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:
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: