I wonder if there is any book and/or article you can recommend on the topic "Irrationality of $\pi^2$ and $\pi^3$" for me to study on. In case you are curious about why I ask these particular exponents, it's because this is a project that my lecturer gave me to study on and then present to the class what I've got on this topic

PS: I do not understand why people vote for my post to be closed

Martin Sleziak
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Leyla Alkan
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  • Why those exponents? Do you understand why $\pi$ is irrational? – Larry B. Dec 29 '17 at 21:08
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    The easy way would be to look into why $\pi$ is *transcendental*. – Ian Dec 29 '17 at 21:09
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    See the [Lindemann–Weierstrass theorem](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem#Transcendence_of_e_and_π). Since $\pi$ is transcendental, no integer power of $\pi$ can be rational (or, in fact, algebraic). – dxiv Dec 29 '17 at 21:09
  • It's a project that my lecturer gave me to study on and then present it to the class@LarryB. – Leyla Alkan Dec 29 '17 at 21:10
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    [Hermite's proof](https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational) that $\pi$ is irrational actually shows that $\pi^2$ is (of course, that is a stronger result). I don't know of a special case way to get at $\pi^3$. – lulu Dec 29 '17 at 21:12
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    [Your question should be clear without the title](https://math.meta.stackexchange.com/a/10144). After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title. – Simply Beautiful Art Dec 29 '17 at 22:01
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    Instead of wasting your time for those specific exponents, learn about about transcendental numbers, if you understand the concept of transcendental numbers then you have really learned something worth while. learning anything specifically about irrationality $\pi^2 or \pi^3$ is waste of time. – jimjim Dec 29 '17 at 22:28
  • Irrational Numbers by Niven, https://www.maa.org/press/ebooks/irrational-numbers – Will Jagy Dec 29 '17 at 22:35
  • I will definitely check that, thanks @WillJagy and do you think this book is good enough to learn about the number $\pi?$ – Leyla Alkan Dec 29 '17 at 22:43
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    Yes. For that matter, here is the one-page proof by the same author, that $\pi$ is irrational. https://projecteuclid.org/download/pdf_1/euclid.bams/1183510788 It is not necessarily possible to extend this proof to $\pi^2$ and $\pi^3;$ maybe the book discusses that, not sure. The book does do everything you might need on transcendentals, but that is generally, well, longer. – Will Jagy Dec 29 '17 at 22:48
  • Proofs of the general theorem, by (fairly) elementary means can be found in "101 Great Problems Of Elementary Mathematics" by H. Dorrie, and in the old classic textbook "Trigonometry" by Hobson. – DanielWainfleet Dec 30 '17 at 07:06
  • Are you talking about the proofs for "the powers of $\pi $ are irrational" ? @DanielWainfleet – Leyla Alkan Dec 30 '17 at 08:33
  • The general theorem is that if $A_1,...,A_n$ are non-zero algebraic numbers and if $B_1,..., B_n$ are $ n$ distinct algebraic numbers then $0\ne \sum_{j=1}^n A_j\exp (B_j).$...... Since $0=1+\exp (\pi i)=1\cdot \exp (0)+1\cdot (\exp \pi i),$ therefore $\pi i$ is transcendental (i.e. not algebraic), hence $\pi$ is transcendental..... If $X$ is transcendental then $X^m$ is transcendental for all $m\in \Bbb N.$ – DanielWainfleet Dec 30 '17 at 14:48

1 Answers1


The following references are mostly specific to your question:

[1] Adrien-Marie Legendre, Éléments de Géométrie, avec des notes, Firmin Didot (Paris), 1794, xii + 334 pages.

A proof of the irrationality of ${\pi}^2$ by the use of continued fractions is given on p. 304.

[2] James Whitbread Lee Glaisher, On Lambert’s proof of the irrationality of $\pi,$ and on the irrationality of certain other quantities, pp. 12-16 in Notices and Abstracts of Miscellaneous Communications to the Sections, Report of the Forty-First Meeting of the British Association for the Advancement of Science (August 1871, Edinburgh), John Murray (London), 1872.

The paper is on “.pdf pages” 341-345 of this google books item. The top third of p. 14 discusses Legendre’s proof that ${\pi}^2$ is irrational.

[3] Charles Hermite, Extrait d'une lettre de Mr. Ch. Hermite à Mr. Borchardt [Extract of a letter of Mr. Ch. Hermite to Mr. Borchardt], Journal für die reine und angewandte Mathematik 76 (1873), pp. 342-344.

Hermite shows ${\pi}^2$ is irrational by a method that avoids the use of continued fractions --- a method that very soon afterwards led to his proof that $e$ is transcendental.

[4] Alfred Pringsheim, Ueber die ersten beweise der irrationalität von $e$ und $\pi$ [On the first proof of irrationality of $e$ and $\pi$], Sitzungsberichte der Mathematisch-Physikalischen Classe der K.B. Akademie der Wissenschaften zu München 28 (1898), 325-337.

If you can read German (I can’t), I believe this could be of use. Lambert’s proof that ${\pi}^2$ is irrational is mentioned on p. 326 (line 9).

[5] Sylvain Wachs, Contribution à l'étude de l'irrationalité de certains nombres [Contribution to the study of the irrationality of certain numbers], Bulletin des Sciences Mathématiques (2) 73 (1949), pp. 77-95.

