At a Glance - I Like Abstract Stuff; Why Should I Care?
The integers form something called a group, and single-variable polynomials with integer coefficients also form a ring. Fortunately, there's no creepy little girl climbing out of your television set for this one.
The rational numbers form a field, as do the rational expressions. A field is a collection of things in which the things "behave very nicely." In a field we can add, subtract, multiply, and divide. There's something called 0 (you may have heard of it) that we can add to any element of the field without changing the value of that element, and there's something called 1, which may also ring a bell, that we can multiply by any element of the field without changing the value of the element. There are also a few other things that must be true in a field. You'll have a field day with this one.
While there are infinitely many rational numbers and infinitely many rational expressions, it's also possible to have a field with only a finite number of things. This is always nice, as it gives us something to count when we're bored without going out of our minds. Using clock arithmetic (also known as modular arithmetic) on a clock that has a prime number of numbers (for example, a clock with the seven numbers 0, 1, 2, 3, 4, 5, 6) we have a finite collection of numbers that acts like the infinite collection of rational numbers. Good luck trying to meet someone on time when they ask you to be somewhere at 7, though.