I've always found how topology and algebra treat the differences between integers and rational numbers to be instructive at developing an intuition for each.<p>As sets the integers and rational numbers are identical. They can be put in one-to-one correspondence with each other.<p>To distinguish them further we must impose additional structure and examine their differences.<p>Algebra is the study of operators over sets. This means from an algebraic perspective the difference between the rational numbers and the integers are the operators supported. The integers support addition, subtraction, and multiplication. The rationals are formed by adding division.<p>On the other hand, topology is the study of how to formalize the intuitive notions of "close" and "far." From a topological perspective, the distinguishing feature of the integers is that each integer is very far from the other, or equivalently that the integers are very sparse. Every integer has two other integers between which there are no other integers. On the other hand the rational numbers are very dense. They are the result of taking the integers and "squishing" them together. Between and two rational numbers there are an infinite number of other rational numbers.