Is there a "positive" definition for irrational numbers? 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
An irrational number is a real number that can be expressed as an infinite
 simple continued fraction.
This question is discussed in some detail on a different answer of mine: Are there real numbers that are neither rational nor irrational? See also What's an example of a number that is neither rational nor irrational?.

A small quote, with the actual definition:


  Thus there is a measurable gap between the irrationality measures of rational and irrational numbers, and this yields an alternative "constructive" definition of irrational: let $x\in\Bbb I$, read "$x$ is irrational", if $|x-p/q|&lt;q^{-2}$ has infinitely many coprime solutions.


Let me add, to forestall confusion about the word "infinite" in the above: I mean that the set of solutions is denumerable, which is to say that there is a function which takes each solution $(p,q)$ to another solution $(p',q')$ such that $(p,q)&lt;(p',q')$ under some suitable total order (such as lexicographic order).

The definition is equivalent to Eric Schmidt's, but the extra comments about application to intuitionistic logic are worth a read if that is your interest.