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

Answer

An irrational number is a real number that can be expressed as an infinite
simple continued fraction.

Attribution
Source : Link , Question Author : barak manos , Answer Author : Robert Israel

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