From MR0033299 (11,418a) review by Jan Popkin: By considering a larger class of similar integrals the author intends to obtain more general results. He gives applications by showing in this manner the irrationality of such numbers as ${\pi}^2,$ $\log A$ $(A \neq 1),$ $e^A,$ where $A$ denotes a positive integer. [The paper contains some misprints and other mistakes; the most serious one at the end of § 6, where the quantity $M$ introduced depends on $n.$ In view of various papers giving generalizations of Niven's method it is perhaps of interest to remark that there exists a close connection between this method and the classical proofs for the irrationality of $\pi$ and ${\pi}^2$ of Lambert, Hermite and others. Take for instance the integral $[\cdots]$

[6] Yosikazu Iwamoto, A proof that ${\pi}^2$ is irrational, Journal of the Osaka Institute of Science and Technology. Part I: Mathematics and Physics 1 (1949), pp. 147-148.

[7] Ivan Morton Niven, Irrational Numbers, The Carus Mathematical Monographs #11, Mathematical Association of America, 1956, xii + 164 pages.

The Alternative proof of Corollary 2.6 on pp. 19-21 gives a proof that ${\pi}^2$ is irrational.

[8] Kustaa Aadolf Inkeri, The irrationality of ${\pi}^2$, Nordisk Matematisk Tidskrift 8 #1 (1960), pp. 11-16 and 63.

[9] John Douglas Dixon, $\pi$ is not algebraic of degree one or two, American Mathematical Monthly 69 #7 (August-September 1962), p. 636.

Regarding this result, see Proof of $\pi$ not being a quadratic irrational number.

[10] Theodor Estermann, A theorem implying the irrationality of ${\pi}^2$, Journal of the London Mathematical Society (1) 41 #3 (1966), 415-416.

[11] Jaroslav Hančl, A simple proof of the irrationality of ${\pi}^4$, American Mathematical Monthly 93 #5 (May 1986), pp. 374-375.

[12] Darrell Desbrow, On the irrationality of ${\pi}^2$, American Mathematical Monthly 97 #10 (December 1990), pp. 903-906.

[13] Michael David Spivak, Calculus, 3rd edition, Publish or Perish, 1994, xiv + 670 pages.

In Chapter 16, Theorem 1 (stated and proved on pp. 323-324) is the irrationality of ${\pi}^2.$ This same result probably appears in either or both earlier editions (1967, 1980), but I have not verified this.

[14] Miklós Laczkovich, On Lambert's proof of the irrationality of $\pi$, American Mathematical Monthly 104 #5 (May 1997), pp. 439-443.

Corollary 2 at the top of p. 441:${\pi}^2$ is irrational.”

[15] Pierre Eymard and Jean-Pierre Lafon, The Number $\pi$, translated by Stephen Stewart Wilson, American Mathematical Society, 2004, x + 322 pages.

Section 4.2.3 on pp. 136-137 is titled “Niven’s proof of the irrationality of ${\pi}^2$”.

[16] Paul Joel Nahin, Dr. Euler’s Fabulous Formula, Princeton University Press, 2006, xxii + 380 pages.

Chapter 3: The Irrationality of ${\pi}^2$ (pp. 92-113) gives a very detailed presentation of the proof in Carl Ludwig Siegel’s book Transcendental Numbers.

[17] Li Zhou and Lubomir Markov, Recurrent proofs of the irrationality of certain trigonometric values, American Mathematical Monthly 117 #4 (April 2010), 360-362.

(2nd sentence of the paper) We also discuss applications of our technique to simpler irrationality proofs such as those for $\pi,$ ${\pi}^2,$ and certain values of exponential and hyperbolic functions.

[18] Timothy W. Jones, Discovering and Proving that $\pi$ is irrational, American Mathematical Monthly 117 #6 (June-July 2010), pp. 553-557.

Niven’s proof of the irrationality of ${\pi}^2$ is discussed on p. 556.

[19] Timothy W. Jones, The powers of $\pi$ are irrational, viXra:1102.0058, 19 October 2010, 17 pages.

[20] Jürgen Müller and Tom Müller, Niven’s irrationality method revisited, manuscript, undated, 3 pages.

(3rd paragraph of the paper, on p. 1) In this note we take a new look at the classic analytic irrationality proofs for ${\pi}^2$ and the integer powers of $e,$ showing that the required approximation polynomials are generated by one single integral expression. Our approach makes it obvious how the polynomials come into existence, why they have integer coefficients and that the irrationality proofs for ${\pi},$ ${\pi}^2$ and $e^k$ are only different special cases derived from the same general formula.

[21] lhf, Direct proof that $\pi$ is not constructible, Mathematics Stack Exchange, 30 January 2012.

This might also be of interest.

Dave L. Renfro
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  • Great, thanks for your effort and the links as well! – Leyla Alkan Dec 30 '17 at 00:31
  • btw, if you remember/find any article later, can you add them to your answer section again? I'll be much appreciated @DaveL.Renfro – Leyla Alkan Dec 30 '17 at 00:47
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    OK, I've added some more references. (Instead of working on something I need to, but find rather boring, I spent a couple of hours this morning doing this.) – Dave L. Renfro Dec 30 '17 at 14:12
  • I'm impressed, thanks a lot for all this effort you made finding them. I am more than glad. @DaveL.Renfro – Leyla Alkan Dec 30 '17 at 23:37
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    @Sobolev: Actually, not all that much was copy and paste. I formatted all the bibliographic entries to my standard style, plus I tried to get as complete information as I could find, such as full names and **[10]** is "issue #3" (which the publisher's web page doesn't tell you --- actually, they say "issue #1", but that's just their default for all the issues in an old volume when no one at the publishers bothered to find out this information). For a much longer reference list, see [here](https://math.stackexchange.com/questions/677927/bibliography-for-singular-functions). – Dave L. Renfro Jan 03 '18 at 15:06