It is known, that $\phi := \frac{sqrt(5)-1}{2}$, is the number, that is hardest to approximate by rationals (cf e.g. the section **properties of the golden ratio $\phi$** here: <http://en.wikipedia.org/wiki/Continued_fraction#A_property_of_the_golden_ratio_.CF.86>).


In the section **Infinite continued fractions** on the same WIKI page, one finds the following "explanation" of how the terms of an infinite continued fraction relate to its "degree of irrationality": 



> 
> *The larger a term is in the continued fraction, the closer the corresponding convergent is to the irrational number being approximated. Numbers like π have occasional large terms in their continued fraction, which makes them easy to approximate with rational numbers. Other numbers like e have only small terms early in their continued fraction, which makes them more difficult to approximate rationally. The golden ratio ϕ has terms equal to 1 everywhere—the smallest values possible—which makes ϕ the most difficult number to approximate rationally. In this sense, therefore, it is the "most irrational" of all irrational numbers* 
> 
> 
> 


That explanation is somewhat fuzzy: "small terms in the beginning" and "occasionally large terms" are not mutually exclusive; so that explanation can't be used for sorting continued fractions according to decreasing irrationality in the sense of rational approximability. 



> 
> **Question:** 
>  is it possible to compare the degree of irrationality of infinite (periodic) continued fractions on basis of their terms? 
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> 
> 


Any suggestions or pointers to the literature would be great.



Comparing the Rational Approximability of Infinite Continued Fractions
A frequently used measurement is ``the largest partial quotient''. Almost all numbers (in the sense of measure theory) have unbounded partial quotients, but many interesting numbers do not (rationals and quadratic irrationals, for example). This is, in my experience, the most usable such gradation of approximability.


Another measurement is to consider the average of the first $n$ partial quotients. Again, for almost all numbers this is unbounded as $n$ increases, but this measurement goes deeper into the well-approximable numbers.


Another measurement is to consider the geometric mean of the first $n$ partial quotients. For almost all numbers, this is finite (as $n\to\infty$). Unfortunately, for almost all numbers the limit of geometric means is the same (Khinchin's constant), so this doesn't allow one to distinguish numbers very well.


