We now have the rational numbers, and we can do addition and multiplication with rational numbers. Addition and multiplication have a bunch of properties on the rational numbers, which I've mentioned as they have come up, but now I'm going to list some of them in particular again.
- Addition has closure: if x and y are rational numbers, x+y is a rational number.
- Addition is commutative: x+y=y+x for all x and y.
- Addition is associative: (x+y)+z=x+(y+z) for all x, y, and z.
- There exists an additive identity 0: x+0=x for all x.
- Every number x has an additive inverse −x: x+(−x)=0 for all x.
- Multiplication has closure: if x and y are rational numbers, xy is a rational number.
- Multiplication is commutative: xy=yx for all x and y.
- Multiplication is associative: (xy)z=x(yz) for all x, y, and z.
- There exists a multiplicative identity 1: 1x=x for all x.
- Every number x except 0 has a multiplicative inverse 1/x: x × 1/x=1 for all x≠0.
- Multiplication is distributive over addition: x(y+z)=xy+xz for all x, y, and z.
In abstract algebra, any set on which addition and multiplication are defined and which has all of these listed properties is called a field. In abstract algebra, the sets don't even have to be sets of numbers, and addition and multiplication can be defined differently, as long as all of these properties exist.
Using the definitions of addition and multiplication that we have been using, the integers are not a field, because the integers do not have multiplicative inverses. However, it's possible to take different definitions of addition and multiplication which result in subsets of the integers being a field.
It's possible to derive a bunch of properties about addition and multiplication directly from the properties that define a field. In the context of the rational numbers, these are all properties that you probably would take for granted anyway, but it's useful to build a framework to show that they are true more generally.
I'm not sure how relevant fields or other concepts from abstract algebra actually are to mathematical analysis, but the textbook called attention to the fact that the rational numbers are a field so I am too.