The square root of any positive non-square integer is irrational.
The square root of any positive non-square integer is irrational.
Proof:
Suppose that is a positive non-square integer.
If were rational then the set is not empty.
Let be the first in . We see because is a on-square.
That means that . Use the floor function: .
That is a contradiction. Do you see it?
Is a positve integer?
What is ?