an element of N (natural) but not of Q (rational).

Question

So I am looking for a natural irrational number.

Answer 1

A rational number is any number that can be expressed as p/q, where p and q are integers and q is nonzero.  Any natural number, n,  can be written as n/1, where n is a nonnegative integer.   So, every natural number is also rational.  In other words, the set of natural numbers is a subset of the set of rational numbers.  

 

There is no such thing as a "natural irrational' number.