do you mean Z or Q ?
members of Z are signed natural numbers
i mean if a belongs to Z -a or a belongs to N
as only positive numbers got a square root you may be had made a little confusion or mistake!
anyway there is a litle thing wrong in speaking of percentage of an infinite set
some will tell and make you the demonstration that they are as much even (or odd) numbers than natural numbers by making a bijection between those two set
though i believe they are fifty percent of the numbers that are odd (or even )
if i where you i'll tried to find if sqr(n) rational => n=m^2 (so n is a perfect square)
I'm not sure of that but i would bet
you'll easyly find the demonstration on this forum or in the net that sqr(2) is irrational maybe (i'd bet but i would'nt cut my head off) this demonstration could be addapted to others numbers which are not perfect square??
so if i were true for every numbers A they will be only entire-part(sqr(A)) numbers <=A that are perfect square so if i trust my feeling for wich sqr(them) are rational
so for a given number A you could find the 'percentage' but for entire N it will be zero at the limit (if suc a limit have a sence in term of percentage) and the half of zero for Z!
and according to the second phrase of is post the question was posed!
though there is a number a) in the third line dealing whith numbers only in the interval you mentionned the formulation was not to very clear: to "(or the same)" in the same phrase i suposed the first line was an expression of a mathèmatical question (refered in the second question (third line) by "the same")
but as the question is about how many sqrt(n) are rational I would'nt see were would be the plot to ask such a quetion if the elementary result was known because the question will be then much more elementary than the theorème involved and would had not much to do about rational or irational but how many perfect square express in percentage can you find below a certain number (in the question were dealing) which is a very elementary question when your supose to deal whith irationality!
but of course i'm not very aware of what's going in college nowadays