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Topic: Historical remarks: Irrationality of e and Liouville
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Dave L. Renfro

Posts: 4,484
Registered: 12/3/04
Historical remarks: Irrationality of e and Liouville
Posted: Jul 22, 2005 12:19 PM
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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in his 1844 publications], but his notes [unpublished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

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

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

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

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


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

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]

APPROXIMANTS, Springer-Verlag, 1991.
[MR 92c:01002; Zbl 714.01001]

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.

7'th Edition, two parts, American Mathematical Society, 1999.
[MR 22 #12066; Zbl 91.01402; JFM 18.0051.01; JFM 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]

TRIGONOMETRY, 7'th edition, Dover Publications, 1928/1957.
[MR 19,876h; Zbl 78.13205]

[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]

Hobson's essay is on-line at

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]

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

[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]

[9] Jesper Lützen, JOSEPH LIOUVILLE (1809-1882): MASTER OF PURE
AND APPLIED MATHEMATICS, Springer-Verlag, 1990.
[MR 91h:01067; Zbl 701.01015]

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]

[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]

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


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.

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

The book this essay is in can be found on-line by searching
for 'Young' using "Identify works by author and title" at

[15] Jan Stevens, "Zur irrationalität von Pi", Mitteilungen der
Mathematischen Gesellschaft in Hamburg 18 (1999), 151-158.
[MR 2000i:11114; Zbl 1045.11049]

[16] Dirk Jan Struik, A SOURCE BOOK IN MATHEMATICS, 1200-1800,
Harvard University Press, 1969. [MR 39 #11; Zbl 205.29202]

[17] Rolf Wallisser, "On Lambert's proof of the irrationality of Pi",
pp. 521-530 in Franz Halter-Koch and Robert F. Tichy (editors),
Gruyter & Co., 2000. [MR 2001h:01022; Zbl 973.11005]

************** 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
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".

ANALYSIS, Classroom Resource Materials, Mathematical
Association of America, 2004, xii + 231 pages.
[Zbl 1059.26001]

[20] Xavier Gourdon and Pascal Sebah, "Irrationality proofs",
Numbers, Constants and Computation web pages.

[21] Eli Maor, e: THE STORY OF A NUMBER, Princeton University
Press, 1994. [MR 95a:01002; Zbl 805.01001]

ON NUMBER THEORY, Springer--Verlag, 2000, xii + 375 pages.
[MR 2002d:11001; Zbl 947.11001]

[23] Marty Ross, "Irrational thoughts", The Mathematical Gazette
88 #511 (March 2004), 68-78.


Dave L. Renfro

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