Assume √41 is rational, so that there exist integers p, q such that

and p/q is irreducible. Taking squares, this would mean

This tells us that 41 is a factor of p², which in turn means 41 is a factor of p because 41 is prime. We can consequently write p = 41n for some integer n. But then

which says 41 also divides q² and hence divides q as well. In other words, p/q is reducible, so we have a contradiction.