# 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