TITLE: Are some numbers more irrational than others?
QUESTION [42 upvotes]: Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly approximable by rationals. But I wonder if there is another sense in which one number is more irrational than another.
Consider the following well known irrationals: 
$\sqrt{2}$,
$\varphi$,
$\log_2{3}$,
$e$,
$\pi$,
$\zeta(3)$.
The proofs of irrationality of these numbers increase in difficulty from grade-school arguments, to calculus, to advanced methods. Other probable irrationals such as $\gamma$ most likely have very difficult proofs.
Can this notion be made precise? Is there a well defined way in which, for example, $\pi$ is more irrational than $e?$

REPLY [5 votes]: In response to a question about the comparative irrationality of real numbers, I wrote, "A theorem only has a difficult or long proof until one finds an easy or short proof."  Mark Sapir replied, "There are only [a] finite number of proofs of length 10^10 so your last statement is wrong."  I would argue, instead, that there are many contentious and, perhaps, wrong assumptions and philosophical misconceptions implicit in Sapir's sentence.  For example, one could state and prove the theorem, "The sum of the first three odd numbers is nine."  One could state and prove another theorem, "The sum of first five odd numbers is 25." One could state and prove another theorem, "The sum of first 100 odd numbers is 10,000." Continuing, one gets to statements of length greater than 10^10, and whose proofs would be even longer.  Of course, someone might have the clever idea of proving inductively that, for all positive integers n, the sum of the first n odd numbers is n^2, and then one has a short proof of infinitely many theorems.  It is exactly this kind of "exponential collapse," often called "progress in science," that I meant when I wrote a proof is long until one finds a short proof.  I would be very interested to know if someone can really prove that there exist theorems that do not have short proofs.  It is more likely that there are only finitely many theorems, and they all have short, simple, and elegant proofs.  This, of course, is simply another description of Erdos' famous book.