In fact, since there are only a countably infinite number of symbolic representations but an uncountable number of irrational numbers, nearly all of them cannot be written down at all!
If you mean representations using finite sequences, sure. But the set of all infinite sequences (e.g., of digits) is uncountable.
That's the point. There is uncountably many infinite sequences, but we only have countably many symbolic representations, so we can't write most real numbers down even with infinite time.
Representing a number as an infinite sequence of digits isn't helpful if you are trying to write down the number. You can't write an infinite number of digits after all
There's no "may" about it!
By definition an irrational number has an infinitely long decimal expantion that does not repeat, so none of them can be written down (other than in symbolic representations like pi)