# Why are some mathematical constants irrational by their continued fraction while others aren't?

Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore hyperlinks because of my low rep.), a condition for irrationality of that continued fraction is given (see 'generalized continued fractions'). It is said that a given continued fraction converges to an irrational limit if $b_n > a_n$ in the continued fraction $b_0 + a_0/(b_1 + a_1/(b_2 + a_2/(...b_n+a_n)))$ for some sufficiently large $n$. In the webpage I provided you with, however, the degree of the polynomial $a_n$ of the continued fractions is bigger than the one of $b_n$. Therefore, all values of $b_n$ will never exceed all values of $a_n$, even after some large $n$.

My question is: Why does the degree of $a_n$ need to be smaller than the degree of $b_n$ in order for a continued fraction representation of a constant to be irrational? I think I read somewhere it had to do with something like 'Tietschzes criterion' (but I'm not sure). (bonus question: Does anyone know where a proof of this 'criterion for irrationality' can be found?)

Thanks,

Max Muller

• no longer interested in references? :P – BBischof Sep 27 '10 at 15:22
• @ BBischof :P . I thought that when I was asking: "Does anyone know where a proof of this 'criterion for irrationality' can be found?" I was already, though implicitly, asking for references ;) – Max Muller Sep 27 '10 at 15:27

## 3 Answers

You can find the proofs here, taken from Chrystal's Algebra - which is one of the best references on continued fractions. I suspect that if you read Chrystal then almost all of your questions will be answered. See also chapter 9 of Fowler's "The mathematics of Plato's Academy"

• Thanks mister Dubuque. I think I found that link too somewhere (I think it's note 13 of the wikipedia page for generalized continued fractions), but I couldn't and still can not view the file on my computer... I guess I need an update of my VCL-media player...? – Max Muller Sep 27 '10 at 15:32
• @Max Muller: Its a .PS = PostScript file. You can read it by downloading Ghostscript or a PS to PDF converter. Better, download a copy of Chrystal's Algebra. – Bill Dubuque Sep 27 '10 at 15:40
• Wow.. it's amazing that book that costs 52 dollars on Amazon can be viewed for free on the internet... (archive.org/details/algebraanelemen08chrygoog), but I guess this applies to quite a few (dated) math books (Divergent Series - Hardy, for example..) By the way, do you think this is still the best book to learn about continued fractions? I've seen more books on this subject by Wall and Khinchin, but all are pretty dated. – Max Muller Sep 27 '10 at 15:52
• @Max Muller: Yes, Chrystal is the best reference for the algebraic (vs. analytic) theory. Another useful reference is Ch.9 of Fowler's "The mathematics of Plato's Academy" for which you can find online links at the usual ebooks sites, e.g. gigapedia – Bill Dubuque Sep 27 '10 at 15:59

The continued fraction expansion of any possible real number falls in one of 2 possible categories:

• the continued fraction expansion has a finite length if and only if the number is rational.
• the continued fraction expansion never ends if and only if the number is irrational.

"quite a few other mathematical constants are known to have an infinite continued fraction"

Yes, and all such constants are therefore all irrational.

"Catalan's Constant ... known to have an infinite continued fraction"

Do you have a proof? My understanding is that we don't yet know whether or not Catalan's Constant has an finite continued fraction -- therefore we don't yet know whether or not it is rational.

"Why does the degree of an need to be smaller than the degree of bn in order for a continued fraction representation of a constant to be irrational?"

That is required for the sequence to converge on some specific value. If the degree of an is larger, that doesn't mean the result is rational -- it indicates that it doesn't converge to any real number, and therefore isn't a "constant" at all.

Is there a way to make the Wikipedia "generalized continued fraction" article a little less confusing?

• @ David Cary: On your last question: by providing the reader with the correct link, you just did that yourself ;)! As I stated, I can't post anymore hyperlinks in my question, so your answer is pretty helpfull for that matter. I'm not sure whether your understanding on whether or not Catalan's constant has a finite continued fraction is accurate, though. On the bottom of the webpage that the hyperlink will send you to, it is stated that Wadim Zudilin gave a continued fraction of Catalan's Constant in 2003. I think it's infinite, which I why I'm not sure why the C-constant isn't irrational. – Max Muller Sep 28 '10 at 16:48
• (by that result). – Max Muller Sep 28 '10 at 16:49

Though infinite, the Zudilin's continued fraction for the Catalan's constant is not "simple". So, the irrationality theorem does not apply to Zudilin result. The distinction between "simple" and "generalized" continued fractions can be found at

http://en.wikipedia.org/wiki/Continued_fraction

Best wishes,

Dr. Fabio M. S. Lima