Jan Struik, A SOURCE BOOK IN MATHEMATICS, 12001800, Harvard University Press, 1969. [MR 39 #11; Zbl 205.29202] http://www.emis.de/cgibin/Zarchive?an=0205.29202
[17] Rolf Wallisser, "On Lambert's proof of the irrationality of Pi", pp. 521530 in Franz HalterKoch 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/cgibin/MATHitem?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 semihistorical 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. 202203. 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 (17681830). 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. 5758 of Rudio [13], a book that I have not yet seen a copy of. However, Lützen has told me (email 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/cgibin/MATHitem?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/cgibin/MATHitem?0805.01001
[22] Paulo Ribenboim, MY NUMBERS, MY FRIENDS: POPULAR LECTURES ON NUMBER THEORY, SpringerVerlag, 2000, xii + 375 pages. [MR 2002d:11001; Zbl 947.11001] http://www.emis.de/cgibin/MATHitem?0947.11001
[23] Marty Ross, "Irrational thoughts", The Mathematical Gazette 88 #511 (March 2004), 6878.
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Dave L. Renfro



