What happens when you add 1 to infinity? Can you divide an infinite quantity in half? If so, how big are the two parts? How many do you have when you double infinity? Does infinity come in different sizes, so that one infinite quantity is bigger than another?
These and similar puzzling questions about infinity cannot be anwered with the arithmetic that we know, because this arithmetic is for operating on finite quantities. These questions belong in the realm of transfinite arithmetic. Careful reasoning about them will bring you answers that are puzzling, paradoxical, and downright bizarre. (That's what makes the Hotel Infinity such an unusual place.)
See also: Infinity is for Children---And Mathematicians!)