TITLE: Is there a "positive" definition for irrational numbers?
QUESTION [114 upvotes]: An irrational number is a real number that cannot be expressed as a ratio of integers.
Is it possible to formulate a "positive" definition for irrational numbers?
A few examples:

An irrational number is a real number that can be expressed as...
An irrational number is a real number that can be expressed only as...

I cannot find any such definition which does not include the term irrational number to begin with.
Since not all irrational numbers are computable, I believe that the answer is No.
Any idea how to prove it formally?
Thanks

REPLY [2 votes]: A real number $\alpha$ is irrational precisely when the equation $y=\alpha x$ has one solution in integers, namely $(0,0)$.  The number $\alpha$ is rational if and only there are and infinite number of integral solutions.  Here is the picture from C. D. Olds' book Continued Fractions: