sleepingsquirrel's Journal: Beyond irrational numbers (continued fractions)

Lately, I've been casually trying my hand at learning some mathematics, especially number theory (my background is in electrical engineering). While doing some reading, I came across continued fractions, which are of the following form...

a/(b+1/(c+1/(d+1/...

...or in ASCII art...

Arrrgh. My ASCII-ART-fu isn't strong enought to defeat slashdot's lameness filter, see the link above to get a better idea of what a continued fraction looks like

So what I found interesting was that you can express any rational number as a continued fraction with a finite number number of terms (a,b,c,d,...) and any irrational number as a continued fraction with an infinite number of terms that repeat peroidically. For example...

sqrt(2) = 1+(1/(2+1/(2+1/(2+1/(2+...

Well it seems like the next question to ask is, "are there numbers that can't be expressed irrationally?" Or put in another sense, in the sequence: integers, rational numbers, irrational number, is there anything after irrational numbers? Let's make the terms of the continued fraction be the digits of pi...

3+1/(1+1/(4+1/(1+1/(5+1/(9+...

Can this number (is it a number?) be expressed as another continued fraction, but with terms that repeat periodiocally? Or is it a new kind of beast altogether? My quest continues...