Quote by arbol
The set N of natural numbers = {1, 2, 3, 4, ...}.
But there exists one (1) number N, such that
N = 12345678910111213... (where the unit's place is at infinity).

No, there is no such number. All integers have only a finite number of digits. By the way, it is not at all a good idea by using "N" to represent the set of natural numbers and then say that "N" is a number.
A good example of an irrational number then would be
1.234567891011121314...

Now THAT is a perfectly good irrataional number.