No, it doesn't matter how you implement square root. If you were able to tell me whether any execution of the program halts, then you would be able to tell me,

No, it doesn't matter how you implement square root. If you were able to tell me whether any execution of the program halts, then you would be able to tell me, for any `n`, whether or not `n` is irrational (since irrational results are exactly the results that it won't halt on). But this isn't possible: it's an open problem whether or not `pi + e` [is irrational](http://math.stackexchange.com/questions/28243/is-there-a-proof-that-pi-times-e-is-irrational http://math.stackexchange.com/questions/28243/is-there-a-pro...). So if you could always tell me when the square root program halted, you'd be able to tell me whether or not it halted for `(pi + e)^2`. But if you were able to tell me that, you would have solved an open problem in number theory along the way -- you'd have solved a whole class of open problems about whether or not certain expressions are rational. I can't prove to you that no solution of this kind exists, but, intuitively, do you see how big the implications would be if you could solve the halting problem for your square root machine?