ABSTRACT
The squares of numbers seem to be far fewer than all the positive whole numbers (the integers), and if you pick them out from the sequence of integers arranged in their usual ascending order they rapidly thin out. Yet, as the incredulous Galileo (1564-1642) noticed, they can be exhaustively paired off with all the positive integers: each positive integer has a unique square associated with it and each square has a unique positive integer (its positive square root) associated with it. The squares form what is called a proper subset of the positive integers, that is a subset whose membership falls short of the set of positive integers, and is therefore distinct from it. Yet the subset can be mapped without remainder onto the set of all the positive integers. So are there fewer squares or just as many?
