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There are some conflicting sources regarding this. It is a matter of fact that Liouville defined what it was for a number to be approximated to degree $n$ by rational numbers. He then effectively defined a Liouville number to be a number that has a rational number approximation of degree $n$ for any $n$. Then "his theorem" stated: if $x$ is a real algebraic number (of degree $n$), then $x$ is approximated by rational numbers to order/degree at most $n$.

Conversely, then, if, for any number $x$, there exists an approximation to order $n$ for any integer $n$, it mustn't be algebraic, and thus is transcendental (this proves Liouville numbers are transcendental).

My question is: When did he define the idea of rational approximation of a number? When did he state and prove "his theorem"?

Note: Every source I've come across says that this work was done in the book "Sur les classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques". I don't think there's much question there. But when was it?

This source (scroll down to the top of "page 2") states that the theorem was proven in 1844, but I can't access the source because the reference page is not available on Google Books.

This source claims (on that first page) that it was published in 1851.

This page (first couple of paragraphs) says his theorem was "in 1844".

So what's the deal? Did he formulate his theorem in 1844, but didn't publish it until 1851? If that is the case, then how do we know he actually came up with it in 1844 if it wasn't published then?

On a side note, I came across two different citations in other peoples' papers. Most of them are like this:

1851 J. Liouville, Sur des classes très étendues de quantités dont la valeur n'est ni algébrique ni même réductible à des irrationnelles algébriques (J. Math. Pures et Appl., (1) , 16 (1851) , 133-142).

But this page uses both the above citation, and the following one:

1844 a J. Liouville, Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques (C.R. Acad. Sci. Paris, 18(1844), 883-885). ( 13 mai 1844 )

So, it's the same paper, different journals it looks like, and different years. What's the deal?

Other questions: apparently, he found the actual decimal expansion for the first Liouville number somewhere between 1850 and 1851 (this number. I can't seem to find any sources for this, any ideas?

Apparently, in 1844, he proved $e$ was not an algebraic number of degree 2. Again, I can't find any sources. Any ideas?

New edit: Transcendental and Algebraic Numbers By A. O. Gelfond (translated by Leo F. Boron) uses the following citation:

Two years?

So it seems that two papers were published separately. I just wish I knew which paper focused on what ideas exactly.

Another edit: Apparently, he presented his ideas in 1844 but published them with more rigor in 1851:

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  • $\begingroup$ I would consult Dickson's History of the Theory of Numbers (but I won't have access to it until Monday). Alternatively, I'd see what Hardy & Wright says. $\endgroup$ – Gerry Myerson Mar 27 '15 at 11:45
  • $\begingroup$ Dickson's was very helpful as it had a chapter on transcendental numbers. Section 11.7 started with Liouville's theorem, and at the end of the chapter, says "Liouville's work was published in the Journal de Math. (1) 16 (1851), 133-42," so if this book is reputable, then I guess a lot of articles incorrectly state that it was published in 1844. Did he publish two papers with the same name, one in 1844, and one in 1851? Note: the other source didn't have a section on transcendental numbers...not that I saw anyways. $\endgroup$ – Sultan of Swing Mar 27 '15 at 18:56
  • $\begingroup$ It appears that you have come across sites.mathdoc.fr/JMPA/PDF/JMPA_1851_1_16_A5_0.pdf or some other site where the 1851 paper is available. I can't find the Comptes Rendus paper online. You might be able to get them by interlibrary loan. $\endgroup$ – Gerry Myerson Mar 30 '15 at 3:16
  • $\begingroup$ Also, you may have come across this: The principal studies dealing with Liouville’s work as a whole are G. Chrystal, “Joseph Liouville,” in Proceedings of the Royal Society of Edinburgh, 14 (1888), 2nd pagination, 83-91; and G. Loria, “Le mathématician J. Liouville et ses oeuvres,” in Archeion, 18 (1936), 117-139, translated into English as “J. Liouville and His Work,” in Scripta mathematica, 4 (1936), 147-154 (with portrait), 257-263, 301-306. I'm surprised there's no such thing as The Collected Works of Liouville. $\endgroup$ – Gerry Myerson Mar 30 '15 at 3:17
  • $\begingroup$ Well I seem pretty convinced now, as per my update. In the first paragraph of his 1851 publication, he states he presented his ideas in 1844 to the Academy of Sciences, and he is going to reproduce them here... $\endgroup$ – Sultan of Swing Mar 30 '15 at 5:59
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Fortunately, there is a great resource that should be considered the first place to look for all things Liouville. That resource is Jesper Lützen‘s biography “Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics.” In this book, he devotes an entire chapter (XII) to Liouville’s work on transcendental numbers. In it, one can find answers to your questions here such as . . .

  1. Liouville’s interest on transcendental numbers appear to originate from a reading of correspondence between Goldbach and Daniel Bernoulli (published by Fuss in 1843), and various investigations concerning $e$ and $\pi$ in the previous century. (Lützen, page 513). Goldbach and Bernoulli’s correspondence contains a construction that is rather reminiscent of Liouville’s (decimal) transcendental number.
  2. Liouville’s 1844 publication, which you mentioned above, contains the first construction of transcendental numbers and are based on a continued fraction approach, perhaps inspired by Lambert’s work on the irrationality of $e$. Also in 1844, Liouville does present as transcendental (but without proof) his construction for Liouville numbers $\sum_{k=1}^\infty 1/b^{k!}$ for $b>2$. (Lützen, page 523). His 1844 published work does not, however, contain his theorem on Diophantine approximation.
  3. Around 1844, Liouville did write down his theorem on Diophantine approximation in his notes. He later published this theorem in the 1851 work you mentioned. In the 1851 work, he abandons the continued fraction approach, as it is he realizes it is not necessary (Lützen, page 524). He also demonstrates that his Liouville number construction is transcendental, which he did not prove in 1844 (at least in print) (Lützen, page 524).
  4. In 1840, Liouville showed that $e$ is not algebraic of degree $2$. Shortly thereafter, that same year, he also showed that $e^2$ is not algebraic of degree $2$ (Lützen, pages 516-517). These papers are "Sur l'irrationalité du nombre e = 2,718…" (1) 5: 192 and "Addition à la note sur l'irrationnalité du nombre e"(1) 5: 193–194, both appearing in Liouville's Journal de Mathématiques Pures et Appliquées..
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The Compte rendus de l'académie des sciences are now online, you can take a look at both texts from Liouville from

https://gallica.bnf.fr/ark:/12148/bpt6k2977n/f883.item

and

https://gallica.bnf.fr/ark:/12148/bpt6k2977n/f910.item

